Operator Algebras and Their Modules: An Operator Space Approach (London Mathematical Society...

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L O N D O N M AT H E M AT I C A L S O C I E T Y M O N O G R A P H S

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L O N D O N M AT H E M AT I C A L S O C I E T Y M O N O G R A P H SN E W S E R I E S

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Christian Le Merdy

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Operator algebras and theirmodules—an operator spaceapproach

David P. BlecherDepartment of Mathematics, University of Houston

Christian Le MerdyLaboratoire de Mathématiques, Université de Besançon

C L A R E N D O N P R E S S • O X F O R D 2 0 0 4

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Preface

A major trend in modern mathematics, inspired largely by physics, is toward‘noncommutative’ or ‘quantized’ phenomena. This thrust has influenced mostbranches of the science. In the vast area of functional analysis, this trend hasappeared notably under the name of operator spaces. This young field lies at theborder between linear analysis, operator theory, operator algebras, and quantumphysics. It has useful applications in all of these directions, and in turn derivesits inspiration and power from these sources. Perhaps the importance of operatorspace theory may be best stated as follows: it is a variant of Banach spaces, whichis particularly appropriate for solving problems concerning spaces or algebras ofoperators on Hilbert space arising in ‘noncommutative mathematics’.

An operator space, loosely speaking, is a linear space of bounded operatorsbetween two Hilbert spaces. Thus the category of operator spaces includes oper-ator algebras, selfadjoint (that is, C∗-algebras) or otherwise. Since any normedlinear space E may be regarded as a subspace of a commutative C∗-algebra (forexample, the continuous scalar functions on the unit ball of E∗), operator spacesalso include all Banach spaces. In addition, most of the important modules overoperator algebras are operator spaces. With this in mind, it is natural to seek totreat the subjects of C∗-algebras, nonselfadjoint operator algebras, and modulesover such algebras (such as Hilbert C∗-modules), together under the umbrella ofoperator space theory. This is the topic of our book. In the last decade or two, ithas become very apparent that it can be a useful perspective. Indeed, operatorspace theory, as opposed to Banach space theory, is a sensitive enough mediumto reflect accurately important noncommutative phenomena such as the spatialtensor product. The underlying operator space structure also captures, very pre-cisely, many of the profound relations between the algebraic and the functionalanalytic structures involved.

Our main goal is to illustrate how a general theory of operator algebras, andtheir modules, naturally develops out of the operator space methodology. Weemphasize both the uniform (or ‘norm’), and the dual (or ‘weak∗’), aspects of thetheory. A second goal, or prevailing theme, is the systematic study of algebraicstructure in spaces of Hilbert space operators. For example, we are interestedin the structural features characterizing the objects which operator algebraistsare interested in, how rigid such structures are, how they behave with respect toduality, and so on. A third goal, and this is one of the most inspiring aspects ofthe subject at large, is to highlight the rich interplay between spectral theory,operator theory, C∗-algebra and von Neumann algebra techniques, and the influx

viii Preface

of important ideas from related disciplines, such as pure algebra, Banach spacetheory, Banach algebras, and abstract function theory. Finally, our fourth goal ispedagogical: to assemble the basic concepts, theory, and methodologies, neededto equip a beginning researcher in this area.

Our book falls roughly into three parts. Each chapter begins with some wordsof introduction, and so we will only very briefly describe their contents here. Part1—Chapters 1–3—presents the basic theory of operator spaces, operator alge-bras, and operator modules. We also introduce much of our notation here. Part2—Chapters 4–7—presents more advanced topics associated with these subjects,and describes more technical results. Chapter 4 discusses, for example, the non-commutative Shilov boundary, injective envelopes, operator space multipliers,and M -ideals, and applications of these topics to ‘operator algebraic structure’.Chapter 5 is devoted to the ‘isomorphic’ (as opposed to ‘isometric’) aspectsof the theory. This includes completely isomorphic characterizations of variousclasses (operator algebras, operator modules, Q-algebras), as well as examples of‘operator algebra structures’. In Chapter 6, we discuss various tensor productsinvolving operator algebras, such as the maximal tensor product, or Pisier’s δtensor norm. We give various applications, for example to dilation theory, or tofinite rank approximation (nuclearity, semidiscreteness, etc.). In Chapter 7 wecollect some criteria which ensure that an operator algebra is selfadjoint. Part3—Chapter 8—develops the theory of Hilbert C∗-modules and the related no-tion of triple systems, largely from an operator space perspective. In this chapterwe also describe some of the beautiful two-way interplay between C∗-modulesand the theory in the earlier chapters. Finally, a short appendix contains somefrequently needed facts from operator theory, Banach space theory, and Banachand C∗-algebras. We include proofs of many of these facts.

Each chapter ends with a lengthy ‘Notes and historical remarks’ section,consisting of attributions, discussion of the literature, observations, additionalproofs, complementary results, and so on. We apologize for inaccuracies or omis-sions here, of which there are sure to be many. In all cases, the reader shouldconsult the original papers for further details, and other perspectives.

This book was begun in 1999, during a year-long visit of the second author toHouston. We wish to thank the Universities of Besancon and Houston, the CNRS,and the National Science Foundation, for their support. We are also indebtedto several colleagues for very many kindnesses, and for teaching us much of thismaterial, in particular William Arveson (who started it all), Edward Effros, PaulMuhly, Vern Paulsen, Gilles Pisier, Zhong-Jin Ruan, Allan Sinclair, and RogerSmith. We thank Matthias Neufang, Bojan Magajna, and Damon Hay for manyvery helpful suggestions, and Oxford University Press and the LMS Series editorsfor an excellent job of processing our book.

Houston D. P. B.Besancon C. L-M.June, 2004

Contents

1 Operator spaces 11.1 Notation and conventions 11.2 Basic facts, constructions, and examples 41.3 Completely positive maps 161.4 Operator space duality 221.5 Operator space tensor products 271.6 Duality and tensor products 381.7 Notes and historical remarks 45

2 Basic theory of operator algebras 492.1 Introducing operator algebras and unitizations 492.2 A few basic constructions 572.3 The abstract characterization of operator algebras 622.4 Universal constructions of operator algebras 682.5 The second dual algebra 782.6 Multiplier algebras and corners 822.7 Dual operator algebras 882.8 Notes and historical remarks 96

3 Basic theory of operator modules 1023.1 Introduction to operator modules 1023.2 Hilbert modules 1093.3 Operator modules over operator algebras 1153.4 Two module tensor products 1193.5 Module maps 1233.6 Module map extension theorems 1283.7 Function modules 1313.8 Dual operator modules 1363.9 Notes and historical remarks 142

4 Some ‘extremal theory’ 1474.1 The Choquet boundary and boundary representations 1474.2 The injective envelope 1524.3 The C∗-envelope 1564.4 The injective envelope, the triple envelope, and TROs 1614.5 The multiplier algebra of an operator space 1674.6 Multipliers and the ‘characterization theorems’ 1754.7 Multipliers and duality 180

x Contents

4.8 Noncommutative M -ideals 1834.9 Notes and historical remarks 188

5 Completely isomorphic theory of operator algebras 1955.1 Homomorphisms of operator algebras 1955.2 Completely bounded characterizations 2005.3 Examples of operator algebra structures 2095.4 Q-algebras 2155.5 Applications to the isomorphic theory 2245.6 Notes and historical remarks 228

6 Tensor products of operator algebras 2326.1 The maximal and normal tensor products 2326.2 Joint dilations and the disc algebra 2396.3 Tensor products with triangular algebras 2416.4 Pisier’s delta norm 2486.5 Factorization through matrix spaces 2546.6 Nuclearity and semidiscreteness for linear operators 2596.7 Notes and historical remarks 265

7 Selfadjointness criteria 2697.1 OS-nuclear maps and the weak expectation property 2697.2 Hilbert module characterizations 2747.3 Tensor product characterizations 2797.4 Amenability and virtual diagonals 2827.5 Notes and historical remarks 292

8 C∗-modules and operator spaces 2968.1 Hilbert C∗-modules—the basic theory 2978.2 C∗-modules as operator spaces. 3088.3 Triples, and the noncommutative Shilov boundary 3228.4 C∗-module maps and operator space multipliers 3288.5 W ∗-modules 3318.6 A sample application to operator spaces 3488.7 Notes and historical remarks 350

Appendix 359A.1 Operators on Hilbert space 359A.2 Duality of Banach spaces 360A.3 Tensor products of Banach spaces 361A.4 Banach algebras 363A.5 C∗-algebras 364A.6 Modules and Cohen’s factorization theorem 367

References 369

Index 385

1

Operator spaces

In this chapter, we present quickly the background results about operator spaceswhich we shall need, and also establish some notation which will be used through-out this book. The reader with a little mathematical maturity could use thischapter as a minicourse on the basics of operator space theory. Fortunately thelengthy proofs here usually belong to the very well-known results (such as Ruan’stheorem, or the extension/characterization theorems for completely positive orcompletely bounded maps). Thus with the exception of a few such well-knownproofs (which may be found in [149, 314, 337, 385, 102]), we can be quite self-contained. Warning: our proofs in this chapter are often only a good sketch,and some things are left as exercises. The reader should also feel free to skimthrough this chapter, returning later for a specific definition or fact (we try tobe conscientious in later chapters about referencing these by number). Thosemainly interested in the general theory of operator spaces, should consult thefine aforementioned texts for a more comprehensive and leisurely development.And of course usually the original papers contain much additional material.

We will take for granted facts found in any basic graduate level functionalanalysis text. For example, we assume that the reader is comfortable with basicspectral theory, the very basics of the theory of C∗-algebras and Banach alge-bras, and standard facts about the various topologies in Banach spaces or dualspaces. Much of this may be found in the Appendix, together with a few of theunexplained terms below.

1.1 NOTATION AND CONVENTIONS

1.1.1 Our set notation and function notation is standard. We use Ac for thecomplement of a set A. The term ‘scalar’ denotes a number in the complex fieldC. We use n, m, i, j, k for integers, and I, J or α, β, γ for cardinal numbers. Vectorspaces are almost always over the field C unless stated to the contrary. The usualbasis of C

n or 2 is written as (ei)i, and we use this notation too in the other p

sequence spaces. We write IE , or sometimes I when there is no confusion, for the‘identity map’ on a vector space E. An isomorphism, at the very least, is alwaysassumed to be linear, one-to-one, and surjective. If T : E → F , and if W ⊂ Eis a linear subspace, then we write T|W for the map from W to F obtained byrestricting T to W . We often use the symbol q or qW for the canonical surjection

2 Notation and conventions

from E onto E/W . We write x+W , or sometimes x, for the class of x in E/W ,thus x+W = qW (x).

If E is a normed space, we write Ball(E) for the set x ∈ E : ‖x‖ ≤ 1.Expressions such as ‘norm closed’, ‘norm closure’, or ‘xn → x in norm’ (orsimply ‘closed’, ‘closure’, or ‘xn → x’), mean of course ‘with respect to the normtopology’. All topological spaces are assumed to be Hausdorff. We use standardnotation for the standard examples, for example, C(Ω) is the Banach space ofscalar valued continuous functions on a compact space Ω. In the literature theseare often called ‘C(K)-spaces’, and of course are exactly the commutative unitalC∗-algebras (see A.5.4). We use the letters H, K, L for Hilbert spaces. Thus ifthese letters appear in the text without explanation, they will always be Hilbertspaces. We write B(E, F ) for the space of bounded linear operators from Eto F , and B(E) = B(E, E). Indeed whenever C(X, Y ) is a class of operatorsthen we use C(X) for C(X, X). We write E∗ for the dual space of E, namelyE∗ = B(E, C), and we often write E∗ for a predual of E (if such exists). Wewrite iE : E → E∗∗ for the canonical embedding, but will often simply think ofE as a subspace of E∗∗. We abbreviate ‘weak*’ to ‘w∗’ usually. Thus we writew∗-continuous, w∗-topology, w∗-closure, etc. Thus Sw∗

denotes the w∗-closure ofa set S. We say that a net of maps Tt : E → F converges strongly (or point-norm)if Tt(x) → T (x) in the norm topology of F for all x ∈ E. If F is a dual spacethen Tt → T point-w∗ if Tt(x) → T (x) in the w∗-topology of F for all x ∈ E.A multilinear map between dual spaces is said to be separately w∗-continuous ifwhenever one fixes all but one of the variables, then the map is w∗-continuousin the remaining variable. We recommend that the reader review the facts aboutthe w∗-topology presented in the first sections of the Appendix.

An operator T between normed spaces, with ‖T ‖ ≤ 1, is called a contraction.A quotient map T : E → F is a linear map which maps the ‘open unit ball ofE’ onto the ‘open unit ball of F ’. A projection or idempotent on a space E is amap P : E → E satisfying P P = P . However if E is a Hilbert space then wewill mean more, indeed for an operator on a Hilbert space, or more generally foran element of an operator algebra, we always use the term projection to meanan orthogonal (i.e. selfadjoint) idempotent. If K is a closed linear subspace of aHilbert space H then PK is the canonical projection from H onto K.

1.1.2 For emphasis, we list separately here some of our major conventions.First, we usually suppose that all of our normed spaces are complete. This isnot a serious restriction, since the completion of an operator space is againan operator space; and the ‘incomplete’ versions of most results ‘pass to thecompletion’ without difficulty. We make the ‘completeness’ assumption mostlyto avoid having to be constantly making annoying and repetitious remarks aboutresults ‘passing to the completion’. Another convention is our use of the notationXY for sets X, Y . Assume that we have a pairing X × Y → E where E is aBanach space. Write this pairing as the map (x, y) → xy. Then XY usuallydenotes the closure in the norm topology in E of the linear span of the xy, forx ∈ X and y ∈ Y . We write Span(XY ) if we are not taking the closure here.

Operator spaces 3

See also A.6.4 for some important related facts. There is an exception to thisnotation; if K is a subset of a Hilbert space H and if D ⊂ B(H, L) is a set ofoperators then we use [DK] for the norm closure in L of the span of terms xζfor x ∈ D, ζ ∈ K.

If X is a subspace of B(K, H) or of a C∗-algebra, then we often use thesymbol X (also written as X∗ when there is no possible confusion with thedual space) for the set of ‘adjoints’ or ‘involutions’ x∗ : x ∈ X.1.1.3 (Matrix notation) Fix m, n ∈ N. If X is a vector space, then so isMm,n(X), the set of m×n matrices with entries in X . This may also be thoughtof as the algebraic tensor product Mm,n ⊗ X , where Mm,n = Mm,n(C). Wewrite In for the identity matrix of Mn = Mn,n. We write Mn(X) = Mn,n(X),Cn(X) = Mn,1(X) and Rn(X) = M1,n(X).

If x is a matrix, then xij or xi,j denotes the i-j entry of x, and we write xas [xij ] or [xi,j ]i,j . We write (Eij)ij for the usual (matrix unit) basis of Mm,n

(we allow m, n infinite here too). We write A → At for the transpose on Mm,n,or more generally on Mm,n(X). We will frequently meet large matrices withrow and column indexing that is sometimes cumbersome. For example, a matrix[a(i,k,p),(j,l,q)] is indexed on rows by (i, k, p) and on columns by (j, l, q), and mayalso be written as [a(i,k,p),(j,l,q)](i,k,p),(j,l,q) if additional clarity is needed.

1.1.4 The Hilbert space direct sum will be written as ⊕2, or simply ⊕ (but weuse the latter for some other kinds of direct sums too). We also write H (α) or2α(H) for the Hilbert space direct sum of α copies of H . Here α is a cardinal. This

is called a multiple of H . The Hilbert space tensor product is denoted H ⊗2 K.If S, T are operators on H and K respectively, then we write S⊗T for the usualoperator on H ⊗2 K taking a rank one tensor ζ ⊗ η in H ⊗K to S(ζ)⊗T (η). Inparticular, S ⊗ IK is often called a multiple of S. Indeed, if K is identified with2α for some cardinal α, then we may unitarily identify H ⊗2 K with 2

α(H), andS ⊗ IK with Sα. Here Sα((ζi)) = (Sζi), for (ζi) ∈ 2

α(H). The commutant of asubset S ⊂ B(H) is written as S ′ or [S]′. The C∗-identity is the statement

‖T ∗T ‖ = ‖T ‖2,

valid for any bounded operator T between Hilbert spaces, or any element ofa C∗-algebra. We write Sp(K, H) for the Schatten p class (see also A.1.2 andA.1.3). If H = K is n-dimensional then we write this as Sp

n, thus S1n is the dual

space of Mn. We use WOT for the weak operator topology (see A.1.4), althoughwe usually prefer to use the (finer) w∗-topology (= σ-weak topology, see A.1.2).Which of these two topologies one uses is often a matter of taste, in the situationswe consider. Very frequently, we will need the polarization identity. We state oneform of it: Suppose that E and F are vector spaces, and that Ψ: E × E → F islinear in the second variable and conjugate linear in the first variable. Then

Ψ(y, x) =14

3∑k=0

ik Ψ(x + iky, x + iky), x, y ∈ E. (1.1)

4 Basic facts, constructions, and examples

This is frequently applied when E = F is a ∗-algebra, and Ψ(x, y) = x∗y.

1.1.5 We also use some basic notions from algebra, such as the definitions ofmodules, algebras, ideals, direct sum, tensor product, etc. These may be foundin any graduate algebra text. Our spaces, of course, usually have extra func-tional analytic structure, and in particular possess a (complete) norm. If A is analgebra, then Mn(A) is also an algebra, if one uses the usual formula for multi-plying matrices. We usually refer to a closed two-sided ideal of a normed algebrasimply as an ‘ideal’. One unusual usage: we use the term unital-subalgebra for asubalgebra of a unital algebra A containing the unit (identity) of A. Similarly,a unital-subspace is a subspace containing the ‘unit’ of the superspace. A unitalmap is one that takes the unit to the unit.

We use the very basics of the language of categories, such as the notion ofobject, morphism, and functor. The main categories we are interested in hereare those of Banach spaces and bounded linear maps, operator spaces and com-pletely bounded linear maps, operator algebras and completely contractive ho-momorphisms, C∗-algebras and ∗-homomorphisms, and operator modules andcompletely bounded module maps. These notions will be introduced in detaillater. However it is worth saying that each of these categories (and any otherswe shall meet) has its own notion of ‘isomorphism’ (i.e. when we consider twoobjects as being essentially the same), subobject, embedding, quotient, quotientmap, direct sum, etc. When we use one of these words in later chapters, it isusually understood to be with reference to the category that we are working in.For example, in Chapter 2 we may simply write ‘A ∼= B’, or ‘A ∼= B as operatoralgebras’, and say that ‘A is isomorphic to B’, when we really mean that there isa surjective algebra homomorphism between them which is completely isometric(defined below). Or we may write A → B to indicate that A is ‘embedded’ in Bin the suitable sense of that chapter. For example, in Chapter 2 it means thatthere is a completely isometric algebra homomorphism from A to B.

1.2 BASIC FACTS, CONSTRUCTIONS, AND EXAMPLES

1.2.1 (Completely bounded maps) Suppose that X and Y are vector spacesand that u : X → Y is a linear map. For a positive integer n, we write un forthe associated map [xij ] → [u(xij)] from Mn(X) to Mn(Y ). This is often calledthe (nth) amplification of u, and may also be thought of as the map IMn ⊗ u onMn ⊗ X . Similarly one may define um,n : Mm,n(X) → Mm,n(Y ). If each matrixspace Mn(X) and Mn(Y ) has a given norm ‖ · ‖n, and if un is an isometry forall n ∈ N, then we say that u is completely isometric, or is a complete isometry.Similarly, u is completely contractive (resp. is a complete quotient map) if eachun is a contraction (resp. takes the open ball of Mn(X) onto the open ball ofMn(Y )). A map u is completely bounded if

‖u‖cbdef= sup

‖[u(xij)]‖n : ‖[xij ]‖n ≤ 1, all n ∈ N

< ∞.

Operator spaces 5

Compositions of completely bounded maps are completely bounded, and onehas the expected relation ‖u v‖cb ≤ ‖u‖cb‖v‖cb. If u : X → Y is a completelybounded linear bijection, and if its inverse is completely bounded too, then wesay that u is a complete isomorphism. In this case, we say that X and Y arecompletely isomorphic and we write X ≈ Y .

1.2.2 (Operator spaces) If m, n ∈ N, and K, H are Hilbert spaces, then wealways assign Mm,n(B(K, H)) the norm (written ‖ · ‖m,n) ensuring that

Mm,n(B(K, H)) ∼= B(K(n), H(m)) isometrically (1.2)

via the natural algebraic isomorphism. Recall from 1.1.4 that H (m) = 2m(H) is

the Hilbert space direct sum of m copies of H , for example.A concrete operator space is a (usually closed) linear subspace X of B(K, H),

for Hilbert spaces H, K (indeed the case H = K usually suffices, via the canonicalinclusion B(K, H) ⊂ B(H ⊕K)). However we will want to keep track too of thenorm ‖ · ‖m,n that Mm,n(X) inherits from Mm,n(B(K, H)), for all m, n ∈ N. Wewrite ‖ · ‖n for ‖ · ‖n,n; indeed when there is no danger of confusion, we simplywrite ‖[xij ]‖ for ‖[xij ]‖n. An abstract operator space is a pair (X, ‖ · ‖nn≥1),consisting of a vector space X , and a norm on Mn(X) for all n ∈ N, such thatthere exists a linear complete isometry u : X → B(K, H). In this case we callthe sequence ‖ · ‖nn an operator space structure on the vector space X . Anoperator space structure on a normed space (X, ‖·‖) will usually mean a sequenceof matrix norms as above, but with ‖ · ‖ = ‖ · ‖1.

Clearly subspaces of operator spaces are again operator spaces. We oftenidentify two operator spaces X and Y if they are completely isometrically iso-morphic. In this case we often write ‘X ∼= Y completely isometrically’, or say‘X ∼= Y as operator spaces’. Sometimes we simply write X = Y .

1.2.3 (C∗-algebras) If A is a C∗-algebra then the ∗-algebra Mn(A) has a uniquenorm with respect to which it is a C∗-algebra, by A.5.8. With respect to thesematrix norms, A is an operator space. This may be seen by noting that Mn(A)corresponds to a closed ∗-subalgebra of B(H (n)), when A is a closed ∗-subalgebraof B(H). We call this the canonical operator space structure on a C∗-algebra. Ifthe C∗-algebra A is commutative, then A = C0(Ω) for a locally compact spaceΩ, and then these matrix norms are determined via the canonical isomorphismMn(C0(Ω)) = C0(Ω; Mn). Explicitly, if [fij ] ∈ Mn(C0(Ω)), then:

‖[fij]‖n = supt∈Ω

∥∥[fij(t)]∥∥. (1.3)

To see this, note that by the above one only needs to verify that (1.3) does indeeddefine a C∗-norm on Mn(C0(Ω)).

Proposition 1.2.4 For a homomorphism π : A → B between C∗-algebras, thefollowing are equivalent: (i) π is contractive, (ii) π is completely contractive,and (iii) π is a ∗-homomorphism. If these hold, then π(A) is closed, and π is a


complete quotient map onto π(A); moreover π is one-to-one if and only if it iscompletely isometric.

Proof Apply A.5.8 to the ‘amplifications’ πn.

1.2.5 (Norm of a row or column) Suppose that A is a C∗-algebra, or a spaceof the form B(K, H), for Hilbert spaces H, K. If X is a subspace of A, and ifx1, . . . , xn ∈ X , then we have

∥∥[x1 · · · xn]∥∥

Rn(X)=∥∥∥ n∑

k=1

xkx∗k

∥∥∥ 12

and

∣∣∣∣∣∣∣∣∣∣∣∣∣∣ x1

...xn

∣∣∣∣∣∣∣∣∣∣∣∣∣∣Cn(X)

=∥∥∥ n∑

k=1

x∗kxk

∥∥∥ 12, (1.4)

where the product and involution mean the obvious thing (in the ambient su-perspace). Indeed this follows from the C∗-identity (see 1.1.4).

1.2.6 (Maps into a commutative C∗-algebra) If [aij ] ∈ Mn then

‖[aij ]‖ = sup ∣∣∣∑

ij

aijzjwi

∣∣∣ : z = [zj], w = [wi] ∈ Ball(2n)

.

It is easy to see that this also holds, but with ‘=’ replaced by ‘≥’, and | · |replaced by ‖ · ‖, if aij ∈ B(H). Using these formulae, it is a simple exerciseto see that any continuous functional ϕ : X → C on an operator space X iscompletely bounded, with ‖ϕ‖ = ‖ϕ‖cb. From this and equation (1.3), it followsthat ‖u‖ = ‖u‖cb for any bounded linear map u from an operator space into acommutative C∗-algebra.

1.2.7 The following trivial principle is used very often: If we are given completecontractions v : X → Y and u : Y → Z, and if uv is a complete isometry (resp.complete quotient map) then v is a complete isometry (resp. u is a completequotient map). If, further, Z = X and uv and vu are both equal to the identitymap, then both u and v are surjective complete isometries, and u = v−1.

Theorem 1.2.8 (Haagerup, Paulsen, Wittstock) Suppose that X is a subspaceof a C∗-algebra B, that H and K are Hilbert spaces, and that u : X → B(K, H) isa completely bounded map. Then there exists a Hilbert space L, a ∗-representationπ : B → B(L) (which may be taken to be unital if B is unital), and boundedoperators S : L → H and T : K → L, such that u(x) = Sπ(x)T for all x ∈ X.Moreover this can be done with ‖S‖‖T ‖ = ‖u‖cb.

In particular, if ϕ ∈ Ball(X∗), and if B is as above, then there exist L, π asabove, and unit vectors ζ, η ∈ L, with ϕ = 〈π(·)ζ, η〉 on X.

The very last line clearly follows from the lines above it, and 1.2.6. Also notethat conversely, any linear map u of the form u = Sπ(·)T as above, is completelybounded with ‖u‖cb ≤ ‖S‖‖T ‖. This is an easy exercise using Proposition 1.2.4.

We omit the well-known proof of Theorem 1.2.8 (see the cited texts above).

Operator spaces 7

1.2.9 (Injective spaces) An operator space Z is said to be injective if for anycompletely bounded linear map u : X → Z and for any operator space Y con-taining X as a closed subspace, there exists a completely bounded extensionu : Y → Z such that u|X = u and ‖u‖cb = ‖u‖cb. A similar definition exists forBanach spaces. Thus an operator space (resp. Banach space) is injective if andonly if it is an ‘injective object’ in the category of operator (resp. Banach) spacesand completely contractive (resp. contractive) linear maps.

The following is ‘contained’ in Theorem 1.2.8 (and the remark after it).

Theorem 1.2.10 If H and K are Hilbert spaces then B(K, H) is an injectiveoperator space.

Recall that one version of the Hahn–Banach theorem may be formulated asthe statement that C is injective (as a Banach space). Thus 1.2.10 is a ‘generalizedHahn–Banach theorem’.

Corollary 1.2.11 An operator space is injective if and only if it is linearly com-pletely isometric to the range of a completely contractive idempotent map onB(H), for some Hilbert space H.

Proof (⇒) Supposing X ⊂ B(H), extend IX to a map from B(H) to X .(⇐) Follows from 1.2.10 and an obvious diagram chase.

1.2.12 (Properties of matrix norms) If K, H are Hilbert spaces, and if X is asubspace of B(K, H), then there are certain well-known properties satisfied bythe matrix norms ‖·‖m,n described in 1.2.2. For example, adding (or dropping) arow of zeros or column of zeros does not change the norm of a matrix of operators.By this principle we really only need to specify the norms for square matrices,that is, the case m = n above. Also, switching two rows (or two columns) of amatrix of operators does not change its norm. From this we derive another usefulproperty. Namely, the canonical algebraic isomorphisms

Mn(Mm(X)) ∼= Mm(Mn(X)) ∼= Mmn(X) (1.5)

are isometric, and hence, by iteration, completely isometric. Thus if X is anoperator space then so is Mn(X) (or Mm,n(X)).

As an exercise in operator theory, one may verify that for such X we have:

(R1) ‖αxβ‖n ≤ ‖α‖‖x‖n‖β‖, for all n ∈ N and all α, β ∈ Mn, and x ∈ Mn(X)(where multiplication of an element of Mn(X) by an element of Mn isdefined in the obvious way).

(R2) For all x ∈ Mm(X) and y ∈ Mn(X), we have∣∣∣∣∣∣∣∣[x 00 y

]∣∣∣∣∣∣∣∣m+n

= max‖x‖m, ‖y‖n.

Conditions (R1) and (R2) above are often called Ruan’s axioms. Ruan’s the-orem asserts that (R1) and (R2) characterize operator space structures on a


vector space. This result is fundamental to our subject in many ways. At themost pedestrian level, it is used frequently to check that certain abstract con-structions with operator spaces remain operator spaces. At a more sophisticatedlevel, it is the foundational and unifying principle of operator space theory.

Theorem 1.2.13 (Ruan) Suppose that X is a vector space, and that for eachn ∈ N we are given a norm ‖ · ‖n on Mn(X). Then X is linearly completelyisometrically isomorphic to a linear subspace of B(H), for some Hilbert spaceH, if and only if conditions (R1) and (R2) above hold.

1.2.14 (Quotient operator spaces) If Y ⊂ X is a closed linear subspaceof an operator space, then using Ruan’s theorem one can easily check thatX/Y is an operator space with matrix norms coming from the identificationMn(X/Y ) ∼= Mn(X)/Mn(Y ). Explicitly, these matrix norms are given by theformula ‖[xij+Y ]‖n = inf‖[xij + yij ]‖n : yij ∈ Y . Here xij ∈ X .

1.2.15 (Factor theorem) If u : X → Z is completely bounded, and if Y is aclosed subspace of X contained in Ker(u), then the canonical map u : X/Y → Zinduced by u is also completely bounded, with ‖u‖cb = ‖u‖cb. If Y = Ker(u),then u is a complete quotient map if and only if u is a completely isometricisomorphism. Indeed this follows exactly the usual Banach space case.

1.2.16 (Operator seminorms) An operator seminorm structure on a vectorspace X is a sequence ρ = ρn∞n=1, where ρn is a seminorm on Mn(X), satisfyingaxioms (R1) and (R2) discussed in 1.2.12. In this case, and if N is defined to bex ∈ X : ρ1(x) = 0, by (R1) we have that the kernel of ρn is Mn(N), and ρinduces matrix norms on X/N in the obvious fashion. By Ruan’s theorem, (thecompletion of) X/N is then an operator space.

Let X be a vector space, and let F = Ti : i ∈ I be a set of linear maps,where Ti maps X into an operator space Zi, for each i ∈ I. We suppose thatsupi ‖Ti(x)‖ < ∞, for all x ∈ X . Let N = ∩i Ker(Ti). For each n ∈ N, we definea seminorm on Mn(X) by

[xpq ] −→ supi

‖[Ti(xpq)]‖.

This is fairly clearly an operator seminorm structure for X . Using the facts inthe last paragraph, these seminorms become matrix norms on X/N , and withthese norms (the completion of) X/N is an operator space. Similarly, if I is adirected set then the expressions lim supi ‖[Ti(xpq)]‖ define an operator seminormstructure on X . This yields an operator space as before.

1.2.17 (The ∞-direct sum) This is the simplest direct sum of a family of op-erator spaces Xλ : λ ∈ I, and we will write this operator space as ⊕λ Xλ (or⊕∞

λ Xλ if more clarity is needed). If I = 1, . . . , n then we usually write thissum as X1 ⊕∞ · · · ⊕∞ Xn. If Xλ ⊂ B(Hλ) then ⊕λ Xλ may be regarded as theobvious subspace of B(⊕2

λ Hλ). A tuple (xλ) is in ⊕∞λ Xλ if and only if xλ ∈ Xλ

Operator spaces 9

for all λ, and supλ ‖xλ‖ < ∞. We may identify Mn(⊕λ Xλ) with ⊕λ Mn(Xλ) iso-metrically (and by iteration, completely isometrically). Thus if x ∈ Mn(⊕λ Xλ),then we have ‖x‖n = supλ ‖xλ‖Mn(Xλ). Clearly the canonical inclusion and pro-jection maps between ⊕λ Xλ and its ‘λth summand’ are complete isometries andcomplete quotient maps respectively. If Xλ are C∗-algebras then this direct sumis the usual C∗-algebra direct sum. If the Xλ are W ∗-algebras then this directsum is a W ∗-algebra too, and is easy to work with in terms of the canonicalcentral projections corresponding to the summands.

The ∞-direct sum has the following universal property. If Z is an operatorspace and uλ : Z → Xλ are completely contractive linear maps, then there is acanonical complete contraction Z → ⊕λ Xλ taking z ∈ Z to the tuple (uλ(z)).

If Xλ = X for all λ ∈ I, then we usually write ∞I (X) for ⊕λ Xλ. If I = N

we may simply write ∞(X). Note that we have

Mn(∞I (X)) ∼= ∞I (Mn(X)).

If I = N then one may define a c0-direct sum operator space of operatorspaces X1, X2, . . . . This is simply the subspace of ⊕∞

n Xn consisting of tuples(xn) with limn ‖xn‖ = 0. We write c0(X) for this space if Xn = X for all n.

1.2.18 (Operator valued continuous functions) Let Ω be a compact space, letX be an operator space, and consider the space C(Ω; X) of continuous X-valuedfunctions on Ω (see A.3.2). This is an operator space with matrix norms comingfrom the identification Mn(C(Ω; X)) = C(Ω; Mn(X)). Clearly this ‘canonical’operator space structure is given by the same formula as (1.3), and the naturalembedding C(Ω; X) ⊂ ∞Ω (X) is a complete isometry. Note that if A is a C∗-algebra, then C(Ω; A) is a C∗-algebra, with product as pointwise multiplicationand with f∗(t) = (f(t))∗ for any f ∈ C(Ω; A) and t ∈ Ω.

Similarly if Ω is merely a locally compact space, then C0(Ω; X) is an operatorspace as well, with Mn(C0(Ω; X)) = C0(Ω; Mn(X)) for all n.

1.2.19 (Mapping spaces) If X, Y are operator spaces, then the space CB(X, Y )of completely bounded linear maps from X to Y , is also an operator space, withmatrix norms determined via the canonical isomorphism between Mn(CB(X, Y ))and CB(X, Mn(Y )). Equivalently, if [uij ] ∈ Mn(CB(X, Y )), then

‖[uij]‖n = sup‖[uij(xkl)]‖nm : [xkl] ∈ Ball(Mm(X)), m ∈ N

. (1.6)

Here the matrix [uij(xkl)] is indexed on rows by i and k and on columns byj and l. Applying the above with n replaced by nN , to the space of matricesMN(Mn(CB(X, Y ))) = MnN (CB(X, Y )), yields

Mn(CB(X, Y )) ∼= CB(X, Mn(Y )) completely isometrically. (1.7)

One may see that (1.6) defines an operator space structure on CB(X, Y ) byappealing to Ruan’s theorem 1.2.13 (directly or in the form of 1.2.16). Alterna-tively, one may see it as follows. Consider the set I = ∪n Ball(Mn(X)), and for


x ∈ Ball(Mm(X)) ⊂ I set nx = m. Consider the operator space direct sum (see1.2.17) ⊕∞

x∈I Mnx(Y ). Then the map from CB(X, Y ) to ⊕x∈I Mnx(Y ) taking uto the tuple ((Inx ⊗ u)(x))x ∈ ⊕x Mnx(Y ) is (almost tautologically) a completeisometry. Thus CB(X, Y ) is an operator space.

1.2.20 (The dual of an operator space) The special case when Y = C in 1.2.19is particularly important. In this case, for any operator space X , we obtain by1.2.19 an operator space structure on X∗ = CB(X, C). The latter space equalsB(X, C) isometrically by 1.2.6. We call X∗, viewed as an operator space in thisway, the operator space dual of X . This duality will be studied further in Sections1.4–1.6. By (1.7) we have

Mn(X∗) ∼= CB(X, Mn) completely isometrically. (1.8)

Note that the map implementing this isomorphism is exactly the canonical map(described in A.3.1) from Mn ⊗ X∗ to B(X, Mn).

1.2.21 (Minimal operator spaces) Let E be a Banach space, and consider thecanonical isometric inclusion of E in the commutative C∗-algebra C(Ball(E∗)).Here E∗ is equipped with the w∗-topology. This inclusion induces, via 1.2.3,an operator space structure on E, which is denoted by Min(E). By (1.3), theresulting matrix norms on E are given by

‖[xij ]‖n = sup‖[ϕ(xij)]‖ : ϕ ∈ Ball(E∗)

(1.9)

for [xij ] ∈ Mn(E). Thus every Banach space may be canonically considered tobe an operator space. Since Min(E) ⊂ C(Ball(E∗)), we see from 1.2.6 that forany bounded linear u from an operator space Y into E, we have

‖u : Y −→ Min(E)‖cb = ‖u : Y −→ E‖. (1.10)

From this last fact one easily sees that Min(E) is the smallest operator spacestructure on E. Also, if Ω is any compact space and if i : E → C(Ω) is an isometry,then the matrix norms inherited by E from the operator space structure of C(Ω),coincide again with those in (1.9). This may be seen by applying 1.2.6 to i andi−1. Summarizing: ‘minimal operator spaces’ are exactly the operator spacescompletely isometrically isomorphic to a subspace of a C(K)-space.

According to A.3.1, another way of stating (1.9) is to say that

Mn

(Min(E)

)= Mn⊗E (1.11)

isometrically via the canonical isomorphism.

1.2.22 (Maximal operator spaces) If E is a Banach space then Max(E) is thelargest operator space structure we can put on E. We define the matrix normson Max(E) by the following formula

‖[xij ]‖n = sup‖[u(xij)]‖ : u ∈ Ball(B(E, Y )), all operator spaces Y

.

This may be seen to be an operator space structure on X by using 1.2.16 say; andfrom this formula it is also clear that it is the largest such. Since every Banach

Operator spaces 11

space is isometric to an operator space (see 1.2.21), ‖ · ‖1 is evidently the usualnorm on E. It is clear from this formula that Max(E) has the property that forany operator space Y , and for any bounded linear u : E → Y , we have

‖u : Max(E) −→ Y ‖cb = ‖u : E −→ Y ‖. (1.12)

1.2.23 (Hilbert column and row spaces) If H is a Hilbert space then thereare two canonical operator space structures on H most commonly considered.The first is the Hilbert column space Hc. Informally one should think of Hc as a‘column in B(H)’. Thus if H = 2

n then Hc = Mn,1, thought of as the matrices inMn which are ‘zero except on the first column’. We write this operator space alsoas Cn, and the ‘row’ version as Rn. For a general Hilbert space H there are severalsimple ways of describing Hc more precisely. For example, one may identify Hc

with the concrete operator space B(C, H). Another equivalent description is asfollows (we leave the equivalence as an exercise). If η is a fixed unit vector in H ,then the set H⊗η of rank one operators ζ⊗η is a closed subspace of B(H) whichis isometric to H via the map ζ → ζ ⊗ η. (By convention, ζ ⊗ η maps ξ ∈ H to〈ξ, η〉ζ.) Thus we may transfer the operator space structure on H ⊗ η inheritedfrom B(H) over to H . The resulting operator space structure is independent ofη and coincides with Hc. Indeed from the C∗-identity in Mn(B(H)) applied to‖[ζij ⊗ η]‖, one immediately obtains

‖[ζij ]‖Mn(Hc) =∥∥∥[ n∑

k=1

〈ζkj , ζki〉]∥∥∥ 1

2, [ζij ] ∈ Mn(H). (1.13)

If T ∈ B(H, K) then ‖T ‖ = ‖T ‖cb, where the latter is the norm taken inCB(Hc, Kc). Indeed let [ζij ] ∈ Mn(Hc), and let α ∈ B(2

n, 2n(H)) correspond to

this matrix via the identity Mn(Hc) = Mn(B(C, H)) = B(2n, 2

n(H)). Likewiselet β ∈ B(2

n, 2n(K)) corresponding to [Tζij ]. Then β = (I2n

⊗T ) α, and hence‖β‖ ≤ ‖T ‖‖α‖. This shows that ‖T ‖cb ≤ ‖T ‖. More generally, we have

B(H, K) = CB(Hc, Kc) completely isometrically (1.14)

We give a quick proof of this identity in the Notes for this section.A subspace K of a Hilbert column space Hc is again a Hilbert column space,

as may be seen by considering (1.13). Similarly the quotient Hc/Kc is a Hilbertcolumn space completely isometric to (H K)c, as may be seen by applying1.2.15 to the canonical (completely contractive by (1.14)) projection P from H c

onto (H K)c.We define Hilbert row space similarly. Recalling that H∗ ∼= H is a Hilbert

space too, we identify Hr with the concrete operator space B(H, C). Analoguesof the above results for Hc hold, except that there is a slight twist in the cor-responding version of (1.14). Namely, although B(H, K) = CB(Hr, Kr) iso-metrically, this is not true completely isometrically. Instead there is a canonicalcompletely isometric isomorphism B(H, K) ∼= CB(Kr, Hr). We have


(Hc)∗ ∼= Hr and (Hr)∗ ∼= Hc (1.15)

completely isometrically using the operator space dual structure in 1.2.20. Thefirst relation is obtained by setting K = C in (1.14). Similarly, the second relationfollows e.g. from the line above (1.15).

We write C and R for 2 with its column and row operator space structuresrespectively.

1.2.24 (The operator space R∩C) We let R∩C be 2 with the operator spacestructure defined by the embedding 2 → R ⊕∞ C which takes any x ∈ 2 tothe pair (x, x). Let (ek)k≥1 denote the canonical basis of 2. Then it follows from(1.4) that for any N ≥ 1 and any x1, . . . , xn in MN , we have∥∥∑

k

xk ⊗ ek

∥∥MN (R∩C)

= max∥∥∑

k

x∗kxk

∥∥ 12 ,

∥∥∑k

xkx∗k

∥∥ 12. (1.16)

We also note that if X is an operator space and u : X → 2 is a bounded linearmap, then u is completely bounded from X into R ∩ C if and only if it is bothcompletely bounded from X into R and from X into C. Moreover we have

‖u : X → R ∩ C‖cb = max‖u : X → R‖cb , ‖u : X → C‖cb

. (1.17)

1.2.25 (Opposite and adjoint) If X is an operator space, in B(K, H) say,then we define the adjoint operator space to be the space X = x∗ : x ∈ X(see 1.1.2). As an abstract operator space X is independent of the particularrepresentation of X on H and K. Indeed we can alternatively define X as theset of formal symbols x∗ for x ∈ X , with scalar product λx∗ = (λx)∗, and withmatrix norms ‖[x∗

ij ]‖n = ‖[xji]‖n, where the latter norm is taken in Mn(X). Theadjoint operator space is sometimes denoted by X by some authors. Howeverwe warn the reader that X is not the same as the conjugate operator spaceconsidered in [337].

If X is an operator space then we define the opposite operator space Xop tobe the Banach space X with the ‘transposed matrix norms’ ‖[xij ]‖op

n = ‖[xji]‖n.Note that if A is a C∗-algebra, then these matrix norms on Aop coincide withthe canonical matrix norms on the C∗-algebra which is A with its reversed mul-tiplication. If X is a subspace of a C∗-algebra A, then Xop may be identifiedcompletely isometrically with the associated subspace of the C∗-algebra Aop.

If u : X → Y , then we write uop for u considered as a map from Xop to Y op,and u for the map from X to Y defined by u(x∗) = u(x)∗. These maps arecompletely bounded, completely contractive, completely isometric, etc., if u hasthese properties. There is a ‘conjugate linear complete isometry’ from Xop toX, namely the map x → x∗.

1.2.26 (Matrix spaces) If X is an operator space, and I, J are cardinal numbersor sets, then we write MI,J(X) for the set of I ×J matrices whose finite subma-trices have uniformly bounded norm. Such a matrix is normed by the supremum

Operator spaces 13

of the norms of its finite submatrices. Similarly there is an obvious way to definea norm on Mn(MI,J(X)) by equating this space with MI,J(Mn(X)), and onehas Mn(MI(X)) ∼= Mn.I(X), for n ∈ N.

We are being deliberately careless here, and indeed in the rest of the bookwe often abusively blur the distinction between cardinals and sets. Technicallyif I, J are cardinals, we should fix sets I0 and J0 of cardinality I and J re-spectively, consider matrices [xij ] indexed by i ∈ I0 and j ∈ J0, and writeMI0,J0(X) instead of MI,J(X). However if one chooses different sets I1 and J1

of these cardinalities, then there is an obvious completely isometric isomorphismMI0,J0(X) ∼= MI1,J1(X), so that with a little care our convention should notlead us into trouble. Or we may protect ourselves by fixing one well-ordered setassociated with each cardinal.

We write MI(X) = MI,I(X), CwI (X) = MI,1(X), and Rw

I (X) = M1,I(X). IfI = ℵ0 we simply denote these spaces by M(X), Cw(X) and Rw(X) respectively.Also, M

finI,J(X) will denote the vector subspace of MI,J(X) consisting of ‘finitely

supported matrices’, that is, those matrices with only a finite number of nonzeroentries. We write KI,J(X) for the norm closure in MI,J(X) of M

finI,J(X). We

set KI(X) = KI,I(X), CI(X) = KI,1(X), and RI(X) = K1,I(X). Again wemerely write K(X), R(X) and C(X) for these spaces if I = ℵ0. If X = C thenCw

I (C) = CI(C) = (2I)

c (see 1.2.23 for this notation), and we usually write thiscolumn Hilbert space as CI . Similarly, RI = RI(C) = (2

I)r. We write KI,J for

KI,J(C), and MI,J for MI,J(C).It is fairly obvious that if u : X → Y is completely bounded, then so is the

obvious amplification uI,J : MI,J(X) → MI,J(Y ), and ‖uI,J‖cb = ‖u‖cb. ClearlyuI,J also restricts to a completely bounded map from KI,J(X) to KI,J(Y ). If u isa complete isometry, then so is uI,J . Thus the MI,J(·) and KI,J(·) constructionsare ‘injective’ in some sense.

For cardinals I, J , we leave it as an exercise that MI,J∼= B(2

J , 2I). Via this

identification, KI,J = S∞(2J , 2

I). Thus for any Hilbert spaces K, H we have thatB(K, H) ∼= MI0,J0 for some cardinals I0, J0. We leave it as another exercise that

MI,J(MI0,J0) ∼= MI×I0,J×J0 (1.18)

completely isometrically. Putting these two exercises together, we have estab-lished that for any cardinals I, J , we have

MI,J(B(K, H)) ∼= B(K(J), H(I)) completely isometrically. (1.19)

If X is an operator space then so is MI,J(X). This may be seen by choos-ing a completely isometric embedding X ⊂ B(H), and noting that by the‘injectivity’ mentioned a few paragraphs back, and formula (1.19), we haveMI,J(X) ⊂ MI,J(B(H)) ∼= B(H(J), H(I)) completely isometrically. If X is com-plete then so is MI,J(X), since it is clearly norm closed in MI,J(B(K, H)).

For any operator space X , we have

MI,J(X) = CwI (Rw

J (X)) = RwJ (Cw

I (X)). (1.20)


One way to see this is to first check (1.20) in the case X = B(H) using(1.19), and then use this fact to do the general case. By a similar argument,MI,J(MI0,J0(X)) ∼= MI×I0,J×J0(X) for any operator space X , generalizing (1.18).

1.2.27 (Infinite sums) Suppose that X, Y are subspaces of a (complete) oper-ator algebra or C∗-algebra A ⊂ B(H). Let I be an infinite set. If x ∈ Rw

I (X)and y ∈ CI(Y ), then the ‘product’ xy (defined to be

∑i xiyi if x and y have

ith entries xi and yi respectively) actually converges in norm to an element ofA. To see this, we use the following notation. If z is an element of Rw

I (X) orCI(Y ), and if ∆ ⊂ I, write z∆ for z but with all entries outside ∆ ‘switchedto zero’. Since y ∈ CI(Y ), given ε > 0 there is a finite set ∆ ⊂ I, such that‖y − y∆‖ = ‖y∆c‖ < ε. If ∆′ is a finite subset of I not intersecting ∆ then∥∥∥∑

i∈∆′xiyi

∥∥∥ = ‖x∆′y∆′‖ ≤ ‖x∆′‖‖y∆′‖ ≤ ‖x‖‖y∆′‖ < ‖x‖ε.

Hence the sum converges in norm as claimed. Thus we have

RwI (X) CI(Y ) ⊂ XY and RI(X) Cw

I (Y ) ⊂ XY, (1.21)

where XY is as defined in 1.1.2, a closed subset of A. Also ‖xy‖ ≤ ‖x‖‖y‖ forx, y as above, as may be seen from a computation identical to the first part ofthe second last centered equation.

Proposition 1.2.28 For any operator space X and cardinal I, we have thatCB(CI , X) ∼= Rw

I (X) and CB(RI , X) ∼= CwI (X) completely isometrically.

Proof We prove just the first relation. Define L : RwI (X) → CB(CI , X) by

L(x)(z) =∑

i xizi, for x ∈ RwI (X), z ∈ CI . This map is well defined, by the

argument for (1.21) for example. It is also easy to check, by looking at thepartial sums of this series as in (1.21), that L is contractive. Conversely, for uin CB(CI , X), let x ∈ Rw

I (X) have ith entry u(ei), where (ei) is the canonicalbasis. It is not hard to see that ‖x‖Rw

I (X) ≤ ‖u‖cb, and L(x) = u. Thus L is asurjective isometry. This together with (1.7) yields

Mm(CB(CI , X)) ∼= CB(CI , Mm(X)) ∼= RwI (Mm(X)) ∼= Mm(Rw

I (X))

isometrically. From this one sees that L is a complete isometry.

Proposition 1.2.29 If X and Y are operator spaces then there are canonicalcomplete isometries

KI,J(CB(X, Y )) → CB(X, KI,J(Y )) → CB(X, MI,J(Y )) ∼= MI,J(CB(X, Y )).

In particular, if Y = C, we have CB(X, MI,J) ∼= MI,J(X∗).

Proof Since KI,J(Y ) ⊂ MI,J(Y ), the middle inclusion is evident. There is acanonical map Θ: MI,J(CB(X, Y )) → CB(X, MI,J(Y )), which takes an element

Operator spaces 15

[uij ] from MI,J(CB(X, Y )), to the map x → [uij(x)]. Also there is a canonicalmap CB(X, MI,J(Y )) → MI,J(CB(X, Y )), which takes u to the matrix [πij u],where πij is the projection of MI,J(Y ) onto its i-j entry. It is rather easy to checkthat these maps are mutual inverses, and are both completely contractive. Hencethey are complete isometries. Thus MI,J(CB(X, Y )) ∼= CB(X, MI,J(Y )). Fi-nally, the isometry above taking MI,J(CB(X, Y )) into CB(X, MI,J(Y )), clearlytakes M

finI,J(CB(X, Y )) into CB(X, KI,J(Y )). By density, KI,J(CB(X, Y )) em-

beds completely isometrically in CB(X, KI,J(Y )).

1.2.30 (Interpolation) We recall the complex interpolation method for Banachspaces (e.g. see [33, Chapter 4]). Suppose that (X0, X1) is a compatible couple ofBanach spaces. This means that we are given a topological vector space Z, andone-to-one continuous linear mappings from X0 to Z and X1 to Z. Regard X0

and X1 as subspaces of Z. Their ‘sum’ X0 + X1 ⊂ Z is, by definition, the spaceof all x0 + x1, with x0 ∈ X0 and x1 ∈ X1. This is a Banach space with norm

‖x‖ = inf‖x0‖X0 + ‖x1‖X1 : x0 ∈ X0, x1 ∈ X1, x = x0 + x1.

We let S denote the strip of all complex numbers z with 0 ≤ Re(z) ≤ 1 andwe let F = F(X0, X1) be the space of all bounded and continuous functionsf : S → X0 + X1 such that the restriction of f to the interior of S is analytic,and such that the maps t → f(it) and t → f(1 + it) belong to C0(R; X0) andC0(R; X1) respectively. Then F is a Banach space for the norm

‖f‖F = maxsup

t‖f(it)‖X0 , sup

t‖f(1 + it)‖X1

. (1.22)

For any 0 ≤ θ ≤ 1, the interpolation space Xθ = [X0, X1]θ is the subspace ofX0 + X1 formed by all x such that x = f(θ) for some f ∈ F . This turns out tobe a Banach space for the norm ‖x‖Xθ

= inf‖f‖F : f ∈ F , f(θ) = x

. If we

let Fθ = Fθ(X0, X1) be the subspace of all f ∈ F for which f(θ) = 0, we seethat the mapping f → f(θ) induces an isometric isomorphism

Xθ = F(X0, X1)/Fθ(X0, X1). (1.23)

Assume now that X0 and X1 are operator spaces. Then each interpolationspace Xθ has a ‘natural’ operator space structure. Indeed note from (1.22) thatthe mapping which takes any f ∈ F(X0, X1) to the pair of its restrictions to thelines Re(z) = 0 and Re(z) = 1, induces an isometric embedding

F(X0, X1) ⊂ C0(R; X0) ⊕∞ C0(R; X1).

By 1.2.17 and 1.2.18, we may consider F(X0, X1) as an operator space, the normon Mn(F(X0, X1)) being inherited from C0(R; Mn(X0)) ⊕∞ C0(R; Mn(X1)).

16 Completely positive maps

Then taking the resulting quotient operator space structure on F/F θ and ap-plying (1.23), makes Xθ an operator space. More explicitly, the matrix norms onthe operator space Xθ are given for any [xjk] ∈ Mn(Xθ) by

‖[xjk]‖n = inf

maxsup

t‖[fjk(it)]‖Mn(X0) , sup

t‖[fjk(1 + it)]‖Mn(X1)

,

the infimum taken over all fjk ∈ F(X0, X1) such that fjk(θ) = xjk, for all j, k.Observe that for each n ≥ 1, we have natural one-to-one continuous linear

maps Mn(X0) → Mn(Z) and Mn(X1) → Mn(Z) . Hence (Mn(X0), Mn(X1)) isa compatible couple of Banach spaces. It follows easily from the above discussionthat Mn(F(X0, X1)) = F(Mn(X0), Mn(X1)) isometrically, and hence

Mn(Xθ) =[Mn(X0), Mn(X1)

]θ. (1.24)

This formula readily implies that a key interpolation theorem (see [33, Theorem4.1.2]) extends to completely bounded maps. Namely, let (Y0, Y1) be anothercompatible couple of operator spaces, and let u : X0 + X1 → Y0 + Y1 be a linearmap. If u is completely bounded as a map from X0 into Y0, and from X1 intoY1, then u is completely bounded from Xθ into Yθ, for any θ ∈ (0, 1), with

‖u : Xθ −→ Yθ‖cb ≤ ‖u : X0 −→ Y0‖1−θcb ‖u : X1 −→ Y1‖θ

cb.

1.2.31 (Ultraproducts) Let U be an ultrafilter on a set I and let (Xi)i∈I be afamily of operator spaces. We let NU ⊂ ⊕∞

i Xi be the space of all (xi)i such thatlimU ‖xi‖Xi = 0. By definition, the ultraproduct of the family (Xi)i∈I along Uis the quotient operator space∏

i∈I

Xi/U =(⊕∞

i Xi

)/NU

from 1.2.17 and 1.2.14. It is easy to check that for any x = (xi)i in ⊕iMn(Xi),the norm of its class x modulo Mn(NU ) is equal to

‖x‖ = limU

‖xi‖Mn(Xi).

This implies the ‘injectivity’ of ultraproducts. Namely if (Yi)i∈I is another familyof operator spaces such that Xi ⊂ Yi completely isometrically for each i ∈ I,then

∏i∈I Xi/U ⊂ ∏

i∈I Yi/U completely isometrically.Finally, we observe that if (Ai)i∈I is a family of C∗-algebras, then NU is an

ideal of ⊕∞i Xi and hence their ultraproduct is fairly clearly a C∗-algebra. Note

that if (ai)i and (bi)i belong to ⊕∞i Ai, the product ab of their classes modulo

NU is the class of (aibi)i.

1.3 COMPLETELY POSITIVE MAPS

1.3.1 (Unital operator spaces) Recall that a C∗-algebra A is called unital if itcontains an identity element 1. We say that an operator space X is unital if it

Operator spaces 17

has a distinguished element usually written as e or 1, called the identity of X ,such that there exists a complete isometry u : X → A into a unital C∗-algebrawith u(e) = 1. A unital-subspace of such X is a subspace containing e.

1.3.2 (Operator systems) An operator system is a unital-subspace S of a unitalC∗-algebra A which is selfadjoint, that is, x∗ ∈ S if and only if x ∈ S. A subsystemof an operator system S is a selfadjoint linear subspace of S containing the‘identity’ 1 of S. If S is an operator system, a subsystem of a C∗-algebra A,then S has a distinguished ‘positive cone’ S+ = x ∈ S : x ≥ 0 in A. We alsowrite Ssa for the real vector space of selfadjoint elements x (i.e. those satisfyingx = x∗) in S. Then S has an associated ordering ≤, namely we say that x ≤ y ifx, y are selfadjoint and y−x ∈ S+. By the usual trick, any element of an operatorsystem S is of the form h + ik for h, k ∈ Ssa. Also, if h ∈ Ssa then ‖h‖1 + h and‖h‖1 − h are positive. Thus Ssa = S+ − S+.

A linear map u : S → S ′ between operator systems is called ∗-linear ifu(x∗) = u(x)∗ for all x ∈ S. Some authors say that such a map is selfadjoint. Wesay that u is positive if u(S+) ⊂ S ′

+. By facts at the end of the last paragraph itis easy to see that a positive map is ∗-linear. The operator system Mn(S), whichis a subsystem of Mn(A), has a ‘positive cone’ too, and thus it makes sense totalk about completely positive maps between operator systems. These are themaps u such that un = IMn ⊗ u : Mn(S) → Mn(S′) is positive for all n ∈ N.Indeed the morphisms in the category of operator systems are often taken to bethe unital completely positive maps.

Suppose that S is a subsystem of a unital C∗-algebra. By the Hahn–Banachtheorem and A.4.2 (resp. (A.11)), it follows that Ssa (resp. S+) is exactly the setof elements x ∈ S such that ϕ(x) ∈ R (resp. ϕ(x) ≥ 0) for all ϕ ∈ (Span1, x)∗with ϕ(1) = ‖ϕ‖ = 1. From this it is clear that if u : S1 → S2 is a contractiveunital linear map between operator systems, then u is a positive and ∗-linearmap. Applying this principle to un, we see that a completely contractive unitallinear map between operator systems is completely positive.

Clearly an isomorphism between operator systems which is unital and com-pletely positive, and has completely positive inverse, preserves all the ‘order’.Such a map is called a complete order isomorphism. The range of a completelypositive unital map between operator systems is clearly also an operator system;we say that such a map is a complete order injection if it is a complete orderisomorphism onto its range.

The following simple fact relates the norm to the matrix order, and is anelementary exercise using the definition of a positive operator. Namely, if x is anelement of a unital C∗-algebra or operator system A, or if x ∈ B(K, H), then[

1 xx∗ 1

]≥ 0 ⇐⇒ ‖x‖ ≤ 1. (1.25)

Here ‘≥ 0’ means ‘positive in M2(A)’ (or ‘positive in B(H ⊕ K)’).


1.3.3 It is easy to see from (1.25) that a completely positive unital map ubetween operator systems is completely contractive. (For example, to see thatu is contractive, take ‖x‖ ≤ 1, and apply u2 to the associated positive matrixin (1.25). This is positive, so that using (1.25) again we see that ‖u(x)‖ ≤ 1.)Putting this together with some facts from 1.3.2 we see that a unital map betweenoperator systems is completely positive if and only if it is completely contractive;and in this case the map is ∗-linear. If, further, u is one-to-one, then by applyingthe above to u and u−1 one sees immediately that a unital map between operatorsystems is a complete order injection if and only if it is a complete isometry.

We omit the well-known proofs of the following two results.

Theorem 1.3.4 (Stinespring) Let A be a unital C∗-algebra. A linear mapu : A → B(H) is completely positive if and only if there is a Hilbert space K, aunital ∗-homomorphism π : A → B(K), and a bounded linear V : H → K suchthat u(a) = V ∗π(a)V for all a ∈ A. This can be accomplished with ‖u‖cb = ‖V ‖2.Also, this equals ‖u‖. If u is unital then we may take V to be an isometry; inthis case we may view H ⊂ K, and we have u(·) = PHπ(·)|H .

Theorem 1.3.5 (Arveson’s extension theorem) If S is a subsystem of a unitalC∗-algebra A, and if u : S → B(H) is completely positive, then there exists acompletely positive map u : A → B(H) extending u.

Indeed, if u is unital, then 1.3.5 may be easily seen from 1.2.10 and 1.3.3(although usually one proves 1.2.10 using the completely positive variant).

Lemma 1.3.6 (Arveson) Suppose that X is a unital-subspace of an operatorsystem, and suppose that u : X → B(K) is a unital contraction (resp. com-plete contraction, complete isometry) with range Y . Then there exists a posi-tive map (resp. completely positive map, complete order isomorphism) u betweenthe operator systems X + X and Y + Y , which extends u, namely the mapx1 + x∗

2 → u(x1) + u(x2)∗, for x1, x2 ∈ X.

Proof Note that the ‘contraction’ result here applied to the amplifications un

will imply the ‘complete contraction’ result; and the complete isometry case willthen follow by considering u−1.

Suppose that u is a contraction, and consider u restricted to the operatorsubsystem X ∩ X. This is a unital contraction and therefore is positive and∗-linear by a fact in 1.3.2. From this it is easy to check directly that the formulaabove for u is well defined.

Suppose that x1 + x∗2 ∈ X + X is positive, and that ζ is a unit vector in K,

so that ϕ = 〈·ζ, ζ〉 is a state on B(K). Then ϕ u extends by the Hahn–Banachtheorem to a contractive unital functional ψ on X + X. By the aforementionedfact from 1.3.2, ψ is therefore positive and ∗-linear. Thus

〈(u(x1) + u(x2)∗)ζ, ζ〉 = ψ(x1) + ψ(x2) = ψ(x1 + x∗2) ≥ 0.

Hence u(x1) + u(x2)∗ ≥ 0.

Operator spaces 19

1.3.7 (The diagonal) Because of this last result, if X is a unital operator space(see 1.3.1) then there is an essentially unique operator system, written as X+X,which is spanned by X and its adjoint space X. Indeed, if u : X → B(H) is anyunital complete isometry into B(H) (or into an operator system), then by 1.3.6the operator system u(X)+u(X) is (up to unital complete order isomorphism)independent of the particular u. We usually identify two unital operator spacesup to unital completely isometric isomorphism.

By the same principle, any such X contains a canonical operator system,namely ∆(X) = x ∈ X : u(x)∗ ∈ u(X), where u is any unital completeisometry as in the last paragraph. This is well defined independently of u. Wecall ∆(X) the diagonal of X .

The following follows immediately from a fact in 1.3.2 and the last definition:

Corollary 1.3.8 If S is an operator system, if Y is a unital operator space, andif u : S → Y is a unital contraction, then Ran(u) ⊂ ∆(Y ), and u is positive.

Proposition 1.3.9 (A Kadison–Schwarz inequality) If u : A → B is a uni-tal completely positive (or equivalently unital completely contractive) linear mapbetween unital C∗-algebras, then u(a)∗u(a) ≤ u(a∗a), for all a ∈ A.

Proof By 1.3.4 we have u = V ∗π(·)V , with ‖V ‖ ≤ 1 and π a ∗-homomorphism.Thus u(a)∗u(a) = V ∗π(a)∗V V ∗π(a)V ≤ V ∗π(a)∗π(a)V = u(a∗a).

Corollary 1.3.10 Let u : A → B be a completely isometric unital surjectionbetween unital C∗-algebras. Then u is a ∗-isomorphism.

Proof By 1.3.9 applied to both u and u−1 we have u(x)∗u(x) = x∗x for allx ∈ A. Now use the polarization identity (see (1.1)).

Proposition 1.3.11 Let u : A → B be as in 1.3.9. Suppose that c ∈ A, and thatc satisfies u(c)∗u(c) = u(c∗c). Then u(ac) = u(a)u(c) for all a ∈ A.

Proof Suppose that B ⊂ B(H). We write u = V ∗π(·)V as in Stinespring’stheorem, with V ∗V = IH . Let P = V V ∗ be the projection onto V (H). Byhypothesis V ∗π(c)∗Pπ(c)V = V ∗π(c)∗π(c)V . For ζ ∈ H , set η = π(c)V ζ. Then‖Pη‖2 = 〈V ∗π(c)∗Pπ(c)V ζ, ζ〉 = ‖η‖2. Thus Pη = η, and V V ∗π(c)V = π(c)V .Therefore u(a)u(c) = V ∗π(a)V V ∗π(c)V = V ∗π(a)π(c)V = u(ac).

1.3.12 (Completely positive bimodule maps) An immediate consequence of1.3.11: Suppose that u : A → B is as in 1.3.9, and that there is a C∗-subalgebraC of A with 1A ∈ C, such that π = u|C is a ∗-homomorphism. Then

u(ac) = u(a)π(c) and u(ca) = π(c)u(a) (a ∈ A, c ∈ C).

Theorem 1.3.13 (Choi and Effros) Suppose that A is a unital C∗-algebra,and that Φ: A → A is a unital, completely positive (or equivalently by 1.3.3,completely contractive), idempotent map. Then we may conclude:(1) R = Ran(Φ) is a C∗-algebra with respect to the original norm, involution,

and vector space structure, but new product r1 Φ r2 = Φ(r1r2).


(2) Φ(ar) = Φ(Φ(a)r) and Φ(ra) = Φ(rΦ(a)), for r ∈ R and a ∈ A.(3) If B is the C∗-subalgebra of A generated by the set R, and if R is given the

product Φ, then Φ|B is a ∗-homomorphism from B onto R.

Proof (2) By linearity and the fact that a positive map is ∗-linear (see 1.3.2),we may assume that a, r are selfadjoint. Set

d = d∗ =[

0 rr∗ a

].

Then Φ2(d2) ≥ (Φ2(d))2 by the Kadison–Schwarz inequality 1.3.9, so that[Φ(r2) Φ(ra)Φ(ar) ∗

]≥[

r2 rΦ(a)Φ(a)r ∗

].

Here ∗ is used for a term we do not care about. Applying Φ2 gives[Φ(r2) Φ(ra)Φ(ar) ∗

]≥[

Φ(r2) Φ(rΦ(a))Φ(Φ(a)r) ∗

].

Thus [0 Φ(ra) − Φ(rΦ(a))

Φ(ar) − Φ(Φ(a)r) ∗

]≥ 0,

which implies that Φ(ra) − Φ(rΦ(a)) = 0 and Φ(ar) − Φ(Φ(a)r) = 0.(1) By (2) we have for r1, r2, r3 ∈ R that

(r1 Φ r2) Φ r3 = Φ(Φ(r1r2)r3) = Φ(r1r2r3).

Similarly, r1 Φ (r2 Φ r3) = Φ(r1r2r3), which shows that the multiplication isassociative. It is easy to check that R (with original norm, involution, and vectorspace structure, but new multiplication) satisfies the conditions necessary to bea C∗-algebra. For example:

(r1 Φ r2)∗ = Φ(r1r2)∗ = Φ(r∗2r∗1) = r∗2 Φ r∗1 .

We check the C∗-identity using the Kadison–Schwarz inequality 1.3.9:

‖r∗ Φ r‖ = ‖Φ(r∗r)‖ ≥ ‖Φ(r)∗Φ(r)‖ = ‖r∗r‖ = ‖r‖2,

and conversely,

‖r‖2 = ‖r∗r‖ ≥ ‖Φ(r∗r)‖ = ‖r∗ Φ r‖.

(3) This will follow if we can prove that Φ(r1r2 · · · rn) = r1 Φ r2 · · · Φ rn,for ri ∈ R. This follows in turn by induction on n. Supposing that it is true forn = k, we see that r1 Φ r2 · · · Φ rk+1 equals

Φ((r1 Φ r2 · · · Φ rk)rk+1) = Φ(Φ(r1r2 · · · rk)rk+1) = Φ(r1r2 · · · rkrk+1),

using (2) in the last equality.

Operator spaces 21

It is important to note, and easy to check, that the canonical matrix normsfrom 1.2.3 for the C∗-algebra Φ(A) in the result above, coincide with its canonicalmatrix norms as a subspace of A. This may be seen by the uniqueness of acomplete C∗-norm on a ∗-algebra (which in turn is immediate from A.5.8), andan application of Theorem 1.3.13 to the canonically associated projection Φn onMn(A), for each n ∈ N.

1.3.14 (The Paulsen system) If X is a subspace of B(H), we define the Paulsensystem to be the operator system

S(X) =[

CIH XX

CIH

]=

[λ xy∗ µ

]: x, y ∈ X, λ, µ ∈ C

in M2(B(H)), where the entries λ and µ in the last matrix stand for λIH and µIH

respectively. The following important lemma shows that as an operator system(i.e. up to complete order isomorphism) S(X) only depends on the operatorspace structure of X , and not on its representation on H .

Lemma 1.3.15 (Paulsen) Suppose that for i = 1, 2, we are given Hilbert spacesHi, Ki, and linear subspaces Xi ⊂ B(Ki, Hi). Suppose that u : X1 → X2 is alinear map. Let S i be the following operator system inside B(Hi ⊕ Ki):

Si =[

CIHi Xi

Xi CIKi

].

If u is contractive (resp. completely contractive, completely isometric), then

Θ :[

λ xy∗ µ

]→

[λ u(x)

u(y)∗ µ

]is positive (resp. completely positive and completely contractive, a complete orderinjection) as a map from S1 to S2.

Proof Suppose that z is a positive element of S1. Thus

z =[

a xx∗ b

]where a and b are positive. Since z ≥ 0 if and only if z + ε1 ≥ 0 for all ε > 0, wemay assume that a and b are invertible. Then[

a− 12 0

0 b−12

] [a xx∗ b

][a− 1

2 00 b−

12

]=

[1 a− 1

2 xb−12

b−12 x∗a− 1

2 1

]≥ 0.

Hence by (1.25), we have that ‖a− 12 xb−

12 ‖ ≤ 1. Applying u we obtain that

‖a− 12 u(x)b−

12 ‖ ≤ 1. Reversing the argument above now shows that Θ(z) ≥ 0. So

Θ is positive, and a similar argument shows that it is completely positive if uis completely contractive. By 1.3.3 we have that Θ is completely contractive inthat case. If in addition u is a complete isometry, then applying the above to uand u−1 we obtain the final assertion.

22 Operator space duality

1.4 OPERATOR SPACE DUALITY

An operator space Y is said to be a dual operator space if Y is completelyisometrically isomorphic to the operator space dual (see 1.2.20) X∗ of an operatorspace X . We also say that X is an operator space predual of Y , and sometimeswe write X as Y∗. If X, Y are dual operator spaces then we write w∗CB(X, Y )for the space of w∗-continuous completely bounded maps from X to Y .

Unless otherwise indicated, in what follows the symbol X∗ denotes the dualspace together with its dual operator space structure as defined in 1.2.20. Ofcourse X∗∗ is considered as the dual operator space of X∗.

Proposition 1.4.1 If X is an operator space then X ⊂ X∗∗ completely isomet-rically via the canonical map iX .

Proof Let X ⊂ B(H). By the definitions, iX is completely contractive. To seethat iX is completely isometric, it suffices to find for a given n ∈ N, ε > 0,and [xkl] ∈ Mn(X), an integer m and a completely contractive u : B(H) → Mm

such that ‖[u(xkl)]‖ ≥ ‖[xkl]‖ − ε. For such [xkl] and ε, by (1.2) we may chooseζ1, . . . , ζn, η1, . . . , ηn ∈ H with

∑l ‖ζl‖2 =

∑k ‖ηk‖2 = 1 and∣∣∣∑

k,l

〈xklζl, ηk〉∣∣∣ ≥ ‖[xkl]‖ − ε.

Let K = Span ζ1, . . . , ζn, η1, . . . , ηn, and let u : B(H) → B(K) be the mapu(x) = PKx|K . If m = dim(K), then we may view u as mapping into Mm. It iscompletely contractive, by the remark after 1.2.8 for example. Finally,

‖[u(xkl)]‖ ≥∣∣∣∑

k,l

〈u(xkl)ζl, ηk〉∣∣∣ =

∣∣∣∑k,l

〈xklζl, ηk〉∣∣∣ ≥ ‖[xkl]‖ − ε,

which is the desired inequality.

1.4.2 (Remarks) From 1.4.1 we have for any [xij ] ∈ Mn(X) that

‖[xij ]‖n = sup‖[ϕkl(xij)]‖ : m ∈ N, [ϕkl] ∈ Ball(Mm(X∗)) (1.26)

In other words, the following canonical isomorphism is a complete isometry:

Mn(X) ∼= w∗CB(X∗, Mn) ⊂ CB(X∗, Mn). (1.27)

Another consequence of 1.4.1, is that if X is an operator space which as aBanach space is reflexive, then X ∼= X∗∗ completely isometrically.

1.4.3 (The adjoint map) The ‘adjoint’ or ‘dual’ u∗ of a completely boundedmap u : X → Y between operator spaces is completely bounded from Y ∗ to X∗,with ‖u∗‖cb ≤ ‖u‖cb, as may be seen from the obvious computation. Indeed using(1.26) during this computation or applying 1.4.1, one sees that ‖u‖cb = ‖u∗‖cb.

Direct computations from the definitions also show that if u is a completequotient map then u∗ is a complete isometry. It is slightly harder to see that

Operator spaces 23

u is completely isometric if and only if u∗ is a complete quotient map. Indeed,if u is a complete isometry, then we may regard X ⊂ Y , and then an elementin the open ball of Mn(X∗) ∼= CB(X, Mn) may be ‘extended’ by 1.2.10 to anelement in the open ball of CB(Y, Mn) = Mn(Y ∗). This shows that u∗ is acomplete quotient map. Conversely, if u∗ is a complete quotient map then u∗∗ isa complete isometry, so that u is a complete isometry (using 1.4.1). Finally, onemay see as in the Banach space case, that for complete operator spaces, if u∗ isa complete isometry then u is a complete quotient map (e.g. see A.2.3 in [149]).

Thus u is a complete isometry if and only if u∗∗ is a complete isometry.

1.4.4 (Duality of subspaces and quotients) The operator space versions of theusual Banach duality of subspaces and quotients follow easily from 1.4.3. If Xis a subspace of Y , then applying 1.4.3 to the inclusion map X → Y yieldsthe fact that X∗ ∼= Y ∗/X⊥ completely isometrically via the canonical map.Similarly, the dual of the canonical quotient map Y → Y/X is the canonicalcomplete isometry (Y/X)∗ ∼= X⊥. The predual versions go through too with thesame proofs as in the Banach space case: if X is a w∗-closed subspace of a dualoperator space Y , then (Y∗/X⊥)∗ ∼= (X⊥)⊥ = X as dual operator spaces. Also,(X⊥)∗ ∼= Y/(X⊥)⊥ = Y/X completely isometrically.

1.4.5 (The trace class operator space) If H is a Hilbert space then B(H) isa dual operator space. More precisely, let us equip its predual Banach spaceS1(H) (e.g. see A.1.2) with the operator space structure it inherits from B(H)∗

via the canonical isometric inclusion S1(H) → B(H)∗. Then B(H) = S1(H)∗

completely isometrically. Indeed the canonical map from B(H) to S1(H)∗ is com-pletely contractive by definition. That this map is completely isometric followsfrom the fact, included in the proof of 1.4.1, that for any n ∈ N, ε > 0, and[xkl] ∈ Mn(B(H)), we can find an integer m and a w∗-continuous completelycontractive u : B(H) → Mm such that ‖[u(xkl)]‖ ≥ ‖[xkl]‖ − ε.

Similarly, B(K, H) is the dual operator space of the space S1(H, K) of traceclass operators, the latter regarded as a subspace of B(K, H)∗. Henceforth, whenwe write S1(H, K) we will mean the operator space predual of B(K, H) describedabove. Similarly, we will henceforth also view S1

n = M∗n as an operator space.

Lemma 1.4.6 Any w∗-closed subspace X of B(H) is a dual operator space.Indeed, if Y = S1(H)/X⊥ is equipped with its quotient operator space structureinherited from S1(H), then X ∼= Y ∗ completely isometrically.

Proof This follows from 1.4.4 and 1.4.5.

In particular this shows that any W ∗-algebra equipped with its ‘natural’operator space structure (see 1.2.3) is a dual operator space.

The converse of 1.4.6 is true too, as we see next, so that ‘dual operator spaces’,and the w∗-closed subspaces of some B(H), are essentially the same thing.

Lemma 1.4.7 Any dual operator space is completely isometrically isomorphic,via a homeomorphism for the w∗-topologies, to a w∗-closed subspace of B(H),for some Hilbert space H.


Proof Suppose that W is a dual operator space, with predual X . Let Y = C,and recall from 1.2.19 the construction of a complete isometry

W = CB(X, Y ) −→ ⊕∞x∈I Mnx(Y ) = ⊕x∈I Mnx ,

namely the map J taking w ∈ W to the tuple ([〈w, xij〉])x in ⊕x Mnx . Sincethe maps w → 〈w, xij〉 are w∗-continuous, and since ⊕fin

x M∗nx

is dense in theBanach space predual ⊕1

x M∗nx

of ⊕x Mnx , it is easy to see that J is w∗-continuoustoo. Thus by A.2.5, W is completely isometrically and w∗-homeomorphicallyisomorphic to a w∗-closed subspace of ⊕x Mnx . If the latter is regarded as avon Neumann subalgebra of B(H) say, then W is completely isometrically andw∗-homeomorphically isomorphic to a w∗-closed subspace of B(H).

1.4.8 (W ∗-continuous extensions) If X and Y are two operator spaces and ifu : X → Y ∗ is completely bounded, then its (unique) w∗-continuous extensionu : X∗∗ → Y ∗ provided by A.2.2 is completely bounded, with ‖u‖cb = ‖u‖cb.Indeed recall from A.2.2 that u = i∗Y u∗∗; and this extension clearly satisfiesthe asserted norm equality (using the first paragraph in 1.4.3). Note that sinceu is w∗-continuous, we have u(X∗∗) ⊂ u(X)

w∗. The above also shows that

CB(X, Y ∗) = w∗CB(X∗∗, Y ∗) (1.28)

isometrically via the mapping u → u. In fact (1.28) can easily be made a completeisometry with the help of the later item 1.6.2.

By 1.4.5, the last paragraph applies in particular to B(K, H) valued maps.

1.4.9 (The second dual) Let X be an operator space, and fix n ∈ N. Wewish to compare the spaces Mn(X∗∗) (equipped with its ‘operator space dual’matrix norms as in 1.2.20), and Mn(X)∗∗. First note that they can be canonicallyidentified as topological vector spaces, as may Mn(X∗) and Mn(X)∗ (this is justthe simple fact that if E = F ⊕· · ·⊕F is a finite direct sum of copies of a Banachspace F , which has been assigned a norm compatible with the norm on F , thenE∗ is canonically algebraically and topologically isomorphic to F ∗ ⊕ · · · ⊕ F ∗,the latter with any norm compatible with the norm on F ∗). We will prove in1.4.11 below that this identification is an isometry. As a first easy step, let uscheck that the identity mapping from Mn(X)∗∗ to Mn(X∗∗) implementing thisidentification, is a contraction. For this purpose, let η = [ηij ] ∈ Mn ⊗ X∗∗, andassume that its norm in Mn(X)∗∗ is less than or equal to 1. By Goldstine’s lemmaA.2.1, there is a net (xs)s in Ball(Mn(X)) such that xs → η in the w∗-topologyof Mn(X)∗∗. Let ϕ = [ϕpq] be an element of Ball(Mm(X∗)) for some m ≥ 1. Thefact that xs → η in the w∗-topology of Mn(X)∗∗, is equivalent to the fact thatxs

ij → ηij in the w∗-topology of X∗∗ for all 1 ≤ i, j ≤ n. Hence we deduce that

[〈ηij , ϕpq〉] = lims

[〈ϕpq , xsij〉].

By (1.6) or (1.26), the norm of the latter matrix is dominated by 1. Thus‖[〈ηij , ϕpq〉]‖ ≤ 1. By (1.6) again, we deduce that ‖[ηij ]‖Mn(X∗∗) ≤ 1, whichproves the result.

Operator spaces 25

1.4.10 (The second dual of a C∗-algebra) If A is a C∗-algebra, then the seconddual A∗∗ has two canonical operator space structures. The first is its ‘operatorspace dual’ matrix norms (see 1.2.20); when we write Mn(A∗∗) in the lines below,we will be using these norms. The second are those from 1.2.3, arising from thefact that A∗∗ is a C∗-algebra (see A.5.6). We claim that these two operator spacestructures are the same. To see this we will need to use notation and facts fromA.5.6. In particular we let πu : A → B(Hu) denote the universal representationof A, and we write A†† for the W ∗-algebra πu(A∗∗) (see the proof of A.5.6). Theclaim will follow if we can prove for any fixed n ≥ 1 that

Mn(A)∗∗ ∼= Mn(A∗∗) ∼= Mn(A††) isometrically (1.29)

via the canonical maps. The first of these maps is the contraction from Mn(A)∗∗

to Mn(A∗∗) discussed in 1.4.9. The second map in (1.29) is IMn ⊗ πu, whichis a contraction since according to 1.4.8, the mapping πu is a complete con-traction. To establish (1.29), we need only prove that the resulting contrac-tion ρ : Mn(A)∗∗ −→ Mn(A††) is isometric. It is clearly one-to-one. We regardMn(A)∗∗ as a C∗-algebra by applying A.5.6 to Mn(A). It therefore suffices tocheck that ρ is a ∗-homomorphism. Regarding ρ as valued in B(H (n)

u ), we have〈 ρ(η) ζ, ξ〉 =

∑i,j〈πu(ηij)ζj , ξi〉, for ζ = [ζi], ξ = [ξi] ∈ H

(n)u , and η = [ηij ] as in

1.4.9. Since πu, and the maps η → ηij , are w∗-continuous, it follows that ρ isw∗-continuous too. By the w∗-continuity properties of the involution and prod-uct in a W ∗-algebra (see A.5.1), it suffices to prove that the restriction of ρ toMn(A) is a ∗-homomorphism. Since the latter equals IMn ⊗ πu, we are done.

The last result has many consequences. For example, we can use it to seethat S∞(H)∗ = S1(H) completely isometrically, complementing the observationin 1.4.5. Also we obtain:

Theorem 1.4.11 If X is an operator space then Mm,n(X)∗∗ ∼= Mm,n(X∗∗)completely isometrically for all m, n ∈ N (via an isomorphism extending theidentity map on Mm,n(X)).

Proof First suppose that m = n, and choose a C∗-algebra A with X ⊂ A com-pletely isometrically. Then X∗∗ ⊂ A∗∗ completely isometrically by 1.4.3, hencewe have both Mn(X)∗∗ ⊂ Mn(A)∗∗, and Mn(X∗∗) ⊂ Mn(A∗∗), isometrically.Under the identifications between Mn(A)∗∗ and Mn(A∗∗) and between Mn(X)∗∗

and Mn(X∗∗) discussed above, these two embeddings are easily seen to be thesame. Hence the isometry Mn(A∗∗) = Mn(A)∗∗ provided by 1.4.10, implies thatwe also have Mn(X∗∗) = Mn(X)∗∗ isometrically. The complete isometry followsby iterating the isometric case.

The case n = m may be derived from the above, viewing Mm,n(X) as asubspace of Mk(X) where k = maxm, n. We leave this as an exercise.

1.4.12 (Duality of Min and Max) For any Banach space E, we have

Min(E)∗ = Max(E∗) and Max(E)∗ = Min(E∗). (1.30)


To see this, note that by using (1.8), (1.11) and (1.12), and the basic propertiesof ⊗ seen in A.3.1, we have

Mn(Max(E)∗) ∼= CB(Max(E), Mn) = B(E, Mn) ∼= Mn⊗E∗ ∼= Mn(Min(E∗)).

That is, Max(E)∗ = Min(E∗). Therefore Max(E∗)∗ = Min(E∗∗). However weclaim that Min(E∗∗) = Min(E)∗∗. This claim may be seen using the fact that‘minimal operator spaces’ are completely isometric to subspaces of unital com-mutative C∗-algebras (i.e. of C(K)-spaces), the fact that the second dual of acomplete isometry is a complete isometry (see 1.4.4), and 1.4.10. Hence Max(E∗)and Min(E)∗ are two operator space structures on E∗ with the same operatorspace dual, and therefore they are completely isometric, by 1.4.1.

1.4.13 (The 1-direct sum) For a family Xλ : λ ∈ I of operator spaces, let⊕fin

λ Xλ be the set of ‘finitely supported’ elements of the algebraic direct sumof the Xλ. There is a canonical one-to-one map µ : ⊕fin

λ Xλ → (⊕∞λ X∗

λ)∗,and we may define the 1-direct sum ⊕1

λ Xλ by identifying it with the closure ofµ(⊕fin

λ Xλ) in the dual operator space (⊕∞λ X∗

λ)∗. Evidently, the norm on ⊕1λ Xλ

is exactly the usual ‘1-norm’∑

λ ‖xλ‖; and a tuple (xλ) is in ⊕1λ Xλ if and only

if xλ ∈ X for all λ, and∑

λ ‖xλ‖ < ∞. If Xλ = X for all λ ∈ I then we write1I(X) for ⊕1

λ X .It is easy to argue that the canonical inclusion and projection maps ελ and

πλ between ⊕1λ Xλ and its ‘λth summand’ are complete isometries and complete

quotient maps respectively. Also,

(⊕1λ Xλ)∗ ∼= ⊕∞

λ X∗λ as dual operator spaces. (1.31)

Similarly, the dual of an ∞-direct sum of finitely many operator spaces (or thedual of a c0-direct sum of infinitely many) is completely isometric to the 1-directsum of the duals. We leave these assertions to the reader to check. In particular,

1I(X)∗ = ∞I (X∗) and c0(X)∗ = 1(X∗).

The 1-direct sum has the following useful universal property. Namely, if Zis another operator space and uλ : Xλ → Z are completely contractive linearmaps, then there is a canonical complete contraction u : ⊕1

λ Xλ → Z suchthat u ελ = uλ. One may see this with a diagram chase: u∗

λ maps Z∗ toX∗

λ, so that by the universal property of the ∞-direct sum we obtain a mapv : Z∗ → ⊕λ X∗

λ. Then v∗ : (⊕λ X∗λ)∗ → Z∗∗. Composing v with the canonical

map ⊕1λ Xλ → (⊕λ X∗

λ)∗ gives a map u : ⊕1λ Xλ → Z∗∗. One easily checks that

u ελ = uλ, whence u actually maps into Z, and we obtain the desired result.This universal property may be rephrased as an isometric isomorphism

CB(⊕1i Xi, Z) ∼= ⊕∞

i CB(Xi, Z).

Replacing Z by Mn(Z) and using (1.7) shows that this relation also holds com-pletely isometrically.

Operator spaces 27

The canonical inclusion and projection maps ελ and πλ between ⊕∞λ X∗

λ andits ‘λth summand’ are w∗-continuous. This may be seen by dualizing the inclusionand projection maps between ⊕1

λ Xλ and its summands.Any operator space is a complete quotient of a 1-sum of spaces of the form

S1n = M∗

n. This may be seen, for example, by applying the second last fact in1.4.3 to the map J in the proof of 1.4.7.

Another frequently used property of these direct sums is as follows. Supposethat uλ : Xλ → Yλ is a complete contraction for each λ ∈ I. Then this familyof maps clearly induces a single contraction u = (uλ) : ⊕1

λ Xλ → ⊕1λ Yλ. It

also clearly induces a complete contraction (uλ) : ⊕∞λ Xλ → ⊕∞

λ Yλ; which iscompletely isometric if every uλ is. Applying the first fact in the last sentence tothe family of maps u∗

λ, gives a complete contraction from ⊕∞λ Y ∗

λ to ⊕∞λ X∗

λ, and itis easy to see from the norm density of the ‘finitely supported’ tuples in a 1-directsum, that this map is the dual of the map u above, and hence is w∗-continuous.From this and 1.4.3 we deduce that u is a complete contraction. We have alsoshown that families (vλ) of w∗-continuous complete contractions between dualspaces canonically induce a single w∗-continuous complete contraction betweenthe ∞-direct sums of the spaces.

1.5 OPERATOR SPACE TENSOR PRODUCTS

As we said in the introduction, the reader should feel free to skim through theseresults, returning later when necessary for a definition or fact. In this sectionand the next we tend to leave more details than in previous sections as exercisesfor the interested reader.

1.5.1 (Minimal tensor product) Let X and Y be operator spaces, and let X⊗Ydenote their algebraic tensor product. Any finite rank bounded map between op-erator spaces is ‘composed’ of scalar functionals, and hence is automatically com-pletely bounded by 1.2.6. Thus the correspondences between tensor products andfinite rank mappings discussed in A.3.1 yield embeddings X ⊗ Y → CB(Y ∗, X)and X ⊗ Y ∗ → CB(Y, X). The minimal tensor product X ⊗min Y may then bedefined to be (the completion of) X ⊗Y in the matrix norms inherited from theoperator space structure on CB(Y ∗, X) described in 1.2.19. That is,

X ⊗min Y → CB(Y ∗, X) completely isometrically. (1.32)

That is, if u =∑n

k=1 xk ⊗ yk ∈ X ⊗ Y , then the norm of u in X ⊗min Y equals

sup∥∥∥[∑

k

xk ψij(yk)]∥∥∥

Mn(X), (1.33)

the supremum taken over all finite matrices [ψij ] of norm 1, where ψij ∈ Y ∗. Asimilar formula holds for u ∈ Mn(X ⊗ Y ). From this similar form of (1.33) and(1.26), we see that the matrix norms on X ⊗min Y are also given by the formula

‖[wrs]‖n = sup‖[(ϕkl ⊗ ψij)(wrs)]‖

(1.34)

28 Operator space tensor products

for [wrs] ∈ Mn(X ⊗ Y ), where the supremum is taken over all finite matrices[ϕkl] and [ψij ] of norm 1, where ϕkl ∈ X∗ and ψij ∈ Y ∗, and where ϕkl ⊗ ψij

denotes the obvious functional on X ⊗ Y formed from ϕkl and ψij .We see from (1.34) that ⊗min is commutative, that is

X ⊗min Y = Y ⊗min X

as operator spaces. It is also easy to see from (1.34) that ⊗min is functorial.That is, if Xi and Yi are operator spaces for i = 1, 2, and if ui : Xi → Yi arecompletely bounded, then the map x ⊗ y → u1(x) ⊗ u2(y) on X1 ⊗ X2 has aunique continuous extension to a map u1 ⊗ u2 : X1 ⊗min X2 → Y1 ⊗min Y2, with‖u1 ⊗ u2‖cb ≤ ‖u1‖cb‖u2‖cb. As an exercise, the reader could check that this isactually an equality, but we shall not need this. If, further, the ui are completelyisometric, then so is u1⊗u2. This latter fact is called the injectivity of the tensorproduct. To prove it, since u1 ⊗ u2 = (u1 ⊗ I) (I ⊗ u2), we may by symmetryreduce the argument to the case that Y2 = X2, u2 = IX2 , X1 ⊂ Y1 and that u1 isthis inclusion map. Then the result we want follows from (1.32) and the obviousfact that CB(X∗

2 , X1) ⊂ CB(X∗2 , Y1) completely isometrically.

For any operator spaces X, Y , we have

X ⊗min Y ∗ → CB(Y, X) completely isometrically. (1.35)

To prove this, note by the injectivity discussed above that we may assume thatX = B(H). However, CB(Y, B(H)) = w∗CB(Y ∗∗, B(H)) ⊂ CB(Y ∗∗, B(H))isometrically by (1.28). Since the norm on B(H) ⊗min Y ∗ is induced by theembedding of B(H) ⊗ Y ∗ in the latter space, B(H) ⊗min Y ∗ ⊂ CB(Y, B(H))isometrically. The complete isometry follows by changing H into H (n), and using(1.2), (1.7), and the fact that Mn(X⊗minY ) = Mn(X)⊗minY for operator spacesX, Y (which in turn follows from (1.32) and (1.7)).

1.5.2 (Further properties of ⊗min) Suppose that H1, H2 are Hilbert spaces, andconsider the canonical map π : B(H1)⊗B(H2) → B(H1 ⊗2 H2). This is the maptaking a rank one tensor S⊗T in B(H1)⊗B(H2) to the map on H1⊗2H2 denotedalso by S ⊗ T in 1.1.4. We claim that π actually is a complete isometry whenB(H1)⊗B(H2) is given its norm as a subspace of B(H1)⊗minB(H2). To see this,we choose a cardinal I such that H2 = 2

I , so that we both have MI∼= B(H2) and

H1⊗2H2∼= 2

I(H1). By (1.35) and 1.4.5, B(H1)⊗minB(H2) → CB(S1(H1), MI).However, by 1.2.29, and (1.19), we have

CB(S1(H1), MI) ∼= MI(B(H1)) ∼= B(H1 ⊗2 H2).

This proves the claim.Thus if X and Y are subspaces of B(H1) and B(H2) respectively, then by the

injectivity of this tensor product, we have that X ⊗min Y is completely isometri-cally isomorphic to the closure in B(H1⊗2 H2) of the span of the operators x⊗yon H1 ⊗2 H2, for x ∈ X, y ∈ Y . We remark, in passing, that the above says that

Operator spaces 29

the minimal tensor product coincides with the tensor product of the same nameused in C∗-algebra theory, or with the so-called spatial tensor product. Indeednote that if A and B are C∗-subalgebras of B(H1) and B(H2) respectively, thenA ⊗min B may be identified with the closure of a ∗-subalgebra of B(H1 ⊗2 H2).Thus A ⊗min B is a C∗-algebra. If A and B are also commutative, then so isA ⊗min B, since it is the closure of a commutative ∗-subalgebra.

From the last paragraph and 1.2.2 it is clear that for any operator space X ,

Mn ⊗min X ∼= Mn(X) (1.36)

completely isometrically. By similar reasoning, using also (1.19), we have

KI,J ⊗min X ∼= KI,J(X). (1.37)

Indeed, in the case I = J , both sides of (1.37) correspond to the closure ofM

finI (X) in B(H(I)), if X ⊂ B(H).Similarly, it follows from the second last paragraph, and from the fact that

B((H1 ⊗2 H2)⊗2 H3) ∼= B(H1 ⊗2 (H2 ⊗2 H3)), that ⊗min is associative. That is,

(X1 ⊗min X2) ⊗min X3 = X1 ⊗min (X2 ⊗min X3). (1.38)

Accordingly, this space will be merely denoted by X1⊗minX2⊗minX3. Similarly,one may consider the N -fold minimal tensor product X1⊗min · · ·⊗min XN of anyN -tuple of operator spaces.

Proposition 1.5.3 Let E, F be Banach spaces and let X be an operator space.(1) Min(E) ⊗min X = E⊗X as Banach spaces.(2) Min(E) ⊗min Min(F ) = Min(E⊗F ) as operator spaces.(3) For any compact space Ω we have (with notation as in 1.2.18),

C(Ω) ⊗min X = C(Ω; X) completely isometrically. (1.39)

Proof We have isometric embeddings Min(E)⊗min X ⊂ CB(X∗, Min(E)) andE⊗X ⊂ B(X∗, E) by (1.32) and (A.1). However CB(X∗, Min(E)) = B(X∗, E)by (1.10), which proves (1). The isometry in (2) follows from (1). Thus thecomplete isometry in (2) will follow if Min(E)⊗minMin(F ) is a minimal operatorspace. But this is clear if we use the ‘injectivity’ of ⊗min, the fact that ‘minimaloperator spaces’ are the subspaces of C(K)-spaces, and the observation in 1.5.2that the minimal tensor product of commutative C∗-algebras is a commutativeC∗-algebra, and hence is a C(K)-space.

We now prove (3). Since C(Ω) is a minimal operator space, the relationC(Ω)⊗min X = C(Ω; X) holds isometrically by (1) and (A.2). By definition (see1.2.18), Mn(C(Ω; X)) = C(Ω; Mn(X)). Replacing X by Mn(X) in the previousrelation, we deduce that C(Ω)⊗min Mn(X) = Mn(C(Ω; X)) isometrically for alln ∈ N. However by (1.36), and also by the associativity and commutativity of⊗min, we have

C(Ω) ⊗min Mn(X) ∼= C(Ω) ⊗min Mn ⊗min X ∼= Mn

(C(Ω) ⊗min X

).

This shows that Mn(C(Ω) ⊗min X) = Mn(C(Ω; X)) isometrically.


1.5.4 (Haagerup tensor product) Before we define this tensor product, weintroduce an intimately related class of bilinear maps. Suppose that X , Y , andW are operator spaces, and that u : X ×Y → W is a bilinear map. For n, p ∈ N,define a bilinear map Mn,p(X) × Mp,n(Y ) → Mn(W ) by

(x, y) −→[ p∑

k=1

u(xik, ykj)]

i,j,

where x = [xij ] ∈ Mn,p(X) and y = [yij ] ∈ Mp,n(Y ). If p = n we write this mapas un. If the norms of these bilinear maps (as defined in A.3.3) are uniformlybounded over p, n ∈ N, then we say that u is completely bounded, and write thesupremum of these norms as ‖u‖cb. It is easy to see (by adding rows and columnsof zeroes to make p = n) that ‖u‖cb = supn ‖un‖. We say that u is completelycontractive if ‖u‖cb ≤ 1. Completely bounded multilinear maps of three vari-ables have a similar definition (involving the expression [

∑k,l u(xik, ykl, zlj)]),

and similarly for four or more variables. We remark that if v : X → B(H) andw : Y → B(H) are completely bounded linear maps, then it is easy to see thatthe bilinear map (x, y) → v(x)w(y) is completely bounded in the sense above,and has completely bounded norm dominated by ‖v‖cb‖w‖cb. Indeed this boilsdown to the fact that B(H (n)) is a Banach algebra.

Let X, Y be operator spaces. For n ∈ N and z ∈ Mn(X ⊗ Y ) we define

‖z‖h = inf‖x‖‖y‖, (1.40)

where the infimum is taken over all p ∈ N, and all ways to write z = xy, wherex ∈ Mn,p(X), y ∈ Mp,n(Y ). Here x y denotes the formal matrix product ofx and y using the ⊗ sign as multiplication: namely x y = [

∑pk=1 xik ⊗ ykj ].

We leave it as an exercise (see 9.2.1 in [149] for more details if needed) that theexpressions in (1.40) for all n ∈ N, define an operator seminorm structure onX ⊗ Y in the sense of 1.2.16.

Now let u : X ×Y → W be a bilinear map which is completely contractive inthe sense above. Let u : X ⊗ Y → W be the canonically associated linear map.For z ∈ Mn(X ⊗ Y ), we have by the definitions above that

‖un(z)‖ ≤ ‖z‖h, (1.41)

where the latter quantity is as defined in (1.40). If ϕ and ψ are contractivefunctionals on X and Y respectively, then using 1.2.6 and the fact at the endof the second last paragraph, we see that the bilinear map (x, y) → ϕ(x)ψ(y) iscompletely contractive. Thus from (1.41) and the definition of the Banach spaceinjective tensor norm of z (see A.3.1), we deduce that the latter norm of anelement z ∈ X ⊗ Y is dominated by ‖z‖h. Hence indeed ‖ · ‖h is a norm. By thefact at the end of the last paragraph, together with Ruan’s theorem, we see thatthe completion X ⊗h Y of X ⊗Y with respect to ‖ · ‖h is an operator space. Thisoperator space is called the Haagerup tensor product. Note that the canonicalbilinear map ⊗ : X × Y → X ⊗h Y is completely contractive in the sense above.

Operator spaces 31

Using (1.41) we see that if u : X × Y → W is a bilinear map with associatedlinear map u : X⊗Y → W , then u is completely bounded if and only if u extendsto a completely bounded linear map on X ⊗h Y . Moreover we have

‖u‖cb =∥∥u : X ⊗h Y −→ W

∥∥cb

in that case. The above property means that the Haagerup tensor product lin-earizes completely bounded bilinear maps. A moments thought shows that thisis a universal property. That is, suppose that (W, µ) is a pair consisting of anoperator space W , and a completely contractive bilinear map µ : X × Y → W ,such that the span of the range of µ is dense in W , and which possesses thefollowing property:

Given any operator space Z and given any completely bounded bilinear mapu : X×Y → Z, then there exists a linear completely bounded u : W → Z suchthat u(µ(x, y)) = u(x, y) for all x ∈ X, y ∈ Y , and such that ‖u‖cb ≤ ‖u‖cb.

Then X ⊗h Y ∼= W via a complete isometry v satisfying v ⊗ = µ.

1.5.5 (Further properties of the Haagerup tensor product) By the definitionof the Haagerup tensor product, or by its universal property, it is easy to seethat this tensor product is functorial. That is, if ui : Xi → Yi are completelybounded maps between operator spaces, then u1 ⊗ u2 : X1 ⊗h X2 → Y1 ⊗h Y2 iscompletely bounded, and ‖u1 ⊗ u2‖cb ≤ ‖u1‖cb ‖u2‖cb.

It is also easy to show that the Haagerup tensor product is associative (e.g.see [318]). That is, (X1⊗hX2)⊗hX3

∼= X1⊗h(X2⊗hX3) completely isometrically.Accordingly, this space will be merely denoted by X1 ⊗h X2⊗h X3. The inducednorms on Mn(X1 ⊗ X2 ⊗ X3) may be described by the ‘3-variable’ version of(1.40). From this one may see that X1 ⊗h X2 ⊗h X3 has the universal propertyof ‘linearizing’ completely bounded trilinear maps (see discussion at the end of1.5.4). Similar assertions clearly hold for the N -fold Haagerup tensor productX1 ⊗h · · · ⊗h XN of any N -tuple of operator spaces. Also, the Haagerup tensorproduct is injective. That is, if u1 and u2 in the last paragraph are completelyisometric, then so is u1⊗u2. See the Notes for directions to a selection of proofs ofthis fact. It is much easier to see that the Haagerup tensor product is projective,that is, if u1 and u2 in the last paragraph are complete quotient maps, then so isu1⊗u2. To see this, note that by the functoriality, the map u1⊗u2 is a completecontraction. Let z ∈ Mn(Y1 ⊗ Y2), with ‖z‖h < 1. By definition, we may writez = y1 y2, where y1 ∈ Mn,p(Y1), y2 ∈ Mp,n(Y2) both have norm < 1. Theny1 = (u1)n,p(x1) and y2 = (u2)p,n(x2) for x1 ∈ Mn,p(X1), x2 ∈ Mp,n(X2), bothof norm < 1. Let w = x1 x2 ∈ Mn(X1 ⊗h X2), this matrix has norm < 1, and(u1 ⊗ u2)n(w) = z. By an obvious density argument, this shows that u1 ⊗ u2

above is a complete quotient map.The Haagerup tensor product is not commutative. That is, in general X⊗h Y

and Y ⊗h X are not isometric. We shall see some examples of this later.One final property of the Haagerup tensor product: there are convenient norm

expressions for ‖·‖h. Suppose that A and B are C∗-algebras, or spaces of the form


B(K, H). If X and Y are subspaces of A and B respectively, and if z ∈ X ⊗ Y ,then by the definition in 1.5.4 and (1.4) we may write

‖z‖h = inf∥∥∥ p∑

k=1

aka∗k

∥∥∥ 12∥∥∥ p∑

k=1

b∗kbk

∥∥∥ 12

(1.42)

where the infimum is taken over all ways to write z =∑p

k=1 ak ⊗ bk in X ⊗ Y(or equivalently, by the injectivity of this tensor product, in A ⊗ B).

Proposition 1.5.6

(1) If z ∈ X⊗h Y with ‖z‖h < 1 then we may write z as a norm convergent sum∑∞k=1 ak ⊗bk in X⊗h Y , with ‖∑∞

k=1 aka∗k‖ < 1 and ‖∑∞

k=1 b∗kbk‖ < 1, andwhere the last two sums converge in norm. That is, [a1 a2 · · · ] ∈ R(X) and[b1 b2 · · · ]t ∈ C(Y ).

(2) If x = [a1 a2 · · · ] ∈ R(X) and y = [b1 b2 · · · ]t ∈ Cw(Y ), that is if∑∞k=1 aka∗

k converges in norm and if the partial sums of∑∞

k=1 b∗kbk are uni-formly bounded in norm, then

∑∞k=1 ak ⊗ bk converges in norm in X ⊗h Y .

Similarly if x ∈ Rw(X) and y ∈ C(Y ).

Proof (1) If z is as stated, choose w1 ∈ X ⊗ Y with ‖z − w1‖h < ε2 and

‖w1‖h < 1. By (1.42) we may write w1 =∑n1

k=1 xk ⊗ yk with∑

k xkx∗k ≤ 1

and∑

k y∗kyk ≤ 1. Repeating this argument, we may choose w2 ∈ X ⊗ Y with

‖z−w1 −w2‖h < ε22 , and ‖w2‖h < ε

2 . By (1.42) we write w2 =∑n2

k=n1+1 xk ⊗ yk

with∑

k xkx∗k ≤ ε

2 and∑

k y∗kyk ≤ ε

2 . Continuing so, we obtain for every m ∈ N

a finite rank tensor wm =∑nm

k=nm−1+1 xk ⊗ yk with ‖z − w1 − . . . − wm‖ < ε2m ,∑nm

k=nm−1+1 xkx∗k ≤ ε

2m−1 , and∑nm

k=nm−1+1 y∗kyk ≤ ε

2m−1 . Now it is clear thatthe partial sums of

∑∞k=1 xkx∗

k and∑∞

k=1 y∗kyk converge in norm to elements

with norm close to 1, and that z =∑∞

k=1 xk ⊗ yk as desired.(2) To see that the partial sums of

∑∞k=1 ak ⊗ bk are Cauchy, note that from

(1.42) we have ‖∑m

k=n ak ⊗ bk‖h ≤ ‖∑m

k=n aka∗k‖

12 ‖

∑mk=n b∗kbk‖

12 . Now use the

fact that the partial sums of∑∞

k=1 aka∗k are Cauchy.

The following important result is due to Christensen and Sinclair, and Paulsenand Smith. We will refer to it as the ‘CSPS theorem’. See the Notes section forreferences, and directions to a sample of the available proofs. Note that thesecond part of this result follows from the first part and 1.2.8.

Theorem 1.5.7 (CSPS theorem) Suppose that X and Y are operator spaces,and that u : X × Y → B(K, H) is a bilinear map.

(1) u is completely contractive (as a bilinear map) if and only if there is a Hilbertspace L, and there are completely contractive linear maps v : X → B(L, H)and w : Y → B(K, L), with u(x, y) = v(x)w(y) for all x ∈ X and y ∈ Y .

(2) If X and Y are subspaces of unital C∗-algebras A and B respectively, and ifthe conditions in (1) hold, then there exist Hilbert spaces K1 and K2, unital

Operator spaces 33

∗-representations π : A → B(K1) and ρ : B → B(K2), and contractionsT ∈ B(K, K2), S ∈ B(K2, K1) and R ∈ B(K1, H), such that

u(x, y) = Rπ(x)Sρ(y)T, x ∈ X, y ∈ Y.

1.5.8 (Remarks on the CSPS theorem) An analoguous result to (1) holdsfor multilinear completely bounded maps of three or more variables, and can beproved using 1.5.7 and the discussion on associativity in 1.5.5. Thus if X1, . . . , XN

are operator spaces and if vj : Xj → B(Hj , Hj−1) are completely contractivelinear maps then the N -linear mapping taking (x1, . . . , xN ) ∈ X1 × · · · × XN

to v1(x1) · · · vN (xN ) ∈ B(HN , H0) is completely contractive. Conversely, anycompletely contractive N -linear map has this form.

Likewise, (2) has analoguous formulations for multilinear maps. For example,let u : X × Y × Z → B(K, H) be any completely contractive trilinear map, andassume that X and Z are subspaces of unital C∗-algebras A and B respectively.Then u has a factorization of the form u(x, y, z) = Rπ(x)Ψ(y)ρ(z)T , whereπ : A → B(K1) and ρ : B → B(K2) are unital ∗-representations, Ψ is a completecontraction from Y to B(K2, K1), and R, T are contractions.

If u : X × Y → B(K, H) is a completely contractive bilinear map, then theHilbert space L in (1) may be replaced by the possibly smaller space which isdensely spanned by v(X)∗(H) and w(Y )(K). In particular L may be chosen tobe finite-dimensional if X, Y, K, H are all finite-dimensional.

As a final remark, we stress the case of completely bounded bilinear forms.If we have a completely contractive u : X × Y → C, the CSPS theorem ensuresthat u may be written as u(x, y) = 〈w(y), v(x)〉H , where H is a Hilbert spaceand v : X → Hr and w : Y → Hc are completely contractive maps.

1.5.9 (Self-duality of ⊗h) Let X and Y be operator spaces. Then

X∗ ⊗h Y ∗ → (X ⊗h Y )∗ completely isometrically (1.43)

via the canonical map. Indeed, first assume that X and Y are finite-dimensional.We let J be the canonical map from X∗ ⊗h Y ∗ into (X ⊗h Y )∗, which is asurjection in our finite-dimensional case. By (1.8), it suffices to show that Jn

is an isometric isomorphism from Mn(X∗ ⊗h Y ∗) to CB(X ⊗h Y, Mn), for anyn ≥ 1. We consider z ∈ Mn(X∗ ⊗h Y ∗) and we let u = Jn(z). By 1.5.7 (1)and the third paragraph in 1.5.8, there exist an integer p ≥ 1 and two linearmaps v : X → Mn,p and w : Y → Mp,n such that ‖v‖cb‖w‖cb ≤ ‖u‖cb andu(x⊗y) = v(x)w(y) for any x ∈ X and y ∈ Y . We let ϕ = [ϕij ] ∈ Mn,p(X∗) andψ = [ψij ] ∈ Mp,n(Y ∗) be the two matrices corresponding to v and w respectively.Then

u(x ⊗ y) = v(x)w(y) = [ϕij(x)][ψij(y)] =[∑

k

ϕik(x)ψkj(y)], x ∈ X, y ∈ Y.

Thus z = ϕψ, and hence ‖z‖h ≤ ‖ϕ‖‖ψ‖ = ‖v‖cb‖w‖cb ≤ ‖u‖cb. The converseinequality ‖u‖cb ≤ ‖z‖h may be obtained by reversing the arguments.


In the general case, fix [uij ] ∈ Mn(X∗ ⊗ Y ∗). Write each uij ∈ X∗ ⊗ Y ∗ inthe form

∑mk=1 ϕk ⊗ ψk, for functionals ϕk ∈ X∗, ψk ∈ Y ∗. Let W (resp. Z) be

the span of all these (finite number of) functionals in X∗ (resp. Y ∗), over all iand j too. We may view [uij ] in Mn(W ⊗h Z), and by the injectivity of ⊗h (see1.5.5), its norm in the latter space equals its norm in Mn(X∗ ⊗h Y ∗). By thelast paragraph, and by 1.4.4, its norm also equals the norm of the associatedelement in Mn(((X/W⊥)⊗h (Y/Z⊥))∗). However, by the projectivity of ⊗h (see1.5.5), and 1.4.3, we may view ((X/W⊥) ⊗h (Y/Z⊥))∗ ⊂ (X ⊗h Y )∗ completelyisometrically. The composition of these identifications is easily seen to take [uij ]to the desired matrix in Mn((X ⊗h Y )∗), and so we are done.

1.5.10 (Some relations to Banach space tensor products) The Haagerup tensorproduct of minimal and maximal operator spaces corresponds to well-knownBanach space tensor products (see A.3.5 for a review). Let E, F be Banachspaces. Then for any m ≥ 1, we have isometric identities Rm(Min(E)) = 2

m⊗Eand Cm(Min(F )) = 2

m⊗F , as may be seen from (1.11). Then it is immediatefrom (1.40) and (A.5) that

Min(E) ⊗h Min(F ) = E ⊗γ2 F isometrically. (1.44)

If G is an arbitrary Banach space and H is a Hilbert space then

CB(Min(G), Hr) = Π2(G, H) isometrically, (1.45)

where the latter denotes the Banach space of 2-summing maps from G into H(see A.3.4). To prove this consider u : G → H and assume that u is 2-summing.According to A.3.4, the mapping I2n

⊗ u : 2n⊗G → 2

n(H) has norm less than orequal to π2(u) for any n ≥ 1. Hence∥∥I2n

⊗ I2n⊗ u : 2

n⊗2n⊗G −→ 2

n⊗2n(H)

∥∥ ≤ π2(u).

Using (1.2) and (A.1), we have isometric identities

Mn(Hr) ∼= Mn(B(H, C)) ∼= B(2n(H), 2

n) ∼= 2n⊗2

n(H).

On the other hand, 2n⊗2

n∼= Mn by (A.1) again, so 2

n⊗2n⊗G = Mn(Min(G))

by (1.11). Thus the second last centered equation exactly means that the mapun from Mn(Min(G)) to Mn(Hr) has norm ≤ π2(u). Thus ‖u‖cb ≤ π2(u). Theconverse inequality is easily obtained by reversing the arguments.

Suppose that x in Rm(Max(E)), viewed as an element of

Rm(Max(E)∗∗) ∼= CB(Max(E)∗, Rm) = CB(Min(E∗), Rm),

the last isomorphisms using 1.2.29 and (1.30). Applying (1.45) with G = E∗ wesee that if w : E∗ → 2

m is the linear mapping corresponding to x, then ‖x‖ isequal to π2(w). Therefore the argument before (1.44), and (A.6), show that

Max(E) ⊗h Min(F ) = E ⊗g2 F isometrically. (1.46)

Likewise by (A.7) we have

Max(E) ⊗h Max(F ) = E ⊗γ∗2

F isometrically. (1.47)

Operator spaces 35

1.5.11 (Operator space projective tensor product) As with the Haagerup tensorproduct, it is convenient to first define an intimately related class of bilinearmaps. Suppose that X , Y , and W are operator spaces and that u : X × Y → Wis a bilinear map. We say that u is jointly completely bounded if there exists aconstant K ≥ 0 such that

‖[u(xij , ykl)](i,k),(j,l)‖ ≤ K‖[xij ]‖‖[ykl]‖

for all m, n and [xij ] ∈ Mn(X), and [ykl] ∈ Mm(Y ). The least such K is writtenas ‖u‖jcb. We say that u is jointly completely contractive if ‖u‖jcb ≤ 1. Jointlycompletely bounded multilinear maps of three or more variables are definedsimilarly. We also observe that any completely contractive (in the sense of 1.5.4)bilinear map u is jointly completely contractive. This is immediate from thesimple relation [u(xij , ykl)] = unm([xij ] ⊗ Im, In ⊗ [ykl]).

Let X, Y be operator spaces, and let n ∈ N. For z ∈ Mn(X ⊗ Y ) define

‖z‖ = inf‖α‖‖x‖‖y‖‖β‖, (1.48)

the infimum taken over p, q ∈ N, and all ways to write z = α(x ⊗ y)β, whereα ∈ Mn,pq, x ∈ Mp(X), y ∈ Mq(Y ), and β ∈ Mpq,n. Here we wrote x ⊗ y for the‘tensor product of matrices’, namely x⊗y = [xij ⊗ykl](i,k),(j,l). We omit the easyproof (see [149] if necessary) that this defines an operator seminorm structure onX ⊗ Y . For z ∈ Mn(X ⊗ Y ), and for any jointly completely contractive bilinearmap u : X × Y → W , it is easy to see from the definitions that

‖un(z)‖ ≤ ‖z‖ , (1.49)

where u : X ⊗ Y → W is the associated linear map. From the observation atthe end of the last paragraph, it follows that this is also true if u is completelycontractive. Taking u = ⊗ : X × Y → X ⊗h Y , we deduce that ‖ · ‖h ≤ ‖ · ‖.Thus the quantities in (1.48) are norms. By Ruan’s theorem the completion ofX ⊗ Y with respect to these matrix norms is an operator space, which we callthe operator space projective tensor product, and write as X

⊗ Y .

By (1.48) and (1.49) we see that if u : X × Y → W is a bilinear map withassociated linear map u : X ⊗ Y → W , then u is jointly completely bounded ifand only if u extends to a completely bounded linear map on X

⊗ Y . Moreover,

‖u‖jcb =∥∥u : X

⊗ Y −→ W

∥∥cb

in that case. The above property means that the operator space projective tensorproduct linearizes jointly completely bounded bilinear maps. As for the Haageruptensor product this is a universal property, and characterizes X

⊗ Y uniquely

up to complete isometry. From this it is easy to argue that

CB(X⊗ Y, Z) ∼= CB(X, CB(Y, Z)) ∼= CB(Y, CB(X, Z)) (1.50)


isometrically, and indeed (by replacing Z by Mn(Z)) completely isometrically.In particular,

(X⊗ Y )∗ ∼= CB(X, Y ∗) ∼= CB(Y, X∗) completely isometrically. (1.51)

We now list a sequence of properties of the operator space projective tensorproduct. Again it is very easy to see, copying the idea of the analoguous proofsin 1.5.5, that

⊗ is functorial, and projective. It is also an easy exercise to see that

⊗ is associative, and (1.51) shows that

⊗ is commutative. We use these words in

the sense that we have used them for the other two tensor products. From theuniversal properties of

⊗ and ⊕1

i (see above and 1.4.13), it is easy to check that

Y⊗ (⊕1

i Xi) ∼= ⊕1i (Y

⊗ Xi), (1.52)

for operator spaces Y and Xi.Proposition 1.5.12 Let E, F be Banach spaces and let Y be an operator space.

(1) Max(E)⊗ Y = E⊗Y as Banach spaces.

(2) Max(E⊗F ) = Max(E)⊗ Max(F ) as operator spaces.

Proof The first item follows by computing the duals of these two tensor prod-ucts, using (1.51), (1.12), and (A.4). Then (2) follows from (1) if we can showthat Max(E)

⊗ Max(F ) is a maximal operator space. To do this, observe that

B(Max(E)⊗ Max(F ), W ) = B(E⊗F, W ) = B(E, B(F, W )),

for any operator space W , by (1) and (A.3). The latter space equals

CB(Max(E), CB(Max(F ), W )) = CB(Max(E)⊗ Max(F ), W )

by (1.12) and (1.50). Thus Max(E)⊗ Max(F ) is ‘maximal’.

Proposition 1.5.13 (Comparison of tensor norms) If X and Y are operatorspaces then the various tensor norms on X ⊗ Y are ordered as follows:

‖ · ‖∨ ≤ ‖ · ‖min ≤ ‖ · ‖h ≤ ‖ · ‖ ≤ ‖ · ‖∧.

Indeed the ‘identity’ is a complete contraction X⊗ Y → X ⊗h Y → X ⊗min Y .

Proof The first inequality follows from (1.34) and the definition of ⊗ (seeA.3.1). The fact that ‖ · ‖ ≤ ‖ · ‖∧ follows from the universal property of ⊗(see A.3.3), since the bilinear map ⊗ : X × Y → X

⊗ Y is jointly completely

contractive, and hence contractive. We saw in 1.5.11 the complete contractionX

⊗ Y → X ⊗h Y . For the remaining relation we may assume by the injectivity

of these tensor norms that X and Y are unital C∗-algebras, and then X ⊗min Y

Operator spaces 37

is a C∗-algebra as observed in 1.5.2. We may write x ⊗ y = (x ⊗ 1)(1 ⊗ y),which may be viewed as the product π(x)ρ(y) of two (completely contractive)∗-homomorphisms. Thus the bilinear map ⊗ : X × Y → X ⊗min Y is completelycontractive, which by (1.41) proves the desired relation.

Proposition 1.5.14 If X, Y are operator spaces, if H, K are Hilbert spaces, andif m, n ∈ N, then we have the following complete isometries:

(1) Hr ⊗h X = Hr⊗ X, and X ⊗h Hc = X

⊗ Hc.

(2) Hc ⊗h X = Hc ⊗min X, and X ⊗h Hr = X ⊗min Hr.(3) Cn(X) ∼= Cn ⊗h X = Cn ⊗min X, and Rn(X) ∼= X ⊗h Rn = X ⊗min Rn.

(4) (Hr⊗ X

⊗ Kc)∗ = (Hr ⊗h X ⊗h Kc)∗ ∼= CB(X, B(K, H)).

(5) S∞(K, H) ∼= Hc ⊗min Kr and S∞(K, H) ⊗min X ∼= Hc ⊗h X ⊗h Kr.(6) Mm,n(X) ∼= Cm ⊗h X ⊗h Rn.(7) Mm,n(X ⊗h Y ) ∼= Cm(X) ⊗h Rn(Y ).

(8) Hc⊗ Kc = Hc ⊗h Kc = Hc ⊗min Kc = (H ⊗2 K)c, and similarly for row

Hilbert spaces.

(9) S1(K, H) ∼= Kr⊗ Hc.

(10) CB(S1(2I ,

2J), X) ∼= MI,J(X), if I, J are cardinals.

Proof To prove that X⊗hHc = X⊗ Hc completely isometrically, it suffices by

Proposition 1.5.13 to show that I : X⊗hHc → X⊗ Hc is completely contractive.

This will follow if we can show that any jointly completely contractive mapu : X × Hc → B(L) is completely contractive (in the sense of 1.5.4). Assumethat H = 2

J for some cardinal J . Let v : X → RwJ (B(L)) = B(L(J), L) be the

linear map defined by v(x) = (u(x, ei))i∈J for any x ∈ X . Then v correspondsto u via the identifications

CB(X⊗ CJ , B(L)) = CB(X, CB(CJ , B(L))) = CB(X, Rw

J (B(L)))

provided by (1.50) and Proposition 1.2.28. Thus ‖v‖cb = ‖u‖jcb. Then we definea map w : CJ → CJ (B(L)) ⊂ B(L, L(J)) by w(ζ) = (ζjIL) for ζ = (ζj) ∈ 2

J . Itis clear that ‖w‖cb = 1 and moreover we have a factorization u(x, ζ) = v(x)w(ζ)for any x ∈ X, ζ ∈ H = 2

J . By the easy part of the CSPS theorem 1.5.7, wededuce the desired inequality ‖u‖cb ≤ ‖u‖jcb. The first part of (1) is similar.

To prove (2), we may assume by injectivity of ⊗min and ⊗h that H and Xare finite-dimensional. In that case the result follows from (1) by duality, using(1.15), 1.5.9, (1.51), and (1.32). Item (3) follows from (2) and (1.37). The firstequality in (4) is clear from (1), and the rest is clear from the string

(Hr⊗ X

⊗ Kc)∗ ∼= CB(X

⊗ Kc, Hc) ∼= CB(X, CB(Kc, Hc)),

which equals CB(X, B(K, H)). Here we have used (1.51), (1.50), (1.15), and(1.14). The first equality in (5) is clear from (1.32), (1.15), and (1.14). Then

38 Duality and tensor products

S∞(K, H) ⊗min X = Hc ⊗min X ⊗min Kr by commutativity of ⊗min, and sothe second part of (5) follows from (2). Item (6) is a special case of (5), and (7)follows from (6) by (3) and the associativity of the Haagerup tensor product. Themiddle equality in (8) follows from (2), and the first equality from (1). WritingHc = CJ and Kr = RI for cardinals I, J , the last equality in (8) may be seenfrom (1.37) and (1.18). Clearly (9) follows from (4) with X = C. Lastly, for (10),CB(S1(2

I , 2J), X) is completely isometric to

CB(RI

⊗ CJ , X) ∼= CB(CJ , CB(RI , X)) ∼= Rw

J (CwI (X)) ∼= MI,J(X),

using (9), (1.50), 1.2.28, and (1.20).

1.6 DUALITY AND TENSOR PRODUCTS

This material is usually only used in the final section of later chapters.

1.6.1 (Mapping spaces as duals) If Y is a dual operator space then so isCB(X, Y ), for any operator space X . Indeed by (1.51) an explicit predual forCB(X, Y ) is X

⊗ Y∗. From this, together with the density of the finite rank

tensors in X⊗ Y∗, it follows that a bounded net (ut)t in CB(X, Y ) converges

in the w∗-topology to a u ∈ CB(X, Y ) if and only if ut(x) converges in thew∗-topology to u(x) in Y for all x ∈ X . In particular, setting Y = B(K, H) forHilbert spaces H, K, and using the same principle again, we see that the aboveequivalent conditions are also equivalent to

〈ut(x)ζ, η〉 → 〈u(x)ζ, η〉 for all x ∈ X, ζ ∈ K, η ∈ H. (1.53)

1.6.2 (Dual matrix spaces) If X is a dual operator space then so is Mn(X).Indeed by (1.51) and (1.8) we have

(S1n

⊗ X∗)∗ ∼= CB(X∗, Mn) ∼= Mn(X).

More generally the same proof, but substituting 1.2.29 for (1.8), shows that forcardinals I, J , MI,J(X) is a dual operator space with operator space predualS1(2

I , 2J)

⊗ X∗, and also MI,J(X) ∼= CB(X∗, MI,J).

If I, J are sets, and if I0 and J0 are finite subsets of I and J respectively,write ∆ = I0×J0. The set Λ of such ∆ is a directed set under the usual ordering.For such ∆, and for x ∈ MI,J(X), we write x∆ for the matrix x, but with entriesxij switched to zero if (i, j) /∈ ∆. Then (x∆)∆ is a net indexed by ∆ ∈ Λ, whichwe call the net of finite submatrices of x.

Corollary 1.6.3 (Effros and Ruan) Let X be a dual operator space, and letI, J be cardinals.(1) If (xt)t is a bounded net in MI,J(X), then xt → x ∈ MI,J(X) in the w∗-

topology in MI,J(X), if and only if each entry in xt converges in the w∗-topology in X to the corresponding entry in x.

Operator spaces 39

(2) If Y is a dual operator space, and if u : X → Y is a w∗-continuous com-pletely bounded map, then the amplification uI,J : MI,J(X) → MI,J(Y ) isw∗-continuous. If, further, u is completely isometric, then uI,J is a w∗-homeomorphic complete isometry onto a w∗-closed subspace.

(3) MfinI,J(X) is w∗-dense in MI,J(X). Indeed if I, J are sets, and x ∈ MI,J(X),

then the net of finite submatrices of x converges to x in the w∗-topology.

Proof Using the fact that MI,J(X) = CB(X∗, MI,J) = CB(X∗, B(2J , 2

I)) (see1.6.2), (1) follows from (1.53), and the density of Spanei in 2

I and 2J . Items

(2) and (3) follow immediately from (1) and A.2.5.

Note from 1.6.3 that for a w∗-closed subspace X ⊂ B(K, H), one may defineMI,J(X) to be the w∗-closure of M

finI,J(X) in MI,J(B(K, H)) = B(K(J), H(I)).

1.6.4 (A characterization of dual operator spaces) An operator space X whichis a dual Banach space, is a dual operator space if and only if Mn(X) is a dualBanach space, and the n2 canonical inclusion maps from X into Mn(X) are w∗-continuous, for all n ≥ 2. This follows immediately from the following criterion(which is the one which will be used in the sequel). If X is the dual Banach spaceof W , and if W is equipped with its natural matrix norms as a subspace of X∗

via the natural inclusion, then X is the dual operator space of W if and onlyif the unit ball of Mn(X) is σ(X, W )-closed for every positive integer n. Thatis, if and only if whenever (xt)t is a net in Ball(Mn(X)), x ∈ Mn(X), and thei-j entry of xt converges in the σ(X, W ) topology to the i-j entry of x for alli, j = 1, . . . , n, then x ∈ Ball(Mn(X)).

The ‘only if’ in the last criterion follows from 1.6.2 and 1.6.3 (1). For theother direction, assume that the unit ball of Mn(X) is σ(X, W )-closed for everyn ≥ 1. By 1.5.14 (6) and the selfduality of the Haagerup tensor product (see1.5.9), we have Mn(X)∗ = Rn ⊗h X∗ ⊗h Cn. Since ⊗h is injective we deducethat Rn ⊗h W ⊗h Cn ⊂ Mn(X)∗ isometrically. Hence it is easy to argue that theimage of Ball(Mn(X)) inside (Rn ⊗h W ⊗h Cn)∗ has polar set in Rn⊗h W ⊗h Cn

equal to the unit ball of Rn ⊗h W ⊗h Cn. Also, this image is w∗-closed byhypothesis. Therefore by the bipolar theorem, Ball(Mn(X)) is equal to the unitball of (Rn⊗hW⊗hCn)∗. By 1.5.14 (4) and (1.8), the latter space is isometricallyequal to CB(W, Mn) = Mn(W ∗). Hence Ball(Mn(X)) = Ball(Mn(W ∗)). Sincethis holds for any n ≥ 1, this shows that X∗ = W completely isometrically.

1.6.5 (Normal spatial tensor product) If X and Y are dual operator spaces,with operator space preduals X∗ and Y∗, then CB(Y∗, X) is the dual operatorspace of X∗

⊗ Y∗ by 1.6.1. As in (1.35), we regard X ⊗min Y → CB(Y∗, X), and

we define the normal minimal tensor product X ⊗Y to be the w∗-closure of X⊗Y(or of X⊗minY ) in CB(Y∗, X). Equivalently, if X and Y are w∗-closed subspacesof B(H) and B(K) respectively, then we may define X ⊗Y to be the w∗-closurein B(H ⊗2 K) of the copy of X ⊗Y . This is sometimes referred to as the normalspatial tensor product. If M and N are W ∗-algebras, then M ⊗N as describedabove is the usual von Neumann algebra tensor product (e.g. see [407, IV; 5])


and in particular, B(H)⊗B(K) = B(H ⊗2 K). To see that these two definitionsof X ⊗Y are the same (up to w∗-homeomorphic complete isometry), we use thefollowing argument. Since X and Y are w∗-closed subspaces of B(H) and B(K)respectively, we know by 1.4.6 that X∗ and Y∗ are quotients of S1(H) and S1(K)respectively. By the ‘projectivity’ property of

⊗, we obtain a complete quotient

map Q : S1(H)⊗ S1(K) → X∗

⊗ Y∗. Using the identification (1.51) we see that

Q∗ may be viewed as a w∗-continuous completely isometric embedding

CB(Y∗, X) → CB(S1(K), B(H)) ∼= B(H ⊗2 K),

the last relation from the first paragraph of 1.5.2. Via the canonical identificationof X ⊗ Y with a subset of CB(Y∗, X), we see that the w∗-closure of X ⊗ Y inB(H ⊗2 K) may be identified with the w∗-closure of X ⊗ Y in CB(Y∗, X).

In general, CB(Y∗, X) (or equivalently, (X∗⊗ Y∗)∗) is not equal to X ⊗Y .

However they are equal when X and Y are W ∗-algebras, as was shown by Effrosand Ruan (see the Notes section for a few more details). This is also the casewhen X = B(H). Indeed, for any dual operator space Y and cardinals I, J ,

MI,J ⊗ Y ∼= MI,J(Y ) (1.54)

as dual operator spaces. This follows by the remark after 1.6.3, and an argumentsimilar to the one used for (1.37). Also, MI,J(Y ) ∼= CB(Y∗, MI,J) by 1.2.29.Setting H = 2

I gives B(H)⊗Y ∼= CB(Y∗, B(H)).We leave it as an exercise that the normal spatial tensor product is ‘associa-

tive’, and ‘functorial’ for w∗-continuous completely bounded maps.

1.6.6 (Operator valued measurable functions) Let (Ω, µ) be a measure space(σ-finite for specificity), and let Y be a dual operator space. Since L∞(Ω) isa commutative W ∗-algebra, ‖u‖cb = ‖u‖ for any L∞(Ω)-valued bounded map(see 1.2.6). Using natural conditional expectations, there is a net vλ of finiterank contractive maps on L∞(Ω) such that vλ converges to IL∞ point-w∗. Ifu ∈ B(Y∗, L∞(Ω)) then (vλu)λ is a net of finite rank operators Y∗ → L∞(Ω)converging point-w∗ to u. Since this net is bounded, this implies that vλu → uin the w∗-topology of CB(Y∗, L∞(Ω)) by 1.6.1. Each vλu is finite rank and so‘belongs’ to Y ⊗L∞(Ω). This together with the first few lines of 1.6.5 shows that

B(Y∗, L∞(Ω)) = CB(Y∗, L∞(Ω)) ∼= L∞(Ω) ⊗ Y. (1.55)

Assume that Y∗ is separable. Then the latter operator space has an interestingdescription in terms of Y -valued functions defined on Ω. We say that a functionf : Ω → Y is w∗-measurable if 〈f(· ), ϕ〉 : Ω → C is measurable for any ϕ ∈ Y∗.Using the separability assumption it is easy to see that for such an f , the scalar-valued function ‖f(· )‖Y is measurable. We set ‖f‖∞ =

∥∥‖f(· )‖Y

∥∥L∞(Ω)

and wesay that f is essentially bounded if ‖f‖∞ < ∞. Then ‖f‖∞ is a seminorm on thespace of such functions and we let L∞(Ω; Y ) be the normed space obtained after

Operator spaces 41

taking the quotient by N0 = f : ‖f‖∞ = 0. As in the scalar-valued case, weidentify any w∗-measurable essentially bounded function f with its class moduloN0. Using again the separability of Y∗, we see that for any f ∈ L∞(Ω; Y ),

‖f‖∞ = sup‖〈f(· ), ϕ〉‖L∞ : ϕ ∈ Ball(Y∗)

.

Thus we may define a bounded map Jf : Y∗ → L∞(Ω), by Jf (ϕ) = 〈f(· ), ϕ〉,and we have ‖Jf‖ = ‖f‖∞. It turns out that the resulting isometric embeddingL∞(Ω; Y ) → B(Y∗, L∞(Ω)) ∼= (L1(Ω)⊗Y∗)∗ is onto. This is proved e.g. in [380,Theorem 1.22.13] when Y is a von Neumann algebra and the proof given thereextends to the general case. (See also Proposition IV.7.16 and its proof in [407].)The resulting identity B(Y∗, L∞(Ω)) ∼= L∞(Ω; Y ) shows that the latter space iscomplete and by (1.55), we obtain an isometric identification

L∞(Ω) ⊗ Y = L∞(Ω; Y ).

By the reasoning in 1.5.3 (3), this identity is a complete isometry if L∞(Ω; Y ) hasmatrix norms obtained by equating Mn(L∞(Ω; Y )) and L∞(Ω; Mn(Y )). ThusL∞(Ω; Y ) is a dual operator space. If jY : L∞(Ω)⊗Y → L∞(Ω; Y ) is the w∗-continuous mapping providing this identification, and if we consider finite fami-lies (fk)k in L∞(Ω) and (yk)k in Y , then jY maps

∑k fk ⊗ yk to

∑k fk(·)yk.

1.6.7 (W ∗-continuous extensions: the bilinear case) Let X, Y be operatorspaces, let W be a dual operator space, and let u : X × Y → W be a completelybounded bilinear map. Then u admits a (necessarily unique) separately w∗-continuous extension u : X∗∗ ×Y ∗∗ → W , this extension is completely bounded,and ‖u‖cb = ‖u‖cb. To prove this, we may assume by Lemma 1.4.7 that Wis a w∗-closed subspace of some B(H). By the CSPS theorem (1.5.7), we finda Hilbert space L and two completely bounded maps v : X → B(L, H) andw : Y → B(H, L) such that u(x, y) = v(x)w(y) for all x ∈ X, y ∈ Y . Ac-cording to 1.4.8, v and w admit w∗-continuous extensions v : X∗∗ → B(L, H)and w : Y ∗∗ → B(H, L) with ‖v‖cb = ‖v‖cb and ‖w‖cb = ‖w‖cb. Define abilinear map u : X∗∗ × Y ∗∗ → B(H) by letting u(η, ν) = v(η)w(ν) for anyη ∈ X∗∗, ν ∈ Y ∗∗. Then the easy part of the CSPS theorem ensures that u iscompletely bounded with ‖u‖cb ≤ ‖v‖cb‖w‖cb = ‖v‖cb‖w‖cb ≤ ‖u‖cb. Since u isvalued in W , and W is w∗-closed, its extension u is valued in W as well.

1.6.8 (Normal Haagerup tensor product) If X and Y are two dual operatorspaces, we let (X ⊗h Y )∗σ denote the subspace of (X ⊗h Y )∗ corresponding tothe completely bounded bilinear forms X × Y → C which are separately w∗-continuous. Then we define the normal Haagerup tensor product X ⊗σh Y to bethe operator space dual of (X ⊗h Y )∗σ.

From 1.6.7 one may deduce the following two consequences. First, if X, Y aretwo (not necessarily dual) operator spaces then (X ⊗h Y )∗ ∼= (X∗∗⊗h Y ∗∗)∗σ. Tosee this, consider the map u → u in 1.6.7, in the case that W = Mn. Note that


a map u into Mn is separately w∗-continuous if and only if each ‘entry’ uij isseparately w∗-continuous. It follows that

(X ⊗h Y )∗∗ = X∗∗ ⊗σh Y ∗∗ as dual operator spaces. (1.56)

Second, if X, Y are two dual operator spaces, then the natural action of X ⊗ Yon (X ⊗h Y )∗σ yields a completely isometric embedding

X ⊗h Y ⊂ X ⊗σh Y. (1.57)

To see (1.57), let j : X ⊗ Y → X ⊗σh Y be the canonical mapping, and fix zin X ⊗ Y . It is clear from the definitions that ‖j(z)‖ ≤ ‖z‖h. The converseinequality relies on the selfduality of the Haagerup tensor product. We let ε > 0.By (1.43), we have X ⊗h Y ⊂ (X∗ ⊗h Y∗)∗ isometrically. Hence there exists amap Φ ∈ X∗ ⊗ Y∗ such that ‖Φ‖h = 1 and ‖z‖h ≤ |〈z, Φ〉| + ε. Writing Φ as∑

k ϕk⊗ψk for some ϕk ∈ X∗ and ψk ∈ Y∗, then the bilinear form u : X×Y → C

defined by u(x, y) =∑

k ϕk(x)ψk(y) is clearly separately w∗-continuous andwe have 〈z, Φ〉 = 〈j(z), u〉. Hence |〈z, Φ〉| ≤ ‖j(z)‖‖u‖cb. By (1.43) again, andthe injectivity of ⊗h, we have X∗ ⊗h Y∗ ⊂ X∗ ⊗h Y ∗ ⊂ (X ⊗h Y )∗, so that‖u‖cb = ‖Φ‖h = 1. Since ε was arbitrary, this shows that ‖z‖h ≤ ‖j(z)‖. Hencej is an isometry, and a similar argument shows that it is a complete isometry.

Indeed, X ⊗ Y is w∗-dense in X ⊗σh Y . Let q : (X ⊗h Y )∗∗ → X ⊗σh Ybe the adjoint mapping of the embedding (X ⊗h Y )∗σ ⊂ (X ⊗h Y )∗, and leti : X⊗h Y → (X⊗h Y )∗∗ be the canonical embedding into the second dual. Thenthe embedding in (1.57) is qi. That its range is w∗-dense follows from Goldstine’slemma A.2.1, and the fact that q is a quotient map, and is w∗-continuous.

1.6.9 (Weak* Haagerup tensor product) This is another ‘dual version’ of theHaagerup tensor product. Since will not use this tensor product very much, wecontent ourselves with an abridged development of it. See [70, 150] for muchmore information, for example, there are several other ways to describe thistensor product. If X and Y are operator spaces then we define

X∗ ⊗w∗h Y ∗ = (X ⊗h Y )∗. (1.58)

We now discuss why this is a ‘tensor product’. By the last paragraph of 1.5.8, andwriting the space H there as 2

I , we have that w ∈ (X⊗hY )∗ if and only if w maybe written in the form w(x⊗ y) = 〈ϕ(x), ψ(y)〉, for a cardinal I, ϕ ∈ CB(X, RI)and ψ ∈ CB(Y, CI). Now CB(Y, CI) ∼= Cw

I (Y ∗) and CB(X, RI) = RwI (X∗),

using 1.2.29. Writing ϕ = [ϕi] and ψ = [ψi], we have shown the following as-sertion. Namely, w ∈ (X ⊗h Y )∗ if and only if there exist [ϕi] ∈ Rw

I (X∗) and[ψi] ∈ Cw

I (Y ∗) such that w may be written as w(x ⊗ y) =∑

i ϕi(x)ψi(y) forx ∈ X, y ∈ Y . The last sum converges absolutely in C, as may be seen by theCauchy–Schwarz inequality. We may therefore think of elements w ∈ X∗ ⊗w∗h Y ∗

Operator spaces 43

as tensors, and write w =∑

i ϕi⊗ψi, or w = ϕψ. We call this a weak represen-tation for w, and equate two weak representations if they agree ‘as functionalson X ⊗h Y ’. By the above,

‖w‖ = min‖[ϕi]‖RwI (X∗)‖[ψi]‖Cw

I (Y ∗),

where the minimum is over all weak representations for w.Note that X ⊗h Y ⊂ X ⊗w∗h Y completely isometrically. This is just a

restatement of (1.43). Thus if X is finite-dimensional then X ⊗w∗h Y = X ⊗h Y .It follows readily from the ‘associativity’ of the Haagerup tensor product (see1.5.4), and (1.58), that the weak* Haagerup tensor product is associative too.It is ‘functorial’ too in an appropriate sense. That is, w∗-continuous completelybounded maps will tensor. In fact more is true: Given dual operator spacesXk and Yk, for k = 1, 2, and completely bounded maps uk : Xk → Yk, thereis a canonical completely bounded map u1 ⊗ u2 : X1 ⊗w∗h X2 → Y1 ⊗w∗h Y2

with (u1 ⊗ u2)(∑

i xi ⊗ wi) =∑

i u1(xi) ⊗ u2(wi), for any weak representation∑i xi ⊗ wi. Moreover, ‖u1 ⊗ u2‖cb ≤ ‖u1‖cb‖u2‖cb. We omit the proof of this

perhaps surprising assertion, which may be found in the original paper [70] (orsee [150] for a different proof).

Many of the identities in 1.5.14 have versions appropriate to ⊗w∗h. For ex-ample, for a dual operator space X and a cardinal I we have

CI ⊗w∗h X = CI⊗X ∼= CwI (X) (1.59)

and a similar relation holds for RwI (X). Indeed, the ‘∼=’ here follows from (1.54).

The predual of CwI (X) is RI

⊗ X∗, by 1.2.28 and (1.51). However by 1.5.14 (1),

the latter space equals RI ⊗h X∗, which is the predual of CI ⊗w∗h X .Another useful relation is

RI ⊗w∗h X = RI ⊗h X = RI

⊗ X ∼= CI(X∗)∗ (1.60)

(and there is a matching formula for RI(X∗)∗). To see (1.60), note that if w isin RI ⊗w∗h X , then w has a weak representation

∑i ri ⊗ xi, with [ri] ∈ Rw

J (RI)and [xi] ∈ Cw

J (X), for a cardinal J . However RwJ (RI) = RJ(RI), so that by the

proof of 1.5.6 (2) we have that w ∈ RI ⊗h X . This gives the first equality in(1.60). The second equality is 1.5.14 (1), whereas the third follows from (1.58),since CI(X∗) = CI ⊗h X∗ by 1.5.14 (2).

By analogy with 1.5.14 (7), for cardinals I, J we have

MI,J(X ⊗w∗h Y ) ∼= CwI (X) ⊗w∗h Rw

J (Y ). (1.61)

To see this, notice that the first space here equals

CI ⊗w∗h (X ⊗w∗h Y ) ⊗w∗h RJ∼= (CI ⊗w∗h X) ⊗w∗h (Y ⊗w∗h RJ),

using (1.20), (1.59) and its matching ‘row version’, and the associativity of ⊗w∗h.By (1.59) again, this equals the right-hand side of (1.61).


As an application, we show that for any operator space X we have

KI,J(X)∗∗ ∼= MI,J (X∗∗) as dual operator spaces. (1.62)

By 1.5.14 (5), (1.37), (1.58), and (1.60) and its matching ‘column’ formulation,

KI(X)∗ ∼= (CI ⊗h X ⊗h RI)∗ ∼= (CI ⊗h X)∗ ⊗h CI∼= RI ⊗h X∗ ⊗h CI .

By 1.5.14 (4) and 1.2.29 we have (RI ⊗h X∗⊗h CI)∗ ∼= CB(X∗, MI) ∼= MI(X∗∗).This proves (1.62). Another proof may be given using C∗-algebraic principles.

1.6.10 (A principle for separately w∗-continuous maps) Suppose, for example,that we have a completely bounded trilinear map Φ: X × Y × Z → B(K, H),where X, Y, Z are dual operator spaces. Suppose that Φ is separately w∗-conti-nuous. From the CSPS theorem 1.5.7 we have

Φ(x, y, z) = u(x)v(y)w(z) (1.63)

for completely contractive maps u : X → B(K1, H), v : Y → B(K2, K1), andw : Z → B(K, K2). Here of course H, K, K1, K2 are Hilbert spaces. Then weclaim that K1, K2, and the maps u, v, w may be slightly adjusted, to each bew∗-continuous and still satisfy (1.63).

To see this, we first state an obvious principle, which we leave as an exercise.Namely, if u : X → B(K, H) is a completely contractive w∗-continuous map, ifL is a subspace of H with associated projection PL, and if G is a subspace of K,then the map PLu(·)|G is w∗-continuous and completely contractive too.

Because of this principle, the following procedure applied to the maps oneat a time, from right to left, will make them w∗-continuous, without destroyingw∗-continuity of any of the previously adjusted maps. We begin on the right,with w. We note that for ζ ∈ K, η ∈ H,

〈Φ(x, y, z)ζ, η〉 = 〈w(z)ζ, v(y)∗u(x)∗η〉 = 〈PLw(z)ζ, v(y)∗u(x)∗η〉, (1.64)

where L = [v(Y )∗u(X)∗H ], and PL is the projection onto L. Replacing w byPLw, and v by its restriction to L, we may assume that K2 = L. If (zt)t is abounded net in Z converging in the w∗-topology to z, and if x ∈ X and y ∈ Yare fixed, then Φ(x, y, zt) → Φ(x, y, z) in the w∗-topology. Thus from (1.64),

〈w(zt)ζ, v(y)∗u(x)∗η〉 → 〈w(z)ζ, v(y)∗u(x)∗η〉.

From this, and norm density considerations in L, it is clear that w(zt) → w(z)in the WOT. Since this is a bounded net, it converges in the w∗-topology (seeA.1.4). Thus by A.2.5, w is w∗-continuous. Finally, we replace K2 by its sub-space [w(Z)H ], and replace v by its restriction to this subspace. This ends theprocedure. We now begin the same procedure again, to make the next map vw∗-continuous. Thus we replace K1 by its subspace [u(X)∗H ], and use an ar-gument similar to the above. Once v is w∗-continuous, we replace K1 again by[v(Y )w(Z)H ], and then continue, to make u w∗-continuous.

Operator spaces 45

1.7 NOTES AND HISTORICAL REMARKS

One might say that operator space theory grew out of Stinespring’s theorem[398], via Arveson’s landmark papers [21, 22] and the fundamental work in theseventies (continuing into the early 1980s) of Choi and Effros, Haagerup, Kirch-berg, Wittstock, and others. The eighties saw much activity in the area of com-pletely bounded maps; the vision of Effros (e.g. see [133]) and Paulsen’s text [307]were particularly influential in this development. See also [94], and the muchoverlooked work of Hamana (referenced in our bibliography). However operatorspaces did not become a field in its own right until Ruan’s theorem 1.2.13. Thisresult was proved by Ruan in his thesis [369] (see also [370]) under the directionof Effros, and was inspired by [90,318]. Ruan used this theorem to define operatorspace quotients, operator seminorm structures, and so on. See also the inspiringpapers [142, 143], for example. A couple of years after that, operator space du-ality, and the basic operator space tensor theory, were developed independentlyby Blecher and Paulsen, and Effros and Ruan (e.g. see [42, 66, 145, 146]). (TheHaagerup tensor product of operator spaces had been developed earlier (e.g.in the important paper [318]).) In the early 1990s Pisier lent his considerablestrength and expertise to the subject, as is fabulously testified to in [332, 337].After this the field grew quite considerably, with many brilliant mathematiciansmaking remarkable contributions. For a more complete account of basic operatorspace theory, and for more references to the literature to complement the list inthese Notes and historical remarks sections, the reader should also consult thetexts [149,314,337,385]. See also [433], which is currently being rewritten in anexpanded printed form. There are other good surveys of part of the subject, suchas [338,366].

1.2: Parts of 1.2.4 are folklore, no doubt known from the beginning. Ageneralization is proved in the later result 8.3.2. Observation 1.2.6 is due toLoebl. A related useful fact is that for a bounded linear map into Mn, we have‖T ‖cb = ‖Tn‖ (see [389]). The canonical example of a map which is not com-pletely bounded is the ‘transpose map’ on K. Injectivity of operator systems wasstudied deeply by Choi and Effros (e.g. see [90]), partly inspired by Arveson’sextension theorem (see 1.3.5 and [21]). Later, Theorems 1.2.8 and 1.2.10 wereproved by Wittstock [432, 431], Haagerup [177], and Paulsen [305]. See thosepapers, and [149,314,337,385], for additional historical details. Paulsen’s proofsof these two results is now the standard route. See also [325,335] for a differentapproach extending Theorem 1.2.8 to a Banach space setting. The first isomor-phism in (1.5) has been dubbed the ‘canonical shuffle’ by Paulsen. It is not truethat a one-to-one and surjective completely bounded linear map T : X → Y be-tween complete operator spaces has a completely bounded inverse. Indeed, thereis no ‘open mapping theorem’ in this sense. Of course if the countably infiniteamplification of T maps K(X) onto K(Y ), then there is no obstacle.

That CB(X, Y ) is an operator space was observed first in [143]. The operatorspace dual was called the standard dual by Blecher and Paulsen, and the asso-

46 Notes and historical remarks

ciated duality theory was developed independently by those authors and Effrosand Ruan. Minimal and maximal operator spaces were first considered in [142]and [66] respectively. Formally, Hilbert row and column spaces may first appearin [432]. They played a role in [135,66] and were further developed in [147]. Someof the results in the latter paper had also been noticed by Blecher, and [43] givesalternative routes to these and to other results from [147]. There are other inter-esting operator space structures on a Hilbert space, most notably the (selfdual)operator Hilbert space OH of Pisier [328, 337] which will be mentioned againin Chapter 5 (see 5.3.4). Nearly all of the definitions/results on infinite matrixspaces here may be found in a series of papers by Hamana, and Effros and Ruan(e.g. see [186,143,146]). Interpolation and ultraproducts of operator spaces werefirst considered by Pisier [328,331,327]. See also [436] for related work.

We sketch a simple proof of (1.14). We may write Hc = CJ , Kc = CI , forcardinals I, J . Setting X = CI in 1.2.28 yields CB(CJ , CI) ∼= Rw

J (CI) ∼= MI,J

completely isometrically, where the last equality follows from any of the centeredequations in 1.2.26.

1.3: Many of the results in this section are from Arveson’s original pa-pers [21,22]. These papers established, for example, the role of completely posi-tive maps in the unitary equivalence theory of irreducible sets of compact opera-tors, a generalization of the Shilov boundary to arbitrary operator algebras, andgeneralizations of the Sz. Nagy–Foias dilation theorem to representations of ar-bitrary function algebras. Arveson’s result 1.3.6 is the decisive relation betweenunital operator spaces and operator systems. Paulsen’s lemma 1.3.15 (originallyfrom [305]) yields the decisive relation between general operator spaces and oper-ator systems. Note that we did not use the fact that the map was unital in 1.3.9,indeed many of the results in this section are true without this assumption (e.g.see [36]). There are also direct proofs of 1.3.9 not using Stinespring’s theorem.The result 1.3.10 is an easy variant of the Banach–Stone–Kadison theorem [212].Item 1.3.11 seems to appear first in [301], but is related to Choi’s multiplicativedomain [87]. Other sources for results in this section are [88,90,211,213,307,399].

1.4: The facts in this section are due to Blecher (see [42], which was writtenclose to the date of [66,145], although it appeared much later), with the followingmain exceptions. The fact that X ⊂ X∗∗ completely isometrically was indepen-dently noticed in [66, 145]. Effros and Ruan had noticed 1.4.7 via a differentroute [146].

1.5: The minimal tensor product may have been first considered for opera-tor spaces in [307]. An account of this tensor product from the perspective ofC∗-algebra theory can be found in [407], for example. Relations such as the com-plete isometry X ⊗min Y → CB(Y ∗, X) were first noticed in [66]. The approachpresented here to the minimal tensor product is taken from the latter paper,with some simplifications influenced by the exposition in Chapter 8 of [149]. Thefunctoriality property of the minimal (or spatial) tensor product is one of themost important properties of completely bounded maps: they may be tensored.This may have been first observed in [117], which dates to 1982 or earlier. This

Operator spaces 47

property is not shared by general bounded maps with respect to the naturaltensor products found in C∗-algebra theory.

Haagerup defined the tensor product that bears his name in [176], at least forC∗-algebras. It was studied in the 1980s in papers of Effros, some with coauthors(see [149] for detailed references). In [318] Paulsen and Smith showed that theHaagerup tensor product in the generality presented here, is an operator space(this is usually seen now using Ruan’s theorem, as was first noted in [369]),and proved many of the other facts presented here. The class of bilinear ormultilinear maps described here as ‘completely bounded’ are given a differentname in [149]; instead Effros and Ruan use this term for the class which we call‘jointly completely bounded’. The injectivity of the Haagerup tensor product isalso from [318]. Other proofs were found later by Effros, Paulsen, and others(e.g. see [385, 149, 314, 337] or 2.10 in [43], for a selection). The projectivity of⊗h and

⊗ was observed in [147]. Part (2) of 1.5.6 is also true for uncountable

sums with the same proof. The original references for Theorem 1.5.7 are [93] (forC∗-algebras) and [318] (for general operator spaces). Other proofs may be foundin [149,307,337]. The selfduality relation (1.43) was conjectured by Blecher andproved in full in [147]. Different proofs appear in [43,70]. The latter paper sketchesa route based on Smith’s important notion of strong independence [391], throughmany of the basic results in this section. Observation (1.44) was perhaps firstin [40]. The relation (1.45) is from [147], see also [325] and [310] for related resultsand developments. For further information on g2, we refer the reader to [118] (e.g.see 12.7 there). Relations (1.44), (1.46), and (1.47) are just three examples ofrelationships between the Haagerup tensor product and Pisier’s ‘gamma-norms’introduced in [326]. See [328] and [243] for related developments.

The operator space projective tensor product and its basic properties weredeveloped in [66,145]. Formula (1.48) is from the latter paper. There is a similarformula valid for all z ∈ Mn(X

⊗ Y ), writing z = α(x ⊗ y)β, where the latter

matrices are now countably infinite (see [149]). In [66] it is shown without usingformula (1.48), that there is an operator space X

⊗ Y having all the other

properties mentioned in 1.5.11 and the paragraphs after it. Various forms of therelations in 1.5.14 may be found in [135,66,147,139,43], but [147] is the primaryreference. Note that 1.2.10 follows from 1.5.14 (4), and the injectivity of ⊗h.

1.6: Nearly all the facts about infinite matrices in 1.6.2 and 1.6.3 are explicitlyin [186, 143, 146]. Neufang has generalized 1.6.3 in [292]. The result 1.6.4 is dueto Le Merdy [244], who also showed in that paper that if X is an operatorspace which is a dual Banach space with predual Y , then there may not exist anoperator space structure on Y such that Y ∗ ≈ X completely isomorphically (seealso 2.7.15). The equivalence between the two descriptions of ⊗ in 1.6.5 is relatedto the fact that (X∗

⊗ Y∗)∗ = CB(Y∗, X) equals the normal Fubini product of X

and Y , which was proved independently by Blecher and Ruan (see [43, Theorem2.5], [372]), inspired by the von Neumann algebra case [146]. The Fubini productfor operator spaces was first studied in [186] though. Since the normal Fubini


product is defined in terms of slice maps, such maps are key to understandingwhen X ⊗Y = CB(Y∗, X) holds for dual operator spaces X and Y . For example,the fact that this holds for two von Neumann algebras, which was proved in [146],follows from the slice map theorem of Tomiyama [410]. This fact is of greatimportance in many recent papers on Fourier analysis (e.g. see [378]), since, forexample, one may deduce from it that A(G1 ×G2) ∼= A(G1)

⊗ A(G2) for locally

compact groups G1, G2 (see [146]). A dual operator space X is said to have thedual slice mapping property if X ⊗Y = CB(Y∗, X) holds for all dual operatorspaces Y . This is equivalent to X∗ possessing the ‘operator space approximationproperty’ (see [146,139,233]). The argument at the beginning of 1.6.6 is only partof this result in the case when X is a commutative W ∗-algebra. It extends to allsemidiscrete (=injective) von Neumann algebras. The description of L∞(Ω; Y )as a space of Y -valued w∗-measurable functions is essentially taken from [380,Section 1.22]. The definitions and results in 1.6.8 are due to Effros and Ruan(see [138,150], and references therein). The weak* Haagerup tensor product wasdeveloped by Blecher and Smith in [70] for dual operator spaces. Later thistensor product, and its basic properties, were generalized to all operator spaces(e.g. see [150], and to a lesser extent [5]), under the name extended Haageruptensor product. Spronk has shown in [397] that the original arguments of [70]immediately yield many of these generalizations from [150]. Result 1.6.10 is from[150].

2

Basic theory of operator algebras

2.1 INTRODUCING OPERATOR ALGEBRAS AND UNITIZATIONS

By definition, a concrete operator algebra is a closed subalgebra of B(H), for someHilbert space H . When the context is clear, we often simply write A ⊂ B(H)or A → B(H) to denote this. As for operator spaces, we will have to consideroperator algebras from an abstract point of view. We will start from the obviousobservation that any operator algebra A is both an operator space and a Banachalgebra. Conversely, if A is both an operator space and a Banach algebra, thenwe call A an (abstract) operator algebra if there exist a Hilbert space H and acompletely isometric homomorphism π : A → B(H). In that situation, we oftenidentify A and the concrete operator algebra π(A) and say that A is representedas a subalgebra of B(H), or that A is an operator algebra on H . We often identifyany two operator algebras A and B which are completely isometrically isomor-phic, that is, there exists a complete isometric algebra homomorphism from Aonto B. In this case, we write ‘A ∼= B completely isometrically isomorphically’,or ‘as operator algebras’; or simply ‘A = B’.

Some more notation that will be used throughout: We say that an operatoralgebra A is unital if it has an identity (i.e. a unit) of norm 1. In this case, theunit is often written as e or 1 (or sometimes eA or 1A if necessary). However inthis book we will usually focus on the larger class of operator algebras which areapproximately unital; that is, which possess a contractive approximate identity(cai). The main reason for this choice of focus is because this class includes allC∗-algebras. We repeat from 1.1.5 the convention that a unital-subalgebra is asubalgebra A of a unital algebra B with 1B ∈ A. Of course operator algebrasmay be thought of as the (closed) subalgebras of C∗-algebras. The closure of asubalgebra of an operator algebra is obviously an operator algebra.

By a representation of an operator algebra A on a Hilbert space H , we meana homomorphism π : A → B(H). Of course we are almost always interestedin the completely contractive representations. It should be noticed that unlike∗-representations between C∗-algebras, bounded and completely bounded ho-momorphisms between operator algebras have no automatic rigidity property ingeneral. For instance, a completely contractive homomorphism need not have aclosed range. If A and B are operator algebras, then we will often write A → Bwhen there exists a completely isometric homomorphism j : A → B. Again, aunital map is one that takes the identity to the identity.

50 Introducing operator algebras and unitizations

2.1.1 (C∗-covers) If S is a subset of a C∗-algebra B, then C∗B(S) denotes the

C∗-subalgebra of B generated by S (that is, the smallest C∗-subalgebra of Bcontaining S). A C∗-cover of an operator algebra A is a pair (B, j) consistingof a C∗-algebra B, and a completely isometric homomorphism j : A → B, suchthat j(A) generates B as a C∗-algebra. That is, such that C∗

B(j(A)) = B.

2.1.2 (The diagonal) We defined this for unital operator spaces in 1.3.7. If Ais an operator algebra (possibly nonunital), represented as a subalgebra of B(H)say, then we define the diagonal of A to be the C∗-algebra

∆(A) = a ∈ A : a∗ ∈ A.

Note that if A is a w∗-closed subalgebra of B(H) then so is ∆(A) (by the w∗-continuity properties mentioned in A.1.2), and hence ∆(A) is a W ∗-algebra.

To see that the definition of ∆(A) is (up to ∗-isomorphism) independentof the particular representation of A, consider any isometric homomorphismπ : A → B(K). Then it follows from A.5.8 that π(a)∗ = π(a∗) for any a ∈ ∆(A).Thus π maps ∆(A) ∗-isomorphically (and therefore also completely isometrically,by 1.2.4) onto the corresponding space b ∈ π(A) : b∗ ∈ π(A).

From this, one deduces that a contractive homomorphism from a C∗-algebrainto an operator algebra B, actually maps into ∆(B), and is a ∗-homomorphism.Also, from this we see that a closed subalgebra of a C∗-algebra B may have atmost one involution with respect to which it is a C∗-algebra. If there exists suchan involution, then the subalgebra is a ∗-subalgebra of B.

2.1.3 (Recognizing selfadjoint elements, etc.) Certain important elements inan operator algebra may be characterized intrinsically, and such results are veryoften useful. For the discussion that follows, we fix a closed subalgebra A ofB(H). Then the projections in B(H) which happen to be also in A, are exactlythe idempotent elements in A of norm 1 (see A.1.1). The selfadjoint operators inB(H) which happen to be also in A, are exactly the selfadjoint elements in thediagonal ∆(A). If A is a unital operator algebra, then these selfadjoint elementsare exactly the Hermitian elements of A in the sense of A.4.2. The diagonal ∆(A)is the span of these selfadjoint elements of course.

If a, b are elements in the unit ball of A, and if ab = 1A, then a = b∗ byA.1.1, and a, b ∈ ∆(A). If in fact ab = IH , then b is an isometry on H , and a is acoisometry. If in addition ba = 1A, then b is a unitary in ∆(A), and is a unitaryoperator on H if 1A = IH .

2.1.4 Let A ⊂ B(H) be a unital operator algebra, with unit denoted by e.Then e2 = e and ‖e‖ = 1, consequently e : H → H is a projection, as notedin 2.1.3. Let K ⊂ H denote the range of e. If a ∈ A, then since eae = a, wehave a(K) ⊂ K and a(K⊥) = 0. We may therefore regard A ⊂ B(K) as anoperator algebra on K, and in that representation, the unit e coincides with theidentity operator IK . Also

‖[aij ]‖ = ‖[eaije]‖ = ‖[(aij)|K ]‖Mn(B(K)),

Basic theory of operator algebras 51

for aij ∈ A. This shows that any unital operator algebra may be represented(completely isometrically) as a unital-subalgebra of some B(H).

In 2.1.10 below, we will prove a similar result for approximately unital oper-ator algebras. In fact the next several results establish various useful propertiesof contractive approximate identities in operator algebras.

2.1.5 (Nondegeneracy) Let A be a Banach algebra, let H be a Hilbert space,and let π : A → B(H) be a contractive homomorphism. We say that π is non-degenerate if [π(A)H ] = H . This terminology is consistent with that in A.6.1.Indeed if we consider H as a left Banach A-module by letting aζ = π(a)ζ, forany a ∈ A and any ζ ∈ H , then π is obviously nondegenerate if and only if His a nondegenerate A-module. This is also equivalent, as we mentioned in A.6.1,to saying that π(et) → IH strongly, if (et)t is a cai for A.

If A ⊂ B(H) is a concrete operator algebra, then we say that A is a nondegen-erate subalgebra of B(H), or that ‘A ⊂ B(H) nondegenerately’, if the embeddingfrom A into B(H) is nondegenerate.

Lemma 2.1.6 Suppose that a is an element of a subspace of B(K, H), and that(et)t is a net of contractions in B(K) such that aet → a. Then aete

∗t → a,

ae∗t et → a, and ae∗t → a.

Proof If aet → a then aete∗t a

∗ → aa∗, so that 0 ≤ a(I − ete∗t )a∗ → 0, where

I = IK . Thus by the C∗-identity, a√

I − ete∗t → 0. Multiplying by√

I − ete∗twe see that a(I − ete

∗t ) → 0 as required for the first assertion. Also,

‖ae∗t − a‖ ≤ ‖ae∗t − aete∗t ‖ + ‖aete

∗t − a‖ → 0

since ‖ae∗t − aete∗t ‖ ≤ ‖a − aet‖ → 0. Finally,

‖ae∗t et − a‖ ≤ ‖ae∗t et − aet‖ + ‖aet − a‖ ≤ ‖ae∗t − a‖ + ‖aet − a‖ → 0

by what we just proved.

Lemma 2.1.7 Let A be an operator algebra and suppose that B is a C∗-coverof A. If A is approximately unital, then B is a nondegenerate A-bimodule in thesense of A.6.1. In this case (viewing A ⊂ B):(1) If b ∈ B then there exists an element b0 ∈ B and elements a1, a2 ∈ A with

b = a1b0a2. Moreover if ‖b‖ < 1 then b0, a1, a2 may be chosen of norm < 1.(2) Every cai for A is a cai for B.

Proof Notice that once (2) is established, then B is a nondegenerate BanachA-bimodule, and then (1) follows from Cohen’s factorization theorem (see A.6.2).

Let (et)t be a cai for A, and let a ∈ A. Then eta → a, and using 2.1.6 wehave a∗et → a∗. Thus if b is in the dense ∗-subalgebra of B generated by A, thenbet → b. By density, bet → b for every b ∈ B. Similarly, etb → b.

2.1.8 It follows from the last result that any C∗-cover (B, j) of an approximatelyunital operator algebra A is a unital C∗-algebra if and only if A is unital. Indeed


if A is unital with unit 1A, then j(1A) is a unit for B by 2.1.7 (2). Conversely, ifB is unital with unit 1B, then by 2.1.7 (2) we see that any cai (et) for A satisfies

j(et) = j(et) 1B −→ 1B.

Since j(A) is closed, we have 1B ∈ j(A).As another application of Lemma 2.1.7, we may see using 2.1.5 that if B is a

C∗-cover of an approximately unital operator algebra A, and if π : B → B(H)is a ∗-representation, then π is nondegenerate if and only if its restriction π|A isnondegenerate.

Lemma 2.1.9 Let A be a Banach algebra with cai (et)t, and let π : A → B(H)be a contractive homomorphism. We let p be the projection onto [π(A)H ]. Thenπ(et) → p in the w∗-topology of B(H). Moreover, for a ∈ A we have

π(a) = pπ(a)p, a ∈ A. (2.1)

Proof Let K = [π(A)H ]. If a ∈ A then π(et)π(a) → π(a), and so π(et)∗π(a)converges to π(a) by 2.1.6. Hence π(et)∗ → IK strongly on K. If ζ, η ∈ H , then

〈π(et)ζ, η〉 = 〈pπ(et)ζ, η〉 = 〈ζ, π(et)∗pη〉 −→ 〈ζ, pη〉 = 〈pζ, η〉.

Thus the bounded net (π(et))t converges to p in the WOT. Hence it also con-verges in w∗-topology to p, by A.1.4. Using the last fact and the separate w∗-continuity of the product in B(H) (see A.1.2), we obtain for a ∈ A that

π(a)p = w∗- limt

π(a)π(et) = limt

π(aet) = π(a).

Since pπ(a) = π(a) by the definition of p, the claim (2.1) follows at once.

2.1.10 (Reducing to the nondegenerate case) The relation (2.1) should beinterpreted as follows. Let H, K, π be as above. If we regard B(K) as a subalgebraof B(H) in the natural way (by identifying any T in B(K) with the map T ⊕ 0in B(K ⊕ K⊥) = B(H)), then the homomorphism π is valued in B(K). Sinceπ is nondegenerate when regarded as valued in B(K), this yields a principlewhereby to reduce a possibly degenerate homomorphism to a nondegenerateone. Applying this to completely isometric homomorphisms, we obtain that forany approximately unital operator algebra A, there exist a Hilbert space H anda nondegenerate completely isometric homomorphism π : A → B(H).

2.1.11 (The unitization) Often problems concerning an operator algebra A aresolved by first tackling the case where A is unital; and then in the general caseconsidering the unitization A1 of A. It is this process of unitization which we nowdiscuss. If A is a nonunital operator algebra, then a unitization of A which is alsoan operator algebra may be obtained by regarding A as a subalgebra of B(H) forsome H , and then taking A1 = SpanA, IH. We will prove in Corollary 2.1.15that up to completely isometric isomorphism, this unitization does not depend


on the embedding A ⊂ B(H). Consequently, A1 will be called the unitizationof A, and we usually use it without any reference to a concrete embedding ofA in B(H). We will first establish the more general Theorem 2.1.13, which isan extremely useful extension principle. The following simple lemma is a specialcase:

Lemma 2.1.12 Let A ⊂ B(H) be an operator algebra, and assume that IH /∈ A.Then for any n ≥ 1 and any matrices a ∈ Mn(A) and λ ∈ Mn, we have

‖λ‖Mn ≤∥∥a + λ ⊗ IH

∥∥Mn(B(H))

.

Proof Consider the algebra A1 = SpanA, IH ⊂ B(H). Since IH /∈ A, we maydefine a functional φ on A1 by letting φ(a + λIH) = λ for any a ∈ A and anyλ ∈ C. It is clear that φ is a homomorphism. From the basic theory of charactersfrom any text on Banach algebras, we have that φ is contractive. Hence by 1.2.6,φ is completely contractive. This proves the result.

Theorem 2.1.13 (Meyer) Let A ⊂ B(H) be an operator algebra, and assumethat IH /∈ A. Let π : A → B(K) be a contractive (resp. completely contractive)homomorphism, K being a Hilbert space. We let A1 = SpanA, IH ⊂ B(H),and we extend π to π0 : A1 → B(K) by letting

π0(a + λIH) = π(a) + λIK , a ∈ A, λ ∈ C.

Then π0 is a contractive (resp. completely contractive) homomorphism.

2.1.14 Before launching into the proof of Theorem 2.1.13, we recall a few factsconcerning the so-called Cayley transform of operators on a Hilbert space H.To those familiar with the basic theory of operator semigroups, the followingfacts will be transparent (e.g. see [405] sections III.8 and IV.4). In order to beself-contained, we will prove these facts using some basic results about the Rieszor analytic functional calculus (e.g. see [106] section 2.4). We let D and P denoterespectively the open unit disk and right-hand open half plane in C. We letd : D → P be the conformal map z → 1+z

1−z , and let c(z) = z−1z+1 be its inverse. If −1

is not in the spectrum σ(S) of an operator S on H (resp. 1 /∈ σ(T )), then we definethe Cayley transform (resp. inverse Cayley transform) by c(S) = (S−I)(S+I)−1

(resp. d(T ) = (I + T )(I − T )−1). It follows from the spectral mapping theorem(e.g. see [106, 2.4.4 (iv)]) that 1 /∈ σ(c(S)) (resp. −1 /∈ σ(d(T ))). Thus we seefrom the functional calculus that these transforms are indeed inverses of eachother. For example, c(d(T )) = T providing that 1 /∈ σ(T ).

An operator T ∈ B(H) will be called a strict contraction if ‖T ‖ < 1. In thatcase, the usual Neumann series trick shows that the operator I − T is invertibleand (I−T )−1 =

∑k≥0 T k belongs to the operator algebra generated by I and T .

Next, an operator S ∈ B(H) will be called strictly accretive if Re(S) = 12 (S+S∗)

is positive and invertible. That is, there is a scalar ε > 0 with Re(S) ≥ εI.After pre- and post-multiplying by Re(S)−

12 , it is easy to see that such an S is


invertible. Similarly, for any z ∈ C with Re(z) ≤ 0, it is clear that S − zI isalso strictly accretive, and therefore invertible in B(H). Thus the spectrum ofS is contained in P. We claim that (S + I)−1 belongs to the operator algebraB generated by I and S. To see this, by the basic theory of Banach algebras itis enough to show that χ(S + I) = 0 for every character of B. However by theHahn–Banach theorem any such χ extends to a state ϕ on B(H) (see A.4.2),and ϕ(S + I) = ϕ(S) + 1 = 0 (indeed Re(ϕ(S)) = ϕ(Re(S)) ≥ 0, by (A.11) say).

Using the formula

4〈Re(S)ζ, ζ〉 = ‖(S + I)ζ‖2 − ‖(S − I)ζ‖2, ζ ∈ H, (2.2)

a simple computation shows that:(1) S ∈ B(H) is strictly accretive if and only if −1 /∈ σ(S) and the Cayley

transform c(S) = (S − I)(S + I)−1 is a strict contraction.(2) T ∈ B(H) is a strict contraction if and only if 1 /∈ σ(T ) and the inverse

Cayley transform d(T ) = (I + T )(I − T )−1 is strictly accretive.Indeed (2) may be derived from (1) and the fact above that T = c(d(T )).

Proof (Of Theorem 2.1.13) It is clear that π0 is a homomorphism. We willshow that if the nth amplification πn is contractive, then so is π0

n. Fix T inMn(A1) ⊂ B(2

n(H)), with ‖T ‖ < 1. We will show that ‖π0n(T )‖ < 1.

We may write T uniquely as T = a + λ ⊗ IH , with a ∈ Mn(A) and λ ∈ Mn.By Lemma 2.1.12, we have ‖λ‖ < 1. To simplify the notation, we shall identifyλ and λ⊗ IH in the sequel. Let us apply the assertion (2) of 2.1.14 to λ. We findtwo selfadjoint matrices α, β ∈ Mn such that (I + λ)(I − λ)−1 = α + iβ, andsuch that α is positive and invertible. We may write the preceding identity as

α−1/2(I + λ)(I − λ)−1α−1/2 − iα−1/2βα−1/2 = I. (2.3)

Let us now apply 2.1.14 (2) to T . We find that (I + T )(I − T )−1 is strictlyaccretive, from which we deduce that

θ = α−1/2(I + T )(I − T )−1α−1/2 − iα−1/2βα−1/2 (2.4)

is strictly accretive. According to (2.3) we can write this operator as

θ = I + α−1/2((I + T )(I − T )−1 − (I + λ)(I − λ)−1

)α−1/2.

Now observe that (I + T )(I − T )−1 − (I + λ)(I − λ)−1 may be rewritten as

(I − T )−1((I + T )(I − λ) − (I − T )(I + λ)

)(I − λ)−1,

which is simply 2(I − T )−1a(I − λ)−1. Since (I − T )−1 belongs to the operatoralgebra generated by I and T (see 2.1.14), hence to Mn(A1), and since Mn(A)is an ideal of Mn(A1), we obtain that

(I + T )(I − T )−1 − (I + λ)(I − λ)−1 ∈ Mn(A).

Therefore θ− I belongs to Mn(A). Moreover by a fact in 2.1.14 again, (θ + I)−1

belongs to the operator algebra generated by I and θ, and hence also to Mn(A1).Hence the Cayley transform (θ − I)(θ + I)−1 belongs to Mn(A).


Since π0n is a unital homomorphism and θ + I is invertible, π0

n(θ) + I isinvertible as well. Also

(π0

n(θ) + I)−1

= π0n

((θ + I)−1

), and hence

π0n

((θ − I)(θ + I)−1

)=(π0

n(θ) − I)(

π0n(θ) + I

)−1.

By the assertion (1) of 2.1.14, the operator (θ−I)(θ+I)−1 is a strict contraction.Since it belongs to Mn(A), and since πn is contractive, we deduce that∥∥(π0

n(θ) − I)(

π0n(θ) + I

)−1∥∥ =∥∥πn

((θ − I)(θ + I)−1

)∥∥ < 1.

By 2.1.14 (2) we deduce that π0n(θ), which is the inverse Cayley transform of the

operator(π0

n(θ) − I)(

π0n(θ) + I

)−1, is strictly accretive. By (2.4),

π0n

((I + T )(I − T )−1

)= π0

n

(α1/2θα1/2 + iβ

)= α1/2π0

n(θ)α1/2 + iβ

is strictly accretive. Since π0n is a homomorphism, we have as above that

π0n

((I + T )(I − T )−1

)=(I + π0

n(T ))(

I − π0n(T )

)−1.

By 2.1.14 (1) again, we have ‖π0n(T )‖ < 1, which concludes the proof.

Corollary 2.1.15 (Meyer) Let A ⊂ B(H) be a nonunital operator algebra andlet π : A → B(K) be an isometric (resp. completely isometric) homomorphism.Then the unital homomorphism from SpanA, IH into B(K) extending π is anisometry (resp. complete isometry).

Proof By assumption, IH /∈ A. Let B = π(A) and regard π as valued in B.Since A is isomorphic to B as algebras, B is nonunital. Hence IK /∈ B. Using thenotation from Theorem 2.1.13, we deduce that π0 : SpanA, IH → SpanB, IKis a bijection, with (π0)−1 = (π−1)0. That theorem shows that π0 and (π−1)0

are both (completely) contractive. Thus π0 is a (complete) isometry.

2.1.16 (Unitization of a subalgebra) It is clear from 2.1.15 that if C is a unitaloperator algebra with unit denoted by 1C and if A ⊂ C is a nonunital subalgebra,then A1 may be taken to be SpanA, 1C ⊂ C. In particular if A, B are nonunitaloperator algebras with A ⊂ B, then the units of A1 and B1 may be identifiedand A1 may be viewed as a unital-subalgebra of B1.

2.1.17 (Unitization of an approximately unital algebra) Although the unitiza-tion A1 is now defined unambiguously, it is in general very difficult to describeits norm (or its matrix norms) intrinsically. However if A is an approximatelyunital operator algebra, and if A ⊂ B(H) nondegenerately (see 2.1.10), then forany a ∈ A and any λ ∈ C, we claim that

‖a + λIH‖ = sup‖ac + λc‖ : c ∈ A, ‖c‖ ≤ 1

.

(We note in passing, although we do not need this here, that the right-handside above is the ‘unitization norm’ from A.4.3, and that another equivalent


description of it is given there.) To prove the claim, let (et)t be a cai for A. Sinceet → IH strongly by 2.1.5, a + λIH is the strong limit of aet + λet. Hence

‖a + λIH‖ ≤ supt

‖aet + λet‖.

The other direction is easier, since for any c ∈ A with ‖c‖ ≤ 1,

‖ac + λc‖ = ‖(a + λIH)c‖ ≤ ‖a + λIH‖.

This proves the result.With the same proof we see that more generally, for any integer n ≥ 1 and

for any matrices [aij ] ∈ Mn(A) and [λij ] ∈ Mn, we have∥∥[aij + λijIH ]∥∥ = sup

∥∥[aijc + λijc]∥∥ : c ∈ A, ‖c‖ ≤ 1

. (2.5)

If A is an already unital operator algebra then Meyer’s result shows thatthere is an essentially unique unital operator algebra containing A completelyisometrically as a codimension 1 ideal. Again we write this strictly larger algebraas A1. In fact this ‘unitization’ is just the ‘∞-direct sum’ A ⊕∞ C (see 1.2.17).

Proposition 2.1.18 Let A be an approximately unital operator algebra with acai (et)t and denote the identity of A1 by 1.(1) If ψ : A1 → C is a functional on A1, then limt ψ(et) = ψ(1) if and only if

‖ψ‖ = ‖ψ|A‖.(2) Let ϕ : A → C be any functional on A. Then ϕ uniquely extends to a func-

tional of the same norm on A1.

Proof We first prove (1). Suppose that ψ : A1 → C satisfies limt ψ(et) = ψ(1).For a ∈ A and λ ∈ C, we obtain limt ψ(aet + λet) = ψ(a + λ1), and so∣∣ψ(a + λ1)

∣∣ ≤ supt

∣∣ψ(aet + λet)∣∣.

Consequently,∣∣ψ(a + λ1)∣∣ ≤ ‖ψ|A‖ sup

t‖aet + λet‖ ≤ ‖ψ|A‖‖a + λ1‖,

which proves that ‖ψ‖ = ‖ψ|A‖.Assume conversely that ‖ψ‖ = ‖ψ|A‖. We may assume that this norm is equal

to 1. By 1.2.8 there exist a Hilbert space H , a contractive unital homomorphismπ : A1 → B(H) and two unit vectors ζ, η ∈ H such that ψ(x) = 〈π(x)ζ, η〉 forany x ∈ A1. Let K = [π(A)ζ] ⊂ H and let p be the projection onto K. For anya ∈ A, we have 〈π(a)ζ, η〉 = 〈pπ(a)ζ, η〉, and so∣∣〈π(a)ζ, η〉

∣∣ =∣∣〈π(a)ζ, pη〉

∣∣ ≤ ‖π(a)ζ‖‖pη‖ ≤ ‖a‖‖pη‖.

This shows that ‖ψ|A‖ ≤ ‖pη‖. Hence, by hypothesis, ‖pη‖ = ‖η‖ = 1. Thuspη = η, that is, η ∈ K. Now recall from Lemma 2.1.9 that π(et) → p in thew∗-topology, and thus


ψ(et) = 〈π(et)ζ, η〉 −→ 〈pζ, η〉 = 〈ζ, η〉 = ψ(1).

We now deduce (2) from (1). Let ϕ ∈ A∗. Again we may write ϕ = 〈π(·)ζ, η〉;and Lemma 2.1.9 and the ‘first half’ of the last centered equation shows that(ϕ(et))t converges. Extend ϕ to a map ψ : A1 → C by defining ψ(1) = limt ϕ(et).The ‘only if’ part of (1) ensures that ‖ψ‖ = ‖ϕ‖. On the other hand, the ‘if’ partof (1) shows that this norm preserving extension is unique.

2.1.19 (States) If A is unital then a state on A is a functional ϕ on A ofnorm 1 which satisfies ϕ(1) = 1 (see A.4.2). If A is not unital but has a cai (et),then we define a state on A to be a functional ϕ on A of norm 1 which satisfieslimt ϕ(et) = 1. By the last result, such ϕ extends uniquely to a state on A1.

2.2 A FEW BASIC CONSTRUCTIONS

The class of operator algebras is stable under many of the categorical construc-tions introduced in Chapter 1. We shall now review the simplest of these in thisshort section.

2.2.1 (Direct sums) Let I be a set and let (Ai)i∈I be a family of operatoralgebras. Let A = ⊕i∈IAi be the ∞-direct sum of this family, as studied in 1.2.17.Besides being an operator space, A is a Banach algebra for the pointwise productdefined by setting the product of (ai)i and (bi)i equal to (aibi)i. Here (ai)i and(bi)i are in A. If we represent each Ai ⊂ B(Hi) as an operator algebra acting onsome Hilbert space Hi, the resulting embedding A ⊂ ⊕i∈IB(Hi) ⊂ B

(⊕2

i∈IHi

)is a homomorphism. Hence A is an operator algebra. It is clear that A is unitalif each Ai is unital.

2.2.2 (The minimal tensor product) To define this we use facts from 1.5.1and 1.5.2. Let A ⊂ B(H) and B ⊂ B(K) be two operator algebras. Equip thealgebraic tensor product A ⊗ B with the joint multiplication defined by letting(∑

i

ai ⊗ bi

)(∑j

cj ⊗ dj

)=∑i,j

aicj ⊗ bidj , (2.6)

for any finite families (ai)i, (cj)j in A and (bi)i, (dj)j in B. It is clear that thenatural embedding A ⊗ B ⊂ B(H ⊗2 K) is an algebra homomorphism. Passingto the completion, we see that A ⊗min B is an operator algebra. If A and B areunital with identities denoted by 1A and 1B, then A ⊗min B is unital as well,with identity equal to 1A ⊗ 1B. It is also worthwhile to observe that if A and Bare approximately unital, then the same holds for A ⊗min B. Indeed if (et)t and(fs)s are cai’s for A and B respectively, then by a simple density argument, anyy ∈ A ⊗min B satisfies

lims

limt

(et ⊗ fs)y = lims

limt

y(et ⊗ fs) = y.

58 A few basic constructions

It follows from the ‘functoriality’ property in 1.5.1 that if π : A → B(H) andρ : B → B(K) are completely contractive homomorphisms, then there is a uniquecompletely contractive linear map

π ⊗ ρ : A ⊗min B −→ B(H) ⊗min B(K) ⊂ B(H ⊗2 K)

with (π ⊗ ρ)(a ⊗ b) = π(a) ⊗ ρ(b) for a ∈ A, b ∈ B. By an obvious densityargument one sees that since π ⊗ ρ is a homomorphism on A ⊗ B, it is also ahomomorphism on A ⊗min B.

2.2.3 (Matrix algebras) A special case of the minimal tensor product of par-ticular interest, is the case when B = B(K), or equivalently when B = MI fora cardinal I. If n is a finite integer then Mn(A) ⊂ Mn(B(H)) = B(2

n(H)) is anoperator algebra on 2

n(H), the product of two elements [aij ] and [bij ] of Mn(A)being given by

[aij ][bij ] =[ n∑

k=1

aikbkj

]. (2.7)

We recall from (1.36) that

Mn ⊗min A = Mn(A)

as operator spaces, and this isomorphism is easily seen to be a homomorphismtoo. Thus they are ‘equal as operator algebras’. This result extends with a similarproof to the case when I is an infinite cardinal or set. Indeed recall from 1.2.26that KI(A) is defined to be the closure of the subalgebra M

finI (A) of the algebra

MI(B(H)) = B(2I(H)). Thus KI(A) is an operator algebra on 2

I(H) and

KI ⊗min A = KI(A) as operator algebras. (2.8)

Note that if [aij ] and [bij ] are two (infinite) matrices belonging to KI(A), thentheir product is equal to the matrix [cij ] whose entries are defined by the ex-pression cij =

∑k∈I aikbkj ∈ A, the latter sum being norm convergent. The last

fact follows from the discussion preceding (1.21).On the other hand, MI(A) need not be an algebra at all. However there are

several operator algebras inside MI(A) which occasionally play a role. For exam-ple, the subsets of MI(A) corresponding to Cw

I (RI(A)), RwI (CI(A)), CI(Rw

I (A))and RI(Cw

I (A)) are all operator algebras. We write RwI (CI(A)) as M

wrI (A); and

we write CwI (RI(A)) as M

wcI (A). These two operator algebras contain the other

two algebras in the last list as subalgebras. These facts will not play much of arole for us, and we leave the details as an exercise. Note that M

wcI (A) ∩M

wrI (A)

is also an operator algebra, and it contains the operator algebra A ⊗min MI asmay be seen by inspecting rank 1 tensors in the latter space.

2.2.4 (Operator algebra valued continuous functions) Let Ω be a compactspace, and A an operator algebra. The operator space C(Ω; A) discussed in 1.2.18is an operator algebra for the product defined by (fg)(t) = f(t)g(t). Namely,


if A is a subalgebra of B(H) then C(Ω; A) is a subalgebra of the C∗-algebraC(Ω; B(H)). If A is unital, then C(Ω; A) is unital as well, the identity beingthe constant function equal to the identity of A. According to (1.39), we havea completely isometric identification C(Ω) ⊗min A = C(Ω; A). We notice herethat this identification holds as well ‘as operator algebras’. This is equivalent tosaying that the linear mapping taking a finite sum

∑k gk ⊗ ak ∈ C(Ω) ⊗ A to

the function f ∈ C(Ω; A) defined by f(t) =∑

k gk(t)ak is a homomorphism ifC(Ω)⊗min A and C(Ω; A) are equipped with the multiplications defined in 2.2.2and above respectively.

Similarly if Ω is merely a locally compact space, then the operator spaceC0(Ω; A) is an operator algebra for the pointwise product. As operator algebras,we have that C0(Ω) ⊗min A = C0(Ω; A).

2.2.5 (Uniform algebras) By definition, a (concrete) uniform algebra is a unital-subalgebra of C(Ω), for some compact space Ω. In this book, we will consider anyuniform algebra as endowed with its minimal operator space structure (describedby the formula (1.3)). Then an (abstract) uniform algebra is a unital operatoralgebra which is completely isometrically isomorphic to a concrete uniform al-gebra. In this way we regard uniform algebras as a subclass of the operatoralgebras. A characterization of the algebras in this subclass will be given in 3.7.9(see also the Notes to Section 4.6).

More generally, we will use the term function algebra for an operator algebraA for which there exists a compact space Ω and a completely isometric homo-morphism π : A → C(Ω). Any function algebra is a minimal operator space.

2.2.6 (Disc algebra) This fundamental example of a uniform algebra has twoequivalent definitions. Let us denote by D and T the open unit disc of C andthe unitary complex group T = z ∈ C : |z| = 1 respectively. Then the discalgebra A(D) is the subalgebra of C(D) consisting of all continuous functionsF : D → C, whose restriction to D is analytic. By the maximal modulus theorem,the restriction of functions in A(D) to the boundary T is an isometry. Hence wemay alternatively regard A(D) ⊂ C(T) as a uniform algebra acting on T. In thatrepresentation, A(D) consists of all elements of C(T) whose harmonic extensionto D given by the Poisson integral is analytic. Equivalently, given any f ∈ C(T),we associate Fourier coefficients

f(k) =∫T

f(z)z−kdm(z), k ∈ Z. (2.9)

Then A(D) ⊂ C(T) is the closed subalgebra of all f ∈ C(T) such that f(k) = 0for every k < 0. The (analytic) polynomials form a dense subalgebra of A(D).

2.2.7 (Operator algebra valued analytic functions) Let X be an operator space.For any f ∈ C(T; X), we may define Fourier coefficients f(k) ∈ X by means of(2.9). According to 2.2.6, we let A(D; X) ⊂ C(T; X) denote the subspace of all f

such that f(k) = 0 for every k < 0. Equivalently, as in the scalar case, A(D; X)

60 A few basic constructions

may be defined to be the subspace of C(D; X) consisting of all functions F whoserestriction to D is analytic. We have

A(D) ⊗min X = A(D; X)

completely isometrically. Indeed both spaces may be regarded as the closure ofA(D)⊗X in C(T; X). (The density of A(D)⊗X in A(D; X) follows from a routineargument using Fejer kernels.) If X = B is an operator algebra, it follows fromeither of these definitions that A(D; B) is an operator algebra which is both asubalgebra of C(T; B) and C(D; B). In that case the above identification holds‘as operator algebras’.

2.2.8 (Opposite and adjoint algebras) We will use notations from 1.2.25. If A isan operator algebra, then the operator space Aop from 1.2.25, is also an operatoralgebra for the reversed product defined on A by a b = ba. Indeed, if A isa subalgebra of a C∗-algebra B, then Aop may be identified with the matchingsubalgebra of the C∗-algebra Bop discussed in 1.2.25. We remark in passing thatif A = Aop, and if A is approximately unital, then A is commutative. This followsquickly from one of our later ‘Banach-Stone theorems’ (e.g. see 4.5.13) appliedto the identity map from A to Aop.

Similarly, the adjoint operator space A (written also as A∗ when there is nopossible confusion with the dual space) from 1.2.25, is an operator algebra, withproduct a∗b∗ = (ba)∗, for a, b ∈ A. Indeed if A is a subalgebra of a C∗-algebraB, then A may be identified with the subalgebra a∗ : a ∈ A of B. Note thatif A has a cai (et)t, then (e∗t )t is a cai for A∗.

If π : A → B is a completely contractive homomorphism between operatoralgebras, then the canonical completely contractive mappings πop : Aop → Bop

and π : A → B from 1.2.25 also are homomorphisms. In particular, givena completely contractive representation π : A → B(H), we obtain a canonicalcompletely contractive representation π : A → B(H). It easily follows fromLemma 2.1.9 that if A is approximately unital and π is nondegenerate, then π∗

is nondegenerate as well.

2.2.9 Let X ⊂ B(H) be an operator space acting on some Hilbert space H .Then the operator space defined inside B(H ⊕ H) = M2(B(H)) as[

0 x0 0

]: x ∈ X

is completely isometric to X . It is also a subalgebra of B(H (2)), with trivial (i.e.zero) product. Thus any operator space may be regarded as an operator algebra.

2.2.10 The last construction has a unital version which turns out to be muchmore useful. Again we consider an operator space X ⊂ B(H) acting on someHilbert space H . We define

U(X) =[

λ1 x0 λ2

]: x ∈ X, λ1, λ2 ∈ C

⊂ B(H ⊕ H), (2.10)


where λ1 and λ2 stand for the operators λ1IH and λ2IH respectively. By defini-tion, U(X) may be regarded as a subspace of the Paulsen system S(X) definedin 1.3.14. It therefore follows from 1.3.15 that as an operator space, U(X) onlydepends on X and not on the Hilbert space H on which it acts.

For any x, y ∈ X and any λ1, λ2, µ1, µ2 ∈ C, we have[λ1 x0 λ2

] [µ1 y0 µ2

]=[

λ1µ1 λ1y + µ2x0 λ2µ2

].

Hence U(X) is a subalgebra of B(H (2)), and the multiplication on U(X) doesnot depend on H . Thus U(X) is a unital operator algebra whose definition onlydepends on X . The norm on U(X) also only depends on the norm on X (andnot on the full operator space structure of X), as we shall see below.

Proposition 2.2.11 Let X and Y be operator spaces, and let u : X → Y be alinear contraction (resp. complete contraction). Then the mapping θu from U(X)to U(Y ) defined by

θu

([λ1 x0 λ2

])=

[λ1 u(x)0 λ2

], x ∈ X, λ1, λ2 ∈ C,

is a contractive (resp. completely contractive) homomorphism.

Proof It is clear that θu is a homomorphism. The ‘complete contraction case’follows immediately from 1.3.15. To see the ‘contraction case’, we explicitly com-pute the norm of a matrix in U(X). We have∣∣∣∣∣∣∣∣[λ1 x

0 λ2

]∣∣∣∣∣∣∣∣2 = sup‖λ1ζ + xη‖2 + |λ2|2‖η‖2 : ‖ζ‖2 + ‖η‖2 ≤ 1

.

Here ζ, η ∈ H , where H is a Hilbert space on which X is represented completelyisometrically. It is easily seen that this last quantity equals

sup(

|λ1|√

1 − ‖η‖2 + ‖xη‖)2

+ |λ2|2‖η‖2 : ‖η‖ ≤ 1,

(the one inequality being obvious, and the other is obtainable by letting ζ equaleiθ

√1 − ‖η‖2 xη

‖xη‖ , for real θ). In turn the last centered quantity equals

sup(

|λ1|√

1 − t2 + ‖xη‖)2

+ |λ2|2t2 : t ∈ [0, 1], ‖η‖ = t

= sup(

|λ1|√

1 − t2 + ‖x‖t)2

+|λ2|2t2 : t ∈ [0, 1].

This last expression, for fixed λ1, λ2, may be regarded as a function f(‖x‖). Sincef is an increasing function, the desired result follows easily.

Corollary 2.2.12 If X and Y are two isometric (resp. completely isometric)operator spaces, then U(X) and U(Y ) are isometric (resp. completely isometric)operator algebras.

62 The abstract characterization of operator algebras

Proof We need only prove the isometric case. Given an isometric isomorphismJ : X → Y , apply Lemma 2.2.11 to J and J−1 to obtain that uJ : U(X) → U(Y )is an invertible isometric homomorphism.

2.2.13 (Ultraproducts) Let U be an ultrafilter on a set I, and consider a familyof operator algebras (Ai)i∈I . We let A be their ultraproduct along U , as definedin 1.2.31. Assume that each Ai is a subalgebra of some B(Hi). Then it followsfrom the last few results in 1.2.31 that A is an operator algebra, a subalgebra ofthe C∗-algebra

∏i∈I B(Hi)/U . Clearly A is unital if each Ai is unital.

2.3 THE ABSTRACT CHARACTERIZATION OF OPERATORALGEBRAS

The main result of this section is Theorem 2.3.2, usually referred to as the BRStheorem. This fundamental result gives a criterion for a unital (or more generallyan approximately unital) Banach algebra with an operator space structure, to bean operator algebra. Among other things, the BRS theorem allows one to checkthat abstract constructions with operator algebras remain operator algebras. Wediscuss some of the most important such constructions later in this section.

2.3.1 Recall from 2.2.3 that if A is an operator algebra, and if n ≥ 1 is an inte-ger, then Mn(A) is an operator algebra with product given by (2.7). In particularit is a Banach algebra, so that the product on A is a completely contractive bi-linear map in the sense described in 1.5.4. This is equivalent (since the Haageruptensor product linearizes completely bounded bilinear maps—see 1.5.4) to sayingthat the multiplication mapping m : A ⊗ A → A on A extends to a completelycontractive mapping on the Haagerup tensor product A ⊗h A. That is,∥∥m : A ⊗h A −→ A

∥∥cb

≤ 1. (2.11)

The BRS theorem asserts that this property characterizes operator algebras,at least for unital or approximately unital algebras. We will see in 5.3.6 that thischaracterization may fail without the approximately unital assumption.

Theorem 2.3.2 (BRS theorem) Let A be an operator space which is also an ap-proximately unital Banach algebra. Let m : A⊗A → A denote the multiplicationon A. The following are equivalent:

(i) The mapping m : A ⊗h A −→ A is completely contractive.(ii) For any n ≥ 1, Mn(A) is a Banach algebra. That is,∥∥∥[ n∑

k=1

aikbkj

]∥∥∥Mn(A)

≤∥∥[aij ]

∥∥Mn(A)

∥∥[bij ]∥∥

Mn(A)

for any [aij ] and [bij ] in Mn(A).(iii) A is an operator algebra, that is, there exist a Hilbert space H and a com-

pletely isometric homomorphism π : A → B(H).


Proof The equivalence between (i) and (ii), and that (iii) implies these, wasmentioned in 2.3.1. To see that (i) and/or (ii) implies (iii), we let A be as in (i),and we will first assume that A is unital, with identity e of norm 1.

Fix a complete isometry u0 : A → B(H0), for some Hilbert space H0, andapply Theorem 1.5.7 (1) to the mapping u0m : A ⊗h A → B(H0). We obtain aHilbert space H1, and two completely contractive mappings u1 : A → B(H0, H1)and v1 : A → B(H1, H0), such that u0m(b, a) = v1(b)u1(a) for any a, b ∈ A.Since ‖m‖cb ≤ 1 and ‖u1‖cb ≤ 1, the mapping u1m : A ⊗h A → B(H0, H1)is completely contractive. By Theorem 1.5.7 (1) again, we may factor u1m aswell as a product of two mappings defined on A. Proceeding by induction, weobtain that for any k ≥ 1, there exist a Hilbert space Hk, and two completelycontractive mappings uk : A → B(H0, Hk) and vk : A → B(Hk, Hk−1), such that

ukm(b, a) = vk+1(b) uk+1(a) k ≥ 0, a, b ∈ A. (2.12)

For k ∈ N define a seminorm | · |k on the algebraic tensor product A ⊗ H0 by∣∣∣∑i

ai ⊗ ξi

∣∣∣k

=∥∥∥∑

i

uk(ai)ξi

∥∥∥Hk

for any finite families (ai)i in A and (ξi)i in H0. These seminorms satisfy thefollowing key inequalities. For any integers k ≥ 0, n ≥ 1, m ≥ 1, and for any[bpq] ∈ Mn(A), [aij ] ∈ Mn,m(A), [ξij ] ∈ Mn,m(H0), we have∑

p

∣∣∣∑q,i

m(bpq, aqi) ⊗ ξqi

∣∣∣2k≤

∥∥[bpq]∥∥2

Mn(A)

∑q

∣∣∣∑i

aqi ⊗ ξqi

∣∣∣2k+1

. (2.13)

Indeed according to the definition of | · |k, we have∑p

∣∣∣∑q,i


∣∣∣2k

=∑

p

∥∥∥∑q,i

ukm(bpq, aqi)ξqi

∥∥∥2

=∑

p

∥∥∥∑q,i

vk+1(bpq)uk+1(aqi)ξqi

∥∥∥2

by (2.12)

≤∥∥[vk+1(bpq)]

∥∥2 ∑q

∥∥∥∑i

uk+1(aqi)ξqi

∥∥∥2

=∥∥[vk+1(bpq)]

∥∥2 ∑q

∣∣∣∑i

aqi ⊗ ξqi

∣∣∣2k+1

.

Since vk+1 is completely contractive, this yields (2.13). Applying (2.13) withn = 1 and b11 = e, we see that∣∣∣∑

i

ai ⊗ ξi

∣∣∣k≤∣∣∣∑

i

ai ⊗ ξi

∣∣∣k+1


for any finite families (ai)i in A and (ξi)i in H0. This monotonicity allows us todefine a new seminorm | · |∞ on A ⊗ H0 by letting∣∣∣∑

i

ai ⊗ ξi

∣∣∣∞

= limk→∞

∣∣∣∑i

ai ⊗ ξi

∣∣∣k

= supk

∣∣∣∑i

ai ⊗ ξi

∣∣∣k.

Passing to the limit in (2.13), we clearly obtain that∑p

∣∣∣∑q,i


∣∣∣2∞

≤∥∥[bpq]

∥∥2

Mn(A)

∑q

∣∣∣∑i

aqi ⊗ ξqi

∣∣∣2∞

(2.14)

for any [bpq] ∈ Mn(A), [aqi] ∈ Mn,m(A), and [ξqi] ∈ Mn,m(H0).Now observe that | · |∞ is a prehilbertian seminorm. Indeed each Hk is a

Hilbert space, hence |·|k is prehilbertian for any k ≥ 0. Thus each |·|k satisfies theparallelogram identity. Passing to the limit, we obtain that | · |∞ also satisfies thisidentity, which proves the observation. We may therefore define a Hilbert space Hby first taking the quotient of A⊗H0 by the kernel N = z ∈ A⊗H0 : |z|∞ = 0and then completing the resulting normed space (A⊗H0)/N . Using (2.14) withn = 1, we see that for any b ∈ A, one may define a bounded linear mappingπ(b) : H → H with ‖π(b)‖ ≤ ‖b‖ by letting

π(b)(∑

i

ai ⊗ ξi +N)

=∑

i

m(b, ai) ⊗ ξi +N, ai ∈ A, ξi ∈ H0.

Then applying (2.14) with arbitrary n ≥ 1, we find that the resulting linear map-ping π : A → B(H) is actually a complete contraction. Since m is a multiplicationour mapping π is clearly a homomorphism.

That π is a complete isometry now reduces to showing that∥∥[bpq]∥∥ ≤

∥∥[π(bpq)]∥∥ (2.15)

for n ≥ 1 and [bpq] ∈ Mn(A). Assume that∥∥[bpq]

∥∥ > 1. Since u0 : A → B(H0) isa complete isometry, there exist ξ1, . . . , ξn ∈ H0 such that∑

i

‖ξi‖2 = 1 and∑

p

∥∥∥∑q

u0(bpq)ξq

∥∥∥2

> 1.

Since the sequence of seminorms | · |k is nondecreasing, for any p we have∣∣∣∑q

bpq ⊗ ξq

∣∣∣∞

≥∣∣∣∑

q

bpq ⊗ ξq

∣∣∣0

=∥∥∥∑

q

u0(bpq)ξq

∥∥∥.Using the identity π(b)(e ⊗ ξ +N) = b ⊗ ξ +N for b ∈ A and ξ ∈ H0, we have∑

p

∥∥∥∑q

π(bpq)(e ⊗ ξq +N

)∥∥∥2

> 1.


On the other hand, since ‖e‖ = 1 and each uk is a contraction, we have∑q

∥∥e ⊗ ξq +N∥∥2 = lim

k

∑q

‖uk(e)ξq‖2 ≤∑

q

‖ξq‖2 = 1.

This shows that∥∥[π(bpq)]

∥∥ > 1 and hence the inequality (2.15), which concludesour proof in the unital case.

Finally, we consider the case that A is nonunital, but has a cai (et)t. LetA1 = A ⊕ C be the Banach algebraic unitization of A considered in A.4.3, withunit denoted by e. More generally, we define matrix norms on A1 by letting∥∥[aij + λije]

∥∥Mn(A1)

= sup∥∥[aijc + λijc]

∥∥Mn(A)

: c ∈ A, ‖c‖ ≤ 1

(2.16)

for any integer n ≥ 1 and for any matrices [aij ] ∈ Mn(A) and [λij ] ∈ Mn. By1.2.16 these matrix norms satisfy Ruan’s axioms 1.2.12, and hence define anoperator space structure on A1. Moreover for any [aij ] ∈ Mn(A) we have∥∥[aij ]

∥∥ = limt

∥∥[aijet]∥∥ ≤ sup

‖c‖≤1

∥∥[aijc]∥∥ ≤

∥∥[aij ]∥∥.

Thus we may regard A as a subalgebra of A1 completely isometrically. Given[aij ], [bij ] ∈ Mn(A) and [λij ], [µij ] ∈ Mn, note that by (2.16),∥∥[aij + λije][bij + µije]

∥∥ = sup∥∥[aij + λije][bijc + µijc]

∥∥Mn(A)

: c ∈ Ball(A),

Moreover, given any c ∈ A we have∥∥[aij + λije][bijc + µijc]∥∥

Mn(A)= lim

t

∥∥[aijet + λijet][bijc + µijc]∥∥

Mn(A).

If Mn(A) is a Banach algebra, we deduce that∥∥[aij + λije][bijc + µijc]∥∥

Mn(A)≤ sup

t

∥∥[aijet + λijet]∥∥∥∥[bijc + µijc]

∥∥≤∥∥[aij + λije]

∥∥∥∥[bijc + µijc]∥∥

by (2.16). Taking the supremum over all c ∈ A with ‖c‖ ≤ 1, shows that Mn(A1)is a Banach algebra. By the first part of the proof (the unital case), we see thatA1 is an operator algebra, and hence so too is its subalgebra A.

2.3.3 (Applications) We now apply the BRS theorem to show that variousnatural constructions produce operator algebras. As a first example, we note thatin analogy to 1.2.16, we may define an operator algebra seminorm structure on aunital algebra A. Namely, this is an operator seminorm structure in the sense of1.2.16, such that ‖1‖1 = 1, and such that the nth seminorm is submultiplicativefor every n ∈ N. We let N be the nullspace of ‖·‖1 as usual. An argument similarto the one in 1.2.16, but using the BRS theorem instead of Ruan’s theorem, showsthat (the completion of) A/N is a unital operator algebra.


We apply this principle to suprema and limsup’s of operator algebra struc-tures, just as we did in 1.2.16. Suppose that A is a unital algebra, that Λ is aset, and that for each k ∈ Λ we have a unital operator algebra Bk and a uni-tal homomorphism πk : A → Bk. We suppose that for all a ∈ A, the quantitysup‖πk(a)‖ : k ∈ Λ is finite. We may then define an operator algebra seminormstructure on A by

‖a‖n = supk

‖(πk)n(a)‖n

for a ∈ Mn(A). By the last paragraph, (the completion of) a quotient of A is anoperator algebra with these matrix norms. If Λ is a directed set, then replacingthe sup by lim sup in the last equation yields another operator algebra.

Throughout this book, a closed two-sided ideal of an operator algebra willsimply be called an ideal. Long before the field of operator spaces arose, it wasknown that if J is an ideal of an operator algebra A, then the quotient algebraA/J is isometrically isomorphic to an operator algebra (see the Notes section forreferences). This fact follows as an immediate corollary of the BRS theorem:

Proposition 2.3.4 Let J be an ideal in an operator algebra A. Equip the quo-tient Banach algebra A/J with its quotient operator space structure (see 1.2.14).Then A/J is an operator algebra. That is, there exist a Hilbert space H and acompletely isometric homomorphism π : A/J → B(H).

Proof Assume that A is a subalgebra of B(K). Then J is also an ideal ofthe unital operator algebra A1 = SpanA, IK; and the canonical embeddingA/J ⊂ A1/J is both a complete isometry and a homomorphism. ReplacingA with A1 if necessary, we may therefore assume that A is unital. Then A/Jis unital, and hence it suffices to show that A/J satisfies the assertion (ii) ofTheorem 2.3.2. This is readily seen from the fact that A satisfies (ii), and fromthe definition of the quotient operator space structure.

2.3.5 (Factor theorem) As in 1.2.15, it follows that if π : A → B is a completelybounded homomorphism, and if J is an ideal in A contained in Ker(π), then πdescends to a homomorphism A/J → B with a ‘cb-norm’ which is no larger thanthat of π.

2.3.6 (Interpolation) We shall apply Proposition 2.3.4 to the complex interpo-lation of operator algebras. We first need a few facts concerning interpolation ofBanach algebras. Consider a compatible couple of Banach spaces (A0, A1) andrecall from 1.2.30 the definition of the ‘interpolation space’ Aθ = [A0, A1]θ forθ ∈ (0, 1). If A0 and A1 are Banach algebras, if A0 + A1 is a C-Banach algebrafor some C ≥ 1, and if the embeddings A0 → A0 + A1 and A1 → A0 + A1

are both homomorphisms, then we say that (A0, A1) is a compatible couple ofBanach algebras. These conditions roughly mean that our two Banach algebrasA0 and A1 have a ‘common multiplication’. In this case, this multiplication isboth a contractive bilinear map from A0 × A0 into A0 and from A1 × A1 intoA1; hence by interpolation, the multiplication extends to a contractive bilinear


map Aθ ×Aθ → Aθ for any θ ∈ (0, 1). Each Aθ is therefore a Banach algebra forthis common multiplication.

The Banach algebra structure of Aθ has a useful alternative description interms of quotients. Indeed, consider the Banach space F = F(A0, A1) definedin 1.2.30 (for A0 = X0 and A1 = X1), and observe that F is an algebra for thepointwise multiplication. Then the space Fθ = f ∈ F : f(θ) = 0 ⊂ F is anideal of F and we may therefore consider the quotient algebra F/F θ. Then it isclear that the isometric identification

Aθ = F/Fθ (2.17)

provided by (1.23) holds at the algebraic level. Indeed if x = f(θ) and y = g(θ)are two arbitrary elements of Aθ, with f, g ∈ F , then xy = f(θ)g(θ) = (fg)(θ).

If (A0, A1) is a compatible couple of Banach algebras and if A0 and A1

are operator algebras, then we will say that (A0, A1) is a compatible couple ofoperator algebras. In that case, Aθ = [A0, A1]θ is both a Banach algebra (by theabove discussion), and an operator space (by 1.2.30).

Proposition 2.3.7 Let (A0, A1) be a compatible couple of operator algebras.Then Aθ = [A0, A1]θ is an operator algebra for any θ ∈ (0, 1).

Proof We fix some θ ∈ (0, 1) and apply 1.2.30 and 2.3.6. The embedding

F ⊂ C0(R; A0) ⊕∞ C0(R; A1)

is obviously a homomorphism; hence F is an operator algebra by 2.2.4. Thus Aθ

is an operator algebra by (2.17) and 2.3.4.

2.3.8 (Direct limits) A similar construction to 2.3.3 yields the direct (or in-verse) limit of operator algebras. There are several variants on this, for simplicitywe sketch only the case of unital operator algebras and unital maps. Let Λ be adirected set, suppose that we have operator algebras An for each n ∈ Λ, and sup-pose that for each m ≥ n we have completely contractive unital homomorphismsπmn : An → Am with the property that πmnπnk = πmk. We suppose that πnn isthe identity map, for all n. We consider the disjoint union of the An’s, and takeequivalence classes under the relation x ≡ y if and only if x ∈ An, y ∈ Am, andthere exists a k ∈ Λ with πkn(x) = πkm(y). Of course addition and multiplicationis well-defined on equivalence classes. Define a matrix seminorm structure on theset of equivalence classes by ‖[xij ]‖ = lim supm‖[πmn(xij)]‖, where n is suchthat xij ∈ An for all i, j. We quotient by the nullspace of this seminorm, anddenote the resulting space by lim→ An. Appealing to the BRS theorem in a similarway to 2.3.3, we find that this space is again an operator algebra. We write in forthe canonical completely contractive unital homomorphism An → lim→ An. Weleave as an exercise the fact that the algebra lim→ An has the appropriate ‘direct

limit universal property’ (analoguous to that in [368] or Appendix L in [423]).

68 Universal constructions of operator algebras

2.3.9 (Matrix normed algebras) Let A be a Banach algebra which is also anoperator space. We say that A is a matrix normed algebra if∥∥[aijbkl]

∥∥Mmp(A)

≤∥∥[aij ]

∥∥Mn(A)

∥∥[bkl]∥∥

Mp(A)

for [aij ] ∈ Mn(A) and [bkl] ∈ Mp(A). In this definition, the rows of [aijbkl] areindexed by (i, k), whereas its columns are indexed by (j, l). From 1.5.11 the lattercondition is equivalent to saying that the multiplication mapping m : A⊗A → A

extends to a completely contractive mapping on A⊗ A. That is,∥∥m : A

⊗ A −→ A

∥∥cb

≤ 1.

Since the operator space projective tensor norm dominates the Haagerup tensornorm (1.5.13), any operator algebra is a matrix normed algebra. The conversehowever is far from being true. Indeed note that if A is a Banach algebra, thenMax(A) is a matrix normed algebra, as is clear from the identity 1.5.12 (1). Thusany Banach algebra may be regarded as a matrix normed algebra. However ofcourse most Banach algebras are not isomorphic to an operator algebra (e.g. see5.1.5 for more on this issue).

A good example of a matrix normed algebra is CB(X) for an operator spaceX . With its canonical matrix norms (1.6) it is fairly obvious that CB(X) isa matrix normed algebra. However we shall see in 5.1.9 that if CB(X) is anoperator algebra then X must be a Hilbert space.

2.4 UNIVERSAL CONSTRUCTIONS OF OPERATOR ALGEBRAS

Various classical universal constructions in the C∗-theory have natural exten-sions to the nonselfadjoint framework. In this section we will present some ofthese, and show as an application of the BRS theorem that they may be char-acterized ‘internally’ in terms of certain ‘factorization formulae’. For example,following [67], we will consider the universal operator algebra of a semigroup,and the free product of operator algebras. Some other universal constructionswill appear later on in the book (for example, the maximal tensor product ofoperator algebras will be discussed in Chapter 6).

2.4.1 (Ordering and isomorphism of C∗-covers) Let A be an operator algebra. If(B, j) and (B′, j′) are C∗-covers of A (see 2.1.1), we then declare (B, j) ≤ (B ′, j′)if and only if there is a ∗-homomorphism π : B′ → B such that π j′ = j. ByA.5.8, such a ∗-homomorphism π is automatically surjective and unique (this isbecause the ∗-algebra generated by A is dense in both B and B ′, and π is uniquelydetermined there, and is surjective). We say that (B, j) is A-isomorphic to (B ′, j′)if a π exists as above which is also one-to-one (and therefore a ∗-isomorphismby A.5.8). This is an equivalence relation, and we define C(A) to be the set ofequivalence classes of C∗-covers of A, with the ordering above. In this ordering,we will see next that there is a largest element of C(A). This C∗-cover we think


of as the noncommutative maximal ideal space of A. It is much more difficult(see Section 4.3) to see that there is a smallest element of C(A) in this ordering,known as the C∗-envelope of A.

Proposition 2.4.2 Let A be an operator algebra. Then there exists a C∗-cover(C∗

max(A), j) of A with the following universal property: if π : A → D is anycompletely contractive homomorphism into a C∗-algebra D, then there exists a(necessarily unique) ∗-homomorphism π : C∗

max(A) → D such that π j = π.

Proof It suffices to assume that D = B(H), for a Hilbert space H . Supposethat the cardinality of A is less than or equal to a cardinal β that we choose suchthat βℵ0 = β. Define F to be the set of completely contractive representationsπ : A → B(2

J) where J varies over the cardinals which are less than or equalto β. We write Hπ = 2

J . Define j = ⊕π : π ∈ F, that is, j(a) = ⊕π∈F π(a)for all a ∈ A. This is a completely contractive representation of A on a Hilbertspace Hmax = ⊕π∈F Hπ. In fact j is also completely isometric, as may be seenby standard arguments. The projection of Hmax onto its ‘πth coordinate’ will bewritten as Pπ . We define C∗

max(A), or C∗(A) for brevity, to be the C∗-algebrainside B(Hmax) generated by j(A).

Note that if θ : A → B(H) is any completely contractive representation of A,with dim(H) ≤ β, then there is a unitary U on H such that ρ = U ∗θ(·)U ∈ F .Define ρ : C∗(A) → B(Hρ) to be ρ(T ) = Pρ T|Hρ

. Then ρ is a ∗-homomorphismdefined on C∗(A), and ρ j = ρ. Then θ = Uρ(·)U∗ is a ∗-homomorphismC∗(A) → B(H), and θ j = θ. Clearly θ is the unique such ∗-homomorphism.

Thus we have shown that C∗(A) has the desired universal property, but forextending completely contractive representations of A on Hilbert spaces of di-mension ≤ β. From this fact it is not hard to show that C∗(A) has this universalproperty when H has arbitrary dimension. We briefly sketch the argument. Forπ : A → B(H) as above, we let J0 be a set whose cardinality is equal to dim(H).Then we consider the set G of pairs (J, Kjj∈J), where J is a subset of J0,the Kj’s are mutually orthogonal nonzero subspaces of H with dim(Kj) ≤ β forany j ∈ J , and each Kj is reducing for π(A). Recall that this means that Kj

is invariant for both π(A) and π(A)∗. The set G has a natural order, namely(J, Kjj∈J) ≤ (J ′, K ′

jj∈J′) if J ⊂ J ′ and Kj = K ′j for any j ∈ J . Apply-

ing a routine Zorn’s lemma argument, we see that G has a maximal element(J, Kjj∈J). Since we supposed that Card(A) ≤ β and βℵ0 = β, it is not hardto check that ⊕jKj = H . Indeed if x ∈ H is orthogonal to each Kj, and ifC ⊂ B(H) is the C∗-algebra generated by π(A), then [Cx] is reducing for π(A)and has dimension less than or equal to β. This is a contradiction. We leavethe details to the reader, Thus π may be written as a direct sum of a family ofcompletely contractive representations on Hilbert spaces of dimension ≤ β. Nowit is clear that π has the desired extension π.

2.4.3 (The maximal C∗-algebra) The algebra C∗max(A) in 2.4.2 is called the

maximal or universal C∗-algebra of A. Algebraically, the universal property in


2.4.2 is saying that the functor A → C∗max(A) is the left adjoint to the forgetful

functor from the category of C∗-algebras to the category of operator algebras.If A is unital, then (C∗

max(A), j) is unital and it is easy to check that it isalso characterized by the following universal property: if π : A → D is any unitalcompletely contractive homomorphism into a unital C∗-algebra D, then thereexists a (necessarily unique and unital) ∗-representation π : C∗

max(A) → D suchthat π j = π.

It is also worth noticing that if A is not unital and if C∗max(A

1) is the maximalC∗-algebra of its unitization, then C∗

max(A) coincides with the C∗-subalgebra ofC∗

max(A1) generated by A. Indeed let C be the latter C∗-algebra and let j be

the canonical embedding A1 → C∗max(A

1). We consider a completely contractiverepresentation π : A → B(H). By Meyer’s theorem 2.1.13 there is a completelycontractive unital homomorphism π0 : A1 → B(H) extending π. By the universalproperty of C∗

max(A1), there is a ∗-homomorphism ρ : C∗

max(A1) → B(H) such

that ρ j = π0. The restriction of ρ to C is the desired extension of π.

2.4.4 (The universal representation) We define the universal representationπu of a possibly nonselfadjoint operator algebra A to be the restriction of theuniversal representation (see A.5.5) of C∗

max(A) to A.

2.4.5 (An example) Consider A = T2, the upper triangular 2 × 2 matrices.According to Corollary 2.2.12, A = U(C) is completely isometrically isomorphicto the subalgebra of M2(C([0, 1])) defined as[

λ1 µ√·

0 λ2

]: λ1, λ2, µ ∈ C

⊂ M2(C[0, 1]),

where√· is the square root function on [0, 1]. We henceforth identify A with this

subalgebra. Claim: C∗max(A) = f ∈ M2(C([0, 1])) : f(0) is a diagonal matrix.

We will prove this claim by showing that the last C∗-algebra has the property ofProposition 2.4.2. First we observe that the ∗-algebra E generated by A is densein the space of matrices of the form[

b1 b2

√·b3√· b4

]with bi ∈ C([0, 1]), and the latter space is dense in the algebra of functionsf ∈ M2(C([0, 1])) such that f(0) is a diagonal matrix. Then we consider anondegenerate (completely) contractive representation π : A → B(L) on someHilbert space L. This immediately gives a decomposition of L as a Hilbert spacesum H ⊕ K say, and a contractive operator T : K → H , namely T = π(E12), so

that π maps any[

λ1 µ√·

0 λ2

]in A to

[λ1IH µT

0 λ2IK

]. It is easily checked that

θ :[

b1 b2

√·b3√· b4

]−→

[b1(TT ∗) b2(TT ∗)T

T ∗b3(TT ∗) b4(T ∗T )

], bi ∈ C([0, 1]),


is a ∗-homomorphism from E into B(H ⊕ K) = B(L). Also, θ is bounded; forexample, ‖b2(TT ∗)T ‖ = ‖b2(TT ∗)(TT ∗)

12 ‖ ≤ ‖b2

√·‖C([0,1]) by spectral theory.Hence θ extends to a ∗-homomorphism on the containing C∗-algebra.

2.4.6 (The enveloping operator algebra of a Banach algebra) This is an ana-logue of the well-known enveloping C∗-algebra of an involutive Banach algebra.We shall only consider unital algebras for the sake of simplicity. Let B be a unitalBanach algebra which is also an operator space. We let I be the collection of allunital completely contractive homomorphisms π : B → B(H), H = Hπ being aHilbert space. For any integer n ≥ 1 and any matrix b ∈ Mn(B), we define

|||b|||n = sup∥∥πn(b)

∥∥Mn(B(H))

: π ∈ I.

It is easy to see that each ||| · |||n is a seminorm on Mn(B), and that they definean ‘operator algebra seminorm structure’ on B in the sense of 2.3.3. We let O(B)denote the resulting operator algebra obtained by the procedure of 2.3.3 (namely,the completion of the quotient of B by the nullspace of |||·|||1). We will call O(B)the enveloping operator algebra of B.

2.4.7 (Universal property of O(B)) For B as above, there is a canonical com-pletely contractive and unital homomorphism i : B → O(B), whose range isdense, and (O(B), i) has the following property: for any unital operator alge-bra A and for any unital completely contractive homomorphism π : B → A,there exists a (necessarily unique) unital completely contractive homomorphismπ : O(B) → A such that πi = π. It is easy to see that this property characterizesO(B) uniquely, up to appropriate isomorphism.

Next we use the BRS theorem to give an alternative description of the en-veloping operator algebra O(B) in terms of factorization matrix norms. We willthen see some surprising applications of this.

Proposition 2.4.8 Let B be a unital Banach algebra which is also an operatorspace. For n ∈ N and b ∈ Mn(B), we have (with the notation from 2.4.6)

|||b|||n = inf‖b1‖‖b2‖ · · · ‖bN‖

, (2.18)

the infimum over all integers N ≥ 1 and all possible factorizations of b in theform b = b1b2 · · · bN , with b1 ∈ Mn,k1(B), b2 ∈ Mk1,k2(B), . . . , bN ∈ MkN−1,n(B)for some positive integers k1, k2, . . . , kN−1.

Proof For any N ≥ 1, let ZN = B⊗· · ·⊗B be the N -fold tensor product of Ncopies of B. Let mN : ZN → B be the N -fold multiplication defined by lettingmN (b1 ⊗ · · · ⊗ bN) = b1 · · · bN for any b1, . . . , bN ∈ B, and then extending bylinearity. We write m1 for the identity map on B. Then it follows from (1.40)that the right-hand side of (2.18) is equal to

|b|n = infN≥1

inf‖z‖h : z ∈ Mn(ZN ),

(IMn ⊗ mN

)(z) = b

.


Fix b, b′ ∈ Mn(B), for some n ≥ 1, and let ε > 0 be an arbitrary positivenumber. For some integers N, N ′ ≥ 1, there exist elements z in Mn(ZN ) and z′

in Mn(ZN ′) such that(IMn ⊗mN

)(z) = b,

(IMn ⊗mN ′

)(z′) = b′, ‖z‖h < ε+ |b|n,

and ‖z′‖h < ε + |b′|n. Regarding z z′ as an element of Mn(ZN+N ′), we have(IMn ⊗ mN+N ′

)(z z′) = bb′. Thus

|bb′|n ≤ ‖z z′‖h ≤ ‖z‖h‖z′‖h ≤(ε + |b|n

)(ε + |b′|n

).

Since ε > 0 was arbitrary, we have proved that

|bb′|n ≤ |b|n|b′|n, b, b′ ∈ Mn(B), n ≥ 1. (2.19)

By adding on identity matrices in the tensor decomposition of z (or z ′), we canassume that N = N ′. Then

(IMn ⊗ mN

)(z + z′) = b + b′, and so

|b + b′|n ≤ ‖z + z′‖h ≤ ‖z‖h + ‖z′‖h ≤ 2ε + |b|n + |b′|n.

Since ε is arbitrary, | · |n is subadditive. Similar arguments show that the quan-tities | · |n define an operator algebra seminorm structure in the sense of 2.3.3,simply because the Haagerup matrix norms on each ZN satisfy Ruan’s axioms(see 1.2.12), and because of (2.19). Thus, by 2.3.3, the completion of the quotientof B by the nullspace of the seminorm, is an operator algebra A.

We shall now prove (2.18). First note that the canonical map i : B → Ais a unital completely contractive homomorphism. Thus the universal propertyof O(B) ensures that ||| · |||n dominates | · |n, for each n ≥ 1. On the otherhand, suppose that b ∈ Mn(B), and let π : B → B(H) be a unital completelycontractive homomorphism. If b =

(IMn ⊗ mN

)(z) for some z ∈ Mn(ZN ), then

b ∈ Mn(B) and πn(b) =(IMn ⊗πmN

)(z). Since πmN : B⊗· · ·⊗B → B(H) is the

linear mapping which takes b1⊗· · ·⊗bN to π(b1) · · ·π(bN ) for any b1, . . . , bN ∈ B,by the first paragraph in 1.5.8, we have that πmN is completely contractive onB ⊗h · · · ⊗h B. Hence ‖πn(b)‖ ≤ ‖z‖h. It therefore follows from the definitionsof ||| · |||n and | · |n, that we have |||b|||n ≤ |b|n.

2.4.9 (The semigroup operator algebra) Let G be a discrete semigroup. Thespace 1

G of summable families indexed by G, is a unital Banach algebra withthe convolution product (e.g. see [106]). We define the semigroup operator al-gebra O(G) of G to be the enveloping operator algebra O(Max(1

G)). Thereis an obvious bijective correspondence between the contractive representationsσ : G → B(H), the unital contractive homomorphisms π : 1

G → B(H), and theunital completely contractive homomorphisms π : Max(1

G) → B(H). Accord-ing to these correspondences, the matrix norms on O(G) may be described asfollows. Let (eg)g∈G denote the canonical basis of 1

G, and let

C[G] = Spaneg : g ∈ G

be the semigroup algebra of G. Then for b =∑

g λg ⊗ eg ∈ Mn ⊗ C[G], withλg ∈ Mn, we have


|||b|||n = sup∥∥∥∑

g

λg ⊗ σ(g)∥∥∥

Mn(B(H))

where the supremum runs over all contractive representations σ : G → B(H).

According to 2.4.7, the operator algebra O(G) is characterized by the follow-ing universal property. There exists a contractive representation i : G → O(G)whose range generates O(G) as an operator algebra, such that: for any contrac-tive representation σ : G → A into a unital operator algebra A, there exists a(necessarily unique) unital completely contractive homomorphism σ : O(B) → Asuch that σ i = σ.

2.4.10 If G is a group, then 1G has a natural involution b → b∗. Regarding

G ⊂ 1G via the map g → eg above, then this involution on 1

G is simply the unique∗-linear extension to 1

G of the map g → g−1 on G. Suppose that π is a contrac-tive group homomorphism G → B(H). Since π(g)π(g−1) = π(g−1)π(g) = IH forany g ∈ G, it follows from the last assertion in 2.1.3 that π(g) is unitary, withπ(g−1) = π(g)∗. By the basic theory of representations, π extends to a contrac-tive ∗-homomorphism 1

G → B(H), and extends further to a ∗-homomorphismfrom the full group C∗-algebra of G, into B(H). By 1.2.4 this homomorphism iscompletely contractive. By the universal property in 2.4.9, it follows that O(G)coincides with the full group C∗-algebra of G.

The next proposition is a strengthening of Proposition 2.4.8, in the case thatB = Max(1

G) for a semigroup G. Since its proof is similar to that of Proposition2.4.8, we omit it.

Proposition 2.4.11 Let G be a discrete semigroup and let b ∈ Mn ⊗ C[G] forsome n ≥ 1. Then

|||b|||n = inf‖α0‖‖α1‖ · · · ‖αN‖

,

where the infimum runs over all possible product factorizations of b of the formb = α0b1α1b2 · · · bNαN , where α0 ∈ Mn,k1 , α1 ∈ Mk1,k2 , . . . , αN ∈ MkN ,n arescalar matrices for some k1, . . . , kN ∈ N, and b1 ∈ Mk1(B), . . . , bN ∈ MkN (B)are diagonal matrices with entries in the set eg : g ∈ G.

The latter factorization result is especially interesting in the case when G iseither N0 (the additive semigroup of nonnegative integers), or N

20. Before coming

to this, we need to review two important results from operator theory.

2.4.12 (Nagy’s dilation theorem) Let A(D) be the disc algebra from 2.2.6,and let P ⊂ A(D) denote the dense subalgebra of polynomials. Assume thatT is a contraction on some Hilbert space H . Nagy’s dilation theorem [404, 405]asserts that there exists a Hilbert space K, an isometry J : H → K, and a unitaryoperator U ∈ B(K), such that T k = J∗UkJ for any k ∈ N. Let π : C(T) → B(K)be the ∗-representation obtained by applying the spectral mapping theorem toU . That is, π(f) = f(U) for any f ∈ C(T). Then f(T ) = J∗π(f)J for anypolynomial f . This shows that the mapping f ∈ P → f(T ) extends to a unitalcompletely contractive homomorphism uT : A(D) → B(H).


Any matrix [fij ] ∈ Mn(P) can be viewed as a matrix valued function, namelyas the function f taking z to [fij(z)]. Moreover it follows from 2.2.5 that itsnorm as an element of Mn(A(D)) is equal to sup

‖f(z)‖Mn : z ∈ D

. Any

such function f will be called a ‘matrix valued polynomial’ (in one variable). Iff =

∑k≥0 f(k) zk is the (finite) expansion of f , then uT (f) =

∑k≥0 f(k)⊗ T k.

Thus we have∥∥∥∑k≥0

f(k) ⊗ T k∥∥∥

Mn(B(H))≤ ‖f‖Mn(A(D)) = sup

∥∥∥∑k≥0

f(k) zk∥∥∥

Mn

: z ∈ D

.

This is a matricial version of von Neumann’s inequality (which is the case n = 1).

2.4.13 (Ando’s dilation theorem) Let A(D2) be the bidisc algebra, which maybe defined to be the closure of the set P2 = P⊗P of polynomials in two variables,in C(T2). Equivalently (see 2.2.7), we have

A(D2) = A(D) ⊗min A(D) = A(D; A(D)).

Ando’s dilation theorem (see [9, 405]) extends Nagy’s theorem as follows. IfT, S are two commuting contractions on a Hilbert space H , then there exista Hilbert space K, an isometry J : H → K, and two commuting unitary opera-tors U, V on K, such that T kSl = J∗UkV lJ for any pair of nonnegative integersk and l. By a similar argument to that of 2.4.12, we deduce that the mappingf ∈ P2 → f(T, S) extends to a unital completely contractive homomorphismuT,S : A(D2) → B(H). In particular, this yields a version of von Neumann’sinequality for pairs of commuting contractions.

2.4.14 (The semigroup operator algebras of N0 and N20) If σ : N0 → B(H) is

a contractive representation of G = N0 on a Hilbert space H , then T = σ(1) isa contraction. Conversely, any contraction T ∈ B(H) gives rise to a contractiverepresentation σ : N0 → B(H), defined by letting σ(k) = T k for any k ≥ 0. Ifwe identify the algebra C[N0] with P in the usual way (ek = zk for any k ≥ 0),we deduce from this correspondence, 2.4.12, and the the last centered formulain 2.4.9 that O(N0) coincides with A(D) completely isometrically.

Similarly, since contractive representations of N20 correspond to pairs of com-

muting contractions, it follows from 2.4.13 that O(N20) = A(D2).

In the following result, matrices are assumed to have (finite) sizes for whichthe matrix products make sense.

Theorem 2.4.15

(1) For n ∈ N let f ∈ Mn(P) be a matrix valued polynomial. Then ‖f(z)‖Mn < 1for all z ∈ D if and only if there exists a factorization

f(z) = α0b1(z)α1b2(z) · · · bN (z)αN , z ∈ C,

where α0, α1, . . . , αN are scalar matrices of norm < 1 and b1, . . . , bN arediagonal matrices whose entries are of the form zk for (varying) k ∈ N.


(2) Let n ∈ N and f ∈ Mn(P2) a matrix valued polynomial in two variables zand w. Then ‖f(z, w)‖Mn < 1 for every (z, w) ∈ D

2, if and only if thereexist matrix valued polynomials of one variable f1, . . . , fN , g1, . . . ∈ gN , with

f(z, w) = f1(z)g1(w)f2(z) · · · gN−1(w)fN (z)gN(w), (z, w) ∈ C2,

with ‖fi(z)‖ < 1 and ‖gi(w)‖ < 1 for any 1 ≤ i ≤ N , and any w, z ∈ D.

Proof Taking 2.4.14 into account, the first assertion is a straightforward ap-plication of Proposition 2.4.11. For the second assertion, we only need to provethe ‘only if’ part. We let f ∈ Mn(P2) be a matrix valued polynomial such that‖f(z, w)‖Mn < 1 for any (z, w) ∈ D

2. Applying Proposition 2.4.11 again, andthe second paragraph of 2.4.14, we obtain a factorization of f of the form

f(z, w) = α0b1(z, w)α1b2(z, w) · · · bN (z, w)αN , z, w ∈ C,

where α0, α1, . . . , αN are scalar matrices of norm < 1 and each bi is a diago-nal matrix of the form bi(z, w) = Diag

(zn1wm1 , zn2wm2 , . . . , znkwmk

)for some

nonnegative integers n1, m1, n2, . . . , mk. We write bi(z, w) = b′i(z)b′′i (w), with

b′i(z) = Diag(zn1 , zn2 , . . . , znk

)and b′′i (w) = Diag

(wm1 , wm2 , . . . , wmk

).

Then f has the desired decomposition, with f1 = α0b′1, fi = b′i for any 2 ≤ i ≤ N ,

and gi = b′′i αi for any 1 ≤ i ≤ N .

2.4.16 (Free product of operator algebras) We now turn to another universalconstruction, the unital free product. We refer, for example, to [26, 368,418] forsome background on free products, in particular free products of unital algebrasand free products of unital C∗-algebras.

Let A and B be unital operator algebras, and let F(A, B) denote their unitalalgebra free product. This unital algebra is characterized by the following uni-versal property. There exist unital one-to-one homomorphisms jA : A → F(A, B)and jB : B → F(A, B) such that jA(A) and jB(B) generate F(A, B) as an alge-bra, and such that: for any pair (π, ρ) of unital homomorphisms π : A → C andρ : B → C into a unital algebra C, there exists a (necessarily unique) homomor-phism π ∗ ρ : F(A, B) → C such that (π ∗ ρ) jA = π and (π ∗ ρ) jB = ρ.

Let I be the collection of all pairs i = (π, ρ) of unital completely contractivehomomorphisms π : A → B(H) and ρ : B → B(H), H = Hi being a Hilbertspace. Then for any integer n ≥ 1 and any matrix y ∈ Mn(F(A, B)), we define

|||y|||n = sup∥∥(IMn ⊗ (π ∗ ρ)

)(y)

∥∥Mn(B(Hi))

: i = (π, ρ) ∈ I

.

It is easy to see that each ||| · |||n is a seminorm on Mn(F(A, B)). It turns outthat ||| · |||n is actually a norm. This follows from [26] in the case when A and Bare C∗-algebras. In the general case, let C and D be unital C∗-algebras such thatA ⊂ C and B ⊂ D are unital-subalgebras and observe that by [26, Proposition


2.1], the canonical embedding F(A, B) ⊂ F(C, D) is one-to-one. For any pair of∗-representations π : C → B(H) and ρ : D → B(H) and any y ∈ Mn(F(A, B)),we have

∥∥(IMn ⊗ (π ∗ ρ))(y)

∥∥Mn(B(H))

≤ |||y|||n. Taking the supremum over allsuch possible pairs (π, ρ), we deduce from the above mentioned result in theC∗-algebra case, that y = 0 if |||y|||n = 0.

We let A ∗ B be the completion of F(A, B) in the norm ||| · |||1. Arguing asin 2.4.6, we see that A ∗ B is a unital operator algebra for the matrix norms(induced by) ||| · |||n. We call A ∗B the unital free product of A and B. If A andB are unital C∗-algebras then A ∗ B coincides with the usual full amalgamatedfree product C∗-algebra, as may be seen using the above and A.5.8.

2.4.17 If A and B are two unital operator algebras, then there exists a pair ofunital completely isometric homomorphisms π : A → B(H) and ρ : B → B(H),for a common Hilbert space H . Indeed if A and B are represented as unital-subalgebras of B(H1) and B(H2) respectively, then take H = H1 ⊗2 H2, anddefine π and ρ by letting π(a) = a⊗IH2 , and ρ(b) = IH1⊗b, for any a ∈ A, b ∈ B.It now follows from the centered equation in 2.4.16, that the embeddings jA andjB considered in 2.4.16 are unital completely isometric homomorphisms

jA : A → A ∗ B and jB : B → A ∗ B.

From now on (except in the next item), we will suppress mention of jA and jB,and simply regard A and B as unital-subalgebras of A ∗ B.

2.4.18 (Universal property of the free product) It is clear that the unitalfree product operator algebra A∗B of A and B is characterized by the followingproperty. There exist unital completely isometric homomorphisms jA : A → A∗Band jB : B → A∗B, whose ranges together generate A∗B as an operator algebra,such that: for any Hilbert space H and any pair (π, ρ) of unital completelycontractive homomorphisms π : A → B(H) and ρ : B → B(H), there exists a(necessarily unique) unital completely contractive homomorphism π ∗ ρ fromA ∗ B to B(H) such that (π ∗ ρ) jA = π and (π ∗ ρ) jB = ρ.

One may use the BRS theorem to give the following description of the matrixnorms on unital free products. The proof is similar to that of Proposition 2.4.8(and Proposition 2.4.11), and is therefore omitted.

Proposition 2.4.19 Let A and B be unital operator algebras, let F(A, B) betheir unital algebra free product, and let y ∈ Mn(F(A, B)) for some n ≥ 1. Then

‖z‖Mn(A∗B) = inf‖a1‖‖b1‖‖a2‖ · · · ‖aN‖‖bN‖

,

where the infimum runs over all N ∈ N and all possible factorizations of y of theform z = a1b1a2 · · · aNbN , with a1 ∈ Mn,k1(A), b1 ∈ Mk1,l1(B), a2 ∈ Ml1,k2(A),. . . , aN ∈ MlN−1,kN (A), bN ∈ MkN ,n(B) , where k1, l1, k2, . . . , lN−1, kN ∈ N.

Our next objective is Theorem 2.4.21, for which we will need the following:

Proposition 2.4.20 Let A be a unital operator algebra.


(1) If u : A → B(H) is completely contractive then there exist a Hilbert space K,contractions V : H → K, W : K → H, and a unital completely contractivehomomorphism π : A → B(K), such that u(a) = Wπ(a)V for a ∈ A.

(2) Let B be a second unital operator algebra, and let u : A ⊗h B → B(H)be a completely contractive map. Then there exist a Hilbert space K, twolinear contractions V : H → K, W : K → H, and two unital completelycontractive homomorphisms π : A → B(K) and ρ : B → B(K), such that wehave u(a ⊗ b) = Wπ(a)ρ(b)V for any a ∈ A and b ∈ B.

Proof Part (1) is an obvious consequence of 1.2.8, so that we only need to prove(2). If u : A ⊗h B → B(H) is completely contractive, then Theorem 1.5.7 (2)ensures that we can find Hilbert spaces K1, K2, and contractive linear operatorsV : H → K2, R : K2 → K1 and W : K1 → H , as well as unital completelycontractive homomorphism π : A → B(K1) and ρ : B → B(K2), such that

u(a ⊗ b) = Wπ(a)Rρ(b)V, a ∈ A, b ∈ B. (2.20)

We shall now modify this factorization into the desired form.Note that in (2.20), without loss of generality we may replace K1 and K2 with

2I(K1) and 2

I(K2), for any set I. Indeed it suffices to replace W ∈ B(K1, H) withthe row matrix [W 0 0 · · · ] ∈ B(2

I(K1), H), replace V ∈ B(H, K2) with thecolumn [V 0 0 · · · ]t ∈ B(H, 2

I(K2)), and to replace π(a), R, and ρ(b), with theircorresponding multiple ρ(a) ⊗ I2I

, etc. Choosing a sufficiently large I, we mayassume that K1 and K2 have the same dimension. Thus there exists a unitaryU : K2 → K1. Using this unitary we can write u(a⊗b) = WUU ∗π(a)UU∗Rρ(b)Vfor a ∈ A and b ∈ B. Replacing W by WU , R by U ∗R, and π by U∗π( )U , wetherefore obtain that (2.20) holds true with K = K1 = K2.

Since R ∈ B(K) is a contraction, the operators I − RR∗ and I − R∗R arenonnegative. We may define a unitary Γ ∈ B(K ⊕ K) by the formula

Γ =[

R (I − RR∗)1/2

−(I − R∗R)1/2 R∗

].

Moreover for a ∈ A and b ∈ B, we can write

u(a ⊗ b) =[W 0

] [π(a) 00 π(a)

]Γ[

ρ(b) 00 ρ(b)

] [V0

]=[W 0

]Γ Γ∗

[π(a) 0

0 π(a)

]Γ[

ρ(b) 00 ρ(b)

] [V0

].

Replacing K by K ⊕ K, V by[

V0

], ρ by ρ ⊕ ρ, π by Γ∗(π ⊕ π

)Γ, and W by

[ W 0 ]Γ, we see that (2.20) may be achieved with R = IK .

Besides A and B, the simplest subspace of A ∗B is the linear span of all theproducts ab, for a ∈ A and b ∈ B. More specifically, note that the map (a, b) → abfrom A × B to A ∗ B is bilinear, and therefore yields a canonical linearizationE : A ⊗ B → A ∗ B. The range of this map has an attractive description:

78 The second dual algebra

Theorem 2.4.21 Let A and B be two unital operator algebras. Then the canon-ical map E : A⊗B → A∗B above extends to a complete isometry A⊗hB → A∗B.

Proof Fix y =∑

k ak ⊗ bk ∈ A ⊗ B, with ak ∈ A and bk ∈ B. ThenE(y) =

∑k akbk ∈ A ∗ B. Let θ : A ∗ B → B(H) be a completely isometric

homomorphism for some suitable H , and let π = θ|A and ρ = θ|B be the corre-sponding (completely isometric) homomorphisms on A and B. Then

‖E(y)‖A∗B = ‖θ(E(y))‖ =∥∥∥∑

k

π(ak)ρ(bk)∥∥∥ ≤ ‖y‖A⊗hB

by the fact at the end of the first paragraph in 1.5.4.Conversely, let u : A ⊗h B → B(H) be a completely isometric mapping

for some suitable H . According to the second part of Proposition 2.4.20, wehave a factorization of the form u(a ⊗ b) = Wπ(a)ρ(b)V for some contractionsV ∈ B(H, K) and W ∈ B(K, H), and for some unital completely contractivehomomorphisms π : A → B(K) and ρ : B → B(K). Then

‖y‖A⊗hB = ‖u(z)‖ =∥∥∥∑

k

Wπ(ak)ρ(bk)V∥∥∥

≤∥∥∥∑

k

π(ak)ρ(bk)∥∥∥ ≤ ‖E(y)‖A∗B

by the universal property 2.4.18. This shows that A⊗h B ⊂ A ∗B isometrically.A similar argument shows that A⊗h B ⊂ A ∗B completely isometrically.

2.5 THE SECOND DUAL ALGEBRA

Passing to the second dual is a commonly used trick in C∗-algebra theory, as itallows one to work with von Neumann algebras and their various weak topologies.For similar purposes, it is important to understand the properties of the seconddual A∗∗ of a nonselfadjoint operator algebra A, and thus we devote a sectionto this topic. We will not take for granted any facts about the second dual of aC∗-algebra not proved in this book, and will prove such facts as we go along. Webegin with some background and notation on second duals of Banach algebras.

2.5.1 (Arens product) Let A be a Banach algebra. Then we may equip itssecond dual A∗∗ with two natural products as follows. Consider a ∈ A, ϕ ∈ A∗,and η ∈ A∗∗. We let aϕ and ϕa be the elements of A∗ defined by

〈aϕ, b〉 = 〈ϕ, ba〉 and 〈ϕa, b〉 = 〈ϕ, ab〉

for any b ∈ A. Then we let ηϕ and ϕη be the elements of A∗ defined by

〈ηϕ, b〉 = 〈η, ϕb〉 and 〈ϕη, b〉 = 〈η, bϕ〉

for any b ∈ A. By definition, the left and right Arens products ·λ and ·µ on A∗∗

are given by the following formulae, for η, ν ∈ A∗∗ and ϕ ∈ A∗:


〈η·λ ν, ϕ〉 = 〈η, νϕ〉 and 〈η·µ ν, ϕ〉 = 〈ν, ϕη〉.These two products are called the Arens products. They have useful equivalentformulations in terms of double limits. Namely, let η, ν ∈ A∗∗, and according toGoldstine’s lemma (see A.2.1), let (aα)α and (bβ)β be two nets in A convergingto η and ν in the w∗-topology of A∗∗. From the last centered formula we have

〈η·λ ν, ϕ〉 = 〈η, νϕ〉 = limα〈νϕ, aα〉 = lim

α〈ν, ϕaα〉 = lim

αlimβ

〈ϕ, aαbβ〉.

A similar calculation works for the other product, and thus:

〈η·λ ν, ϕ〉 = limα

limβ

〈ϕ, aαbβ〉 and 〈η·µ ν, ϕ〉 = limβ

limα

〈ϕ, aαbβ〉 (2.21)

for any ϕ ∈ A∗. It is easy to check that A∗∗ is a Banach algebra with eitherproduct ·λ or ·µ, and that these products both extend the product of A.

2.5.2 (Arens regularity) A Banach algebra A is called Arens regular if theleft and right Arens products coincide on A∗∗. In that case, we will speak ofthe product ην of two elements of A∗∗ without any ambiguity, and we drop thenotation ·λ or ·µ. It is easy to see using (2.21) say, that a subalgebra A of anArens regular Banach algebra C, is again Arens regular, and that in this caseA∗∗ becomes a subalgebra of C∗∗.

2.5.3 (A characterization of the Arens product) If a Banach algebra A isArens regular, then the Arens product on A∗∗ is separately w∗-continuous. Tosee this, fix ϕ ∈ A∗ and η ∈ A∗∗, and consider the functional ν → 〈ην, ϕ〉 onA∗∗. By the formulae above, this functional coincides with ϕη ∈ A∗. Hence itis w∗-continuous. Similarly for any ν ∈ A∗∗ and any ϕ ∈ A∗, the functionalη → 〈ην, ϕ〉 coincides with νϕ ∈ A∗, and hence is w∗-continuous. Conversely,if A∗∗ is equipped with a product which extends that of A and is separatelyw∗-continuous, then A is Arens regular and the product on A∗∗ is indeed theArens product. This clearly follows from (2.21).

From this last fact and A.5.7, it is clear that any C∗-algebra A is Arensregular and that the product on the W ∗-algebra A∗∗ is the Arens product.

Corollary 2.5.4 Any operator algebra is Arens regular.

Proof We just saw that any C∗-algebra is Arens regular. The result then followsby the remarks at the end of 2.5.2.

2.5.5 (W ∗-continuous extension of homomorphisms) Let A be an operatoralgebra (or more generally an Arens regular matrix normed algebra), and letπ : A → B(H) be a completely contractive homomorphism. According to 1.4.8,we let π : A∗∗ → B(H) be the unique w∗-continuous mapping extending π, andwe recall that π also is completely contractive. We observe that moreover, πis a homomorphism when A∗∗ is equipped with its Arens product. Indeed, thisfollows immediately from the separate w∗-continuity of the products on B(H)and A∗∗ (see A.1.2 and 2.5.3). If π above maps into a w∗-closed subalgebra Mof B(H) then it is evident that π maps into M too (by w∗-continuity).

80 The second dual algebra

Corollary 2.5.6 If A is an operator algebra, then A∗∗ is also an operator alge-bra. Indeed, there exist a Hilbert space H, and a completely isometric homomor-phism π : A → B(H) whose (unique) w∗-continuous extension π : A∗∗ → B(H)

is a completely isometric homomorphism. In this case, A∗∗ ∼= π(A∗∗) = π(A)w∗

.

Proof Let C be a C∗-algebra containing A as a subalgebra. Dualizing the em-bedding map A → C twice, we obtain by 1.4.3 a completely isometric embeddingA∗∗ → C∗∗. By the remark at the end of 2.5.2, A∗∗ is a subalgebra of C∗∗. Sincethe second dual operator space structure on C∗∗ coincides with the one inheritedfrom its C∗-structure (see 1.4.10), we see that A∗∗ is an operator algebra.

Consider the following commutative diagram of operator algebra embeddings:

A ⊂ C∩ ∩A∗∗ ⊂ C∗∗

Let πu : C → B(H) be the universal representation of C. According to 1.4.10,πu satisfies the conclusion of our proposition. That is, πu : C∗∗ → B(H) is acompletely isometric homomorphism. If π is the restriction of πu to A, then π isthe restriction of πu to A∗∗, and so π is a complete isometry. Then it follows bythe Krein–Smulian theorem, as in the last few lines of the proof of A.5.6, that

π(A)w∗

= π(A∗∗).

2.5.7 It is clear from the above proof and 1.4.10 that for an operator algebraA, the completely isometric identification

Mn(A)∗∗ ∼= Mn(A∗∗)

from Theorem 1.4.11, is also an isomorphism of algebras, for any n ≥ 1.

We now turn to approximately unital algebras and mention a well-knownresult concerning Arens regular Banach algebras.

Proposition 2.5.8 Let A be a Banach algebra and assume that A is Arensregular. Then A has a cai (et)t if and only if A∗∗ has an identity e of norm 1.Similarly, A has a right cai (et)t if and only if A∗∗ has a right identity e of norm1. If A is an operator algebra, and (et)t is a cai or right cai for A, and if e isas above, then et → e in the w∗-topology of A∗∗. If A has both a left and a rightcai, then A has a (two-sided) cai.

Proof Assume that A admits a cai (et)t, and let e ∈ A∗∗ be a cluster point ofthis net for the w∗-topology. Then for any a ∈ A, ae is a cluster point of aet

for the w∗-topology. We are using the separate w∗-continuity of the product onA∗∗ (see 2.5.3). Thus ae = a. Thus, by A.2.1 and separate w∗-continuity again,ηe = η for any η ∈ A∗∗. Similarly eη = η, and so e is an identity (of norm 1).Analoguous arguments give the ‘only if’ direction in the ‘right cai case’.

Next we note that a right identity of norm 1 for an operator algebra isunique if it exists. To see this simply note that such a right identity is by A.1.1


a (selfadjoint) projection. If p, q are projections with pq = p and qp = q, then bytaking adjoints we see that p = q.

If e is an identity or right identity of norm one for A∗∗ and if (et)t is as above,then the previous argument shows that e is the unique w∗-cluster point of (et)t.Thus et → e in the w∗-topology. The last assertion of the proposition followsfrom the other assertions, and the analoguous results for left identities.

Next suppose that A∗∗ has a right identity e of norm 1. By Goldstine’s lemma(A.2.1) we may choose a net (et)t in Ball(A) converging to e in the w∗-topologyof A∗∗. Then for each a ∈ A it is clear that

aet −→ a weakly in A. (2.22)

Let F be the set of finite subsets of A. For any F = a1, a2, . . . , an ∈ F , define

PF = (a1u − a1, a2u − a2, . . . , anu − an) : u ∈ Ball(A),

a subset of A⊕∞A⊕∞ · · ·⊕∞A. By (2.22), 0 lies in the weak closure of PF . SincePF is convex, 0 also lies in its norm closure, by Mazur’s theorem from basic func-tional analysis. Let Λ = F ×N, which we consider as a directed set with respectto the product ordering. Given λ = (F, m) for some F = a1, a2, . . . , an ∈ Fand m ∈ N, there exists a uλ ∈ Ball(A) such that ‖akuλ − ak‖ < 1/m for all1 ≤ k ≤ n. Then (uλ) is evidently a right approximate identity for A.

Corollary 2.5.9 Suppose that A is an approximately unital operator algebra (orArens regular matrix normed algebra), and that π : A → B(H) is a completelycontractive homomorphism. If π : A∗∗ → B(H) is its w∗-continuous extension(see 2.5.5), then π is nondegenerate if and only if π(et) → IH in the w∗-topology,for any cai (et)t of A. This is also equivalent to π being unital.

Proof The first ‘if and only if’ follows from Lemma 2.1.9. For the other, notethat et → eA∗∗ by 2.5.8, so that π(et) → π(eA∗∗) (in the w∗-topology).

As another application of the ‘second dual’ we give the following principle,which is surprisingly useful for nonselfadjoint algebras.

Theorem 2.5.10 Suppose that A and B are (norm closed) subalgebras of anoperator algebra C. Suppose further that each of A and B are approximatelyunital; and that AB = B and BA = A. Then A = B as subsets of C.

Proof We may assume that C is selfadjoint. By symmetry it is enough toshow B ⊂ A. We shall consider second duals and apply Proposition 2.5.8. SinceA ⊂ C and B ⊂ C are subalgebras, we may regard A∗∗ ⊂ C∗∗ and B∗∗ ⊂ C∗∗

as w∗-closed subalgebras. Recall from basic functional analysis (see A.2.3 (4))that A = A∗∗ ∩ C. Thus we only need to show that B ⊂ A∗∗.

Let (et)t be a cai for A and let eA be the unit of A∗∗, so that et → eA in thew∗-topology of C∗∗. Also etb → b for any b ∈ B by the equality AB = B. Sincethe product on C∗∗ is separately w∗-continuous, we obtain that eAb = b for anyb ∈ B. Hence, again by separate w∗-continuity, we have eAη = η for any η ∈ B∗∗.

82 Multiplier algebras and corners

In particular, if eB denotes the unit of B∗∗, then eAeB = eB. By symmetry, wealso have eBeA = eA. Since eA and eB are projections in C∗∗, we deduce thateA = eB by taking adjoints. We can now conclude as follows: for any b ∈ B, wehave b = beB = beA = w∗-limt bet. Since BA ⊂ A we see that b belongs to thew∗-closure of A, and hence to A∗∗.

There is an analoguous version of the last result when AB = A and BA = B.It is interesting that these results are not true for general Banach algebras.

2.6 MULTIPLIER ALGEBRAS AND CORNERS

2.6.1 (Left multipliers) If A is simply an algebra then, according to the al-gebraists, a left multiplier of A is a right A-module map u : A → A. The leftmultiplier algebra LM(A) is the unital algebra of left multipliers of A. Let λ bethe left regular representation of A on itself. Namely, λ(a)(b) = ab for a, b ∈ A;this is a homomorphism from A into the left multiplier algebra of A. If A is aBanach algebra which has a cai, then it follows from A.6.3 that any left multiplieris bounded. Thus Banach algebraists usually define the left multiplier algebraLM(A) of an approximately unital Banach algebra A to be BA(A), the unitalBanach algebra of bounded right A-module maps on A. If, further, A is an op-erator algebra or a matrix normed algebra in the sense of 2.3.9, with cai (et)t,then it follows immediately from the relation u(a) = limt u(et)a, which clearlyholds for all u ∈ BA(A) and a ∈ A, that BA(A) = CBA(A) isometrically. HereCBA(A) is the set of completely bounded right A-module maps. Thus we definethe left multiplier algebra LM(A) of such A to be the pair (CBA(A), λ), whereλ : A → CBA(A) is the left regular representation of A mentioned above. SinceLM(A) is a subalgebra of CB(A) (completely isometrically), it is clearly a ma-trix normed algebra. Note that the usual matrix norms on LM(A) = CBA(A)as defined by (1.6), have a simple description in this case. Namely, due to therelation u(a) = limt u(et)a above, it is fairly evident that

‖[uij]‖Mn(CBA(A)) = sup‖[uij(a)]‖Mn(A) : a ∈ Ball(A)

. (2.23)

Equivalently, if we regard Mn(A) as a right A-module in the obvious way, wehave Mn(LM(A)) = CBA(A, Mn(A)) = BA(A, Mn(A)) isometrically.

Henceforth in this section A is an approximately unital operator algebra. Onewould wish the left multiplier algebra of A to be a unital operator algebra, andfortunately it turns out that CBA(A) with its usual matrix norms discussed inthe last paragraph, is an abstract operator algebra. This may be seen by theBRS theorem, or by the Theorem 2.6.2 below.

Theorem 2.6.2 Let A be an approximately unital operator algebra. Then thefollowing algebras are all completely isometrically isomorphic:(1) η ∈ A∗∗ : ηA ⊂ A,(2) T ∈ B(H) : Tπ(A) ⊂ π(A), for any nondegenerate completely isometric

representation π of A on a Hilbert space H,


(3) CBA(A).In particular, CBA(A) is a unital operator algebra.

Proof Let LM be the algebra defined in (1). According to 2.5.6, LM is anoperator algebra. Any η ∈ A∗∗ with the property that ηA ⊂ A may be clearlyregarded as an element of CBA(A) hence there is a canonical homomorphismθ : LM → CBA(A), which may easily be seen to be completely contractive using(2.23). Let (et)t be a cai for A. If θ(η) = 0 then ηet = 0. Using the separatew∗-continuity of the product in A∗∗ (see 2.5.3), and Proposition 2.5.8, it followsthat η = 0. Thus θ is one-to-one. Given v ∈ CBA(A), let η be a w∗-cluster pointof v(et) in A∗∗. Clearly ‖η‖ ≤ ‖v‖. For a ∈ A, we have

v(a) = limt

v(eta) = limt

v(et)a = ηa,

where again we have used the w∗-continuity of the product. Hence θ(η) = v.Thus θ is an isometric surjection, and a similar proof shows that it is a completeisometry. This proves the completely isometric isomorphism between (1) and (3),and also shows that CBA(A) is a unital operator algebra.

If we write LM(π) for the algebra in (2), then there is a canonical homo-morphism ρ : LM(π) → CBA(A), namely ρ(T )(a) = π−1(Tπ(a)) for a ∈ A. If[Tij ] ∈ Mn(LM(π)), then by (2.23) we have

‖[ρ(Tij)]‖n = sup‖[ρ(Tij)a]‖n

= sup

‖[Tijπ(a)]‖n

≤ ‖[Tij]‖n, (2.24)

where the supremum is taken over a ∈ Ball(A). Thus ρ is completely contractive.To see that ρ is completely isometric, we take ε > 0, and choose a vector ζ inBall(2

n(H)) such that ‖[Tij]‖n ≤ ‖[Tij]ζ‖ + ε. Then

‖[Tij ]‖n ≤ limt

‖[Tijπ(et)]ζ‖ + ε ≤ supt

‖[Tijπ(et)]‖ + ε ≤ ‖[ρ(Tij)]‖n + ε,

the last inequality by (2.24). Thus ρ is completely isometric.To see that ρ is onto, suppose that v ∈ BA(A). Let η = θ−1(v) ∈ LM, and let

π : A∗∗ → B(H) be the w∗-continuous homomorphism extending π as discussedin 2.5.5 say. Then for any a ∈ A,

π(η)π(a) = π(ηa) = π(ηa).

This shows that π(η) ∈ LM(π) and ρ(π(η)) = θ(η) = v. Thus ρ is onto.

2.6.3 (Left multiplier operator algebras) More generally, for A as above, weconsider pairs (D, µ) consisting of a unital operator algebra D and a completelyisometric homomorphism µ : A → D, such that Dµ(A) ⊂ µ(A). Sometimes wewrite µ as µA to indicate the dependence on A. We say that two such pairs(D, µ) and (D′, µ′) are completely isometrically A-isomorphic if there exists acompletely isometric surjective homomorphism θ : D → D′ such that θ µ = µ′.This is an equivalence relation.


One should think of each of the three equivalent algebras in Theorem 2.6.2as a pair (D, µA) as in the previous paragraph. We spell out what the map µA

is in each case. In (1), it is the usual inclusion iA : A → A∗∗. In (2), it is π, andin (3) it is the map λ mentioned above the theorem.

We define a left multiplier operator algebra of A, to be any pair (D, µ) asabove which is completely isometrically A-isomorphic to (CBA(A), λ). We alsowrite any such pair as LM(A). The following is clear from the last proof.

Corollary 2.6.4 Each of the operator algebras in 2.6.2, together with its asso-ciated map µA discussed above, is a left multiplier operator algebra of A. Thatis, they are each completely isometrically A-isomorphic to (CBA(A), λ).

2.6.5 If A is unital, then it is easy to see that LM(A) = A.We also remark that if π : A → B(H) is as in 2.6.2 (2), then we may view

LM(A) as a subset of the second commutant π(A)′′. Indeed if T ∈ B(H) withTπ(A) ⊂ π(A), and if S ∈ π(A)′ and a ∈ A, then STπ(a) = Tπ(a)S = TSπ(a).Since π is nondegenerate we have ST = TS. Thus T ∈ π(A)′′.

Corollary 2.6.6 Let A be an approximately unital operator algebra, and fix u inMn(CBA(A)). Write Lu (resp L′

u) for the operator on Cn(A) (resp. on Mn(A))given by the usual formula for multiplication of matrices. Then(1) ‖u‖

Mn(CBA(A)) = ‖Lu‖cb = ‖L′u‖cb.

(2) Mn(CBA(A)) ∼= CBMn(A)(Mn(A)) ∼= CBMn(A)(Cn(A)) ∼= BA(A, Mn(A))isometrically.

(3) LM(Mn(A)) ∼= Mn(LM(A)) completely isometrically.

Proof We observed after (2.23) that BA(A, Mn(A)) ∼= Mn(CBA(A)). We letLM(A) denote the algebra in (1) of Theorem 2.6.2. Clearly Mn(LM(A)) is iso-morphic to LM(Mn(A)) as operator algebras, using the fact from 2.5.7 thatMn(A)∗∗ = Mn(A∗∗). This yields (3), by the way. Thus by 2.6.2 we haveMn(CBA(A)) ∼= CBMn(A)(Mn(A)) isometrically. It is clear by inspection that‖Lu‖cb = ‖L′

u‖cb for u as above, so that CBMn(A)(Mn(A)) ∼= CBA(Cn(A))isometrically. This proves (1) and (2).

2.6.7 (Right and two-sided multiplier algebras) We may define the right (resp.two-sided) multiplier operator algebra RM(A) (resp. M(A)) in a similar way,and one obtains similar results. For example, these algebras may be taken to beRM(A) = η ∈ A∗∗ : Aη ⊂ A (resp. M(A) = η ∈ A∗∗ : ηA ⊂ A, Aη ⊂ A).However there are one or two pitfalls to be wary of. First, the canonical mapRM(A) →ACB(A) into the algebra of completely bounded left A-module mapsis a completely isometric anti-homomorphism. Thus RM(A) should be identifiedwith ACB(A) with the reverse of the usual multiplication on the latter. This‘twist’ is related to the reason why algebraists often write operators on the rightof the variable. Second, the canonical embedding LM(A) → CBA(A) does notrestrict, as one might first guess, to an embedding of M(A) into the A-bimodulemaps on A. The traditional way for Banach algebraists to circumvent this last


difficulty is to consider two-sided multipliers as a pair of maps in the doublecentralizer algebra (see the Notes on Section 2.6). Instead, we take the followingapproach. We consider pairs (D, µ) consisting of a unital operator algebra Dand a completely isometric homomorphism µ : A → D such that Dµ(A) ⊂ µ(A)and µ(A)D ⊂ µ(A). We define two such pairs to be completely isometricallyA-isomorphic just as we did in 2.6.3. We use the term ‘(two-sided) multiplieroperator algebra of A’, and write M(A), for any pair (D, µ) as above which iscompletely isometrically A-isomorphic to x ∈ A∗∗ : xA ⊂ A and Ax ⊂ A. Infact these may be characterized quite nicely as subalgebras of LM(A). We leavethe following as a (simple algebraic) exercise:

Proposition 2.6.8 Suppose that A is an approximately unital operator algebra.If (D, µ) is a left multiplier operator algebra of A, then the closed subalgebra

d ∈ D : µ(A)d ⊂ µ(A)

of D, together with the map µ, is a (two-sided) multiplier operator algebra of A.

Corollary 2.6.9 If A is a C∗-algebra then M(A) is the diagonal ∆(LM(A)).In particular, M(A) is a C∗-algebra.

Proof If η ∈ LM(A) ⊂ A∗∗ then

Aη ⊂ A ⇔ η∗A ⊂ A ⇔ η∗ ∈ LM(A),

hence η ∈ M(A) if and only if η ∈ ∆(LM(A)).

2.6.10 (Essential ideals) If C is a C∗-algebra then an essential (two-sided)ideal of C is an ideal I of C for which the canonical homomorphism from C toB(I) associated to the product map C × I → I, is one-to-one. It is a pleasant∗-algebraic exercise to show that this is equivalent to saying that K ∩ I = (0)for every nontrivial (two-sided closed) ideal K of C. Any C∗-algebra A is anessential ideal of M(A). Clearly, if C is a C∗-algebra containing A as an essentialideal, then the canonical homomorphism C → BA(A) = CBA(A) = LM(A)is completely contractive. Hence by the last paragraph in 2.1.2, for example, ityields a one-to-one ∗-homomorphism into ∆(LM(A)) = M(A) (see 2.6.9). ThusM(A) is the largest C∗-algebra with A as an essential ideal.

If A is a w∗-dense ideal in a W ∗-algebra M , then M = LM(A) = M(A).Indeed, the complete contraction above from M to CBA(A), is one-to-one bythe w∗-density of A. That it is completely isometric and surjective is easily seenby considering, for any T ∈ CBA(A), a w∗-limit point of (T (et)) in M , where(et) is the approximate identity for A. The other assertions are now easy. In fact,this argument works more generally if M is a matrix normed algebra which is adual space with separately w∗-continuous product, and if A has a cai.

2.6.11 (Multiplier-nondegenerate morphisms) Let A, B be approximately uni-tal algebras. A completely contractive homomorphism π : A → LM(B) will becalled a left multiplier-nondegenerate morphism, if B is a nondegenerate left


module with respect to the natural left module action of A on B via π. Thisis equivalent, by Cohen’s factorization theorem A.6.2, to saying that for anyleft cai (et)t of A, we have π(et)b → b for all b ∈ B; or to saying that anyb ∈ B may be written b = π(a)b′ for some a ∈ A, b′ ∈ B. Similarly for rightmultiplier-nondegenerate morphism; and a (two-sided) multiplier-nondegeneratemorphism is a map π : A → M(B) that has both of these properties. Thus if πactually maps into B, then π is multiplier-nondegenerate if and only if (π(et))t

is a cai for B, for any cai (et)t for A. Note that the canonical map A → M(A)is multiplier-nondegenerate.

Proposition 2.6.12 If A, B are approximately unital operator algebras, and ifπ : A → M(B) is a multiplier-nondegenerate morphism then π extends uniquelyto a unital completely contractive homomorphism π : M(A) → M(B). Moreoverπ is completely isometric if and only if π is completely isometric.

Proof By 2.6.7 we may regard M(A) and M(B) as subalgebras of A∗∗ and B∗∗

respectively. Let π : A∗∗ → B∗∗ be the (unique) w∗-continuous homomorphismextending π (see 2.5.5). We will prove that π maps M(A) into M(B) and thatπ(·)|M(A) is the unique bounded homomorphism on M(A) extending π.

As usual we let (et)t be a cai for A. Let η ∈ M(A) and let b ∈ B. Thenπ(et)b → b by 2.6.11, hence π(η)π(et)b → π(η)b. Since π is a homomorphismand ηet ∈ A, we find that π(η)b = limt π(ηet)b is the limit of a net of B, hencebelongs to B. Similarly, bπ(η) ∈ B for any b ∈ B, which shows that π(η) ∈ M(B).

Assume now that π : M(A) → M(B) is a bounded homomorphism extendingπ, and let η ∈ M(A) and b ∈ B. Each ηet belongs to A, and this net convergesto η in the w∗-topology by Proposition 2.5.8. Hence

π(η) = w∗- limt

π(ηet).

On the other hand, π(et)b → b, and hence

π(η)b = limt

π(η)π(et)b = limt

π(ηet)b = limt

π(ηet)b.

This shows that π = π|M(A). It remains to prove the last assertion, so supposethat π is completely isometric. Using (2.23) we have for η ∈ M(A) that

‖π(η)‖cb ≥ ‖π(η)(π(a)b)‖ = ‖π(ηa)b‖

if a ∈ Ball(A), b ∈ Ball(B). Taking the supremum over all b ∈ Ball(B) gives

‖π(η)‖cb ≥ ‖π(ηa)‖ = ‖ηa‖.

Taking the supremum over all such a ∈ Ball(A), gives that ‖π(η)‖cb ≥ ‖η‖cb. Soπ is isometric, and similarly it is completely isometric.

2.6.13 The analoguous result for ‘left multiplier-nondegenerate morphisms’, inwhich we replace M(·) by LM(·) in 2.6.12, holds with the same proof.


As a corollary of the last result, and 2.1.7 (2), we note that if B is any C∗-cover of an approximately unital operator algebra A, then LM(A) ⊂ LM(B),RM(A) ⊂ RM(B), and M(A) ⊂ M(B), as unital-subalgebras.

As another corollary, suppose that J is a left ideal of an operator algebraA, that J has a cai, and that π : J → B(H) is a completely contractive non-degenerate representation. We claim that π extends uniquely to a completelycontractive representation π of A on H . Indeed the uniqueness follows from therelation π(a)π(x)ζ = π(a)π(x)ζ = π(ax)ζ, for a ∈ A, x ∈ J, ζ ∈ H . ViewingB(H) as the multiplier algebra of S∞(H), it is easy to see that π is a multiplier-nondegenerate morphism in the sense above. By 2.6.12, π extends to a completelycontractive representation of LM(J) on H . Since LM(J) = CBJ(J), it is clearthat there is a canonical completely contractive homomorphism from A intoLM(J). Composing the last two homomorphisms gives the desired result.

We end this section with some relations between multipliers and ‘corners’.

2.6.14 (Corners) If p is a projection in M(A) then we say that pAp is the1-1-corner of A (with respect to p). In this case pAp is an operator algebra(indeed it is a C∗-algebra if A is one), as is the 2-2-corner (1 − p)A(1 − p).We say that pA(1 − p) is the 1-2-corner of A (with respect to p). Consider thesubset B of M2(A) consisting of all 2 × 2 matrices whose i-j entry belongs tothe i-j-corner of A with respect to p. This set B is a norm closed subalgebra ofM2(A), which we claim is canonically completely isometrically isomorphic to A.To see this, first suppose that A = B(H). Then A ∼= B(K ⊕K⊥), and it is fairlyclear that this is ∗-isomorphic to B in this case, via a ∗-isomorphism π say. By1.2.4, π is completely isometric. For a general subalgebra A ⊂ B(H), the claim isestablished by considering the restriction of π to A. In fact, it is usually helpfulto regard the 2×2 matrix operator algebra B as a subalgebra of the 2×2 matrix∗-algebra corresponding to B(K ⊕ K⊥).

2.6.15 (Corner-preserving maps) Suppose that A and B are approximatelyunital operator algebras, and that p and q are projections in M(A) and M(B)respectively. As in 2.6.14, this allows us to decompose A and B as 2 × 2 matrixoperator algebras. We say that a map π : A → B is corner-preserving, or decou-pled, if π maps each of the four corners of A into the corresponding corner of B.In this case we let πij be the restriction of π to the i-j-corner of A, viewed as amap from the i-j-corner of A into the i-j-corner of B. We call πij the i-j-cornerof π, and sometimes write π = [πij ] to indicate this situation.

Suppose that π : A → M(B) is a contractive multiplier-nondegenerate mor-phism (see 2.6.11) and that p is a projection in M(A). If we define q = π(p),where π is as in 2.6.12, then q is a contractive idempotent, and is consequently,by 2.1.3, a projection in M(B). Also, π is corner-preserving as a map into M(B)(with respect to the corners defined via p and q = π(p)). To see this, note forexample that π(pa(1 − p)) = π(pa(1 − p)) = qπ(a)(1 − q) for any a ∈ A.

2.6.16 (Corner-preserving completely positive maps) Suppose that Φ: A → Bis a unital completely positive map between unital C∗-algebras. Suppose that

88 Dual operator algebras

p is a projection in A and that q = Φ(p) is a projection in B. Then a similarprinciple to 2.6.15 holds. Obviously Φ restricted to the linear span of p and 1−p(which is a two-dimensional C∗-algebra) is a unital ∗-homomorphism. It followsby 1.3.12 that Φ(pa) = qΦ(a) and Φ(ap) = Φ(a)q for all a ∈ A. Hence Φ isagain corner-preserving, and we may write Φ = [ϕij ] just as in 2.6.15. Since Φis completely positive and hence ∗-linear it follows that first, ϕ11 and ϕ22 arecompletely positive, and second, that ϕ12(x∗) = ϕ21(x)∗ for any x ∈ pA(1 − p).In other words, ϕ21 = ϕ

12 in the sense of 1.2.25. In passing we observe thatusing 1.2.10 and 1.3.3, we may extend (the first part of) these results to thecase when Φ: A → B is a unital completely contractive homomorphism betweenunital operator algebras.

2.6.17 We may rephrase 2.6.16 in the language of bimodules. Note that theexistence of projections p ∈ A and q ∈ B, is equivalent to saying that A and Beach contain D2 = C⊕C as a unital C∗-subalgebra. That Φ may be written as amatrix of maps [ϕij ] with respect to these projections p, q, is equivalent to sayingthat Φ is a D2-bimodule map. We can therefore extend the observation 2.6.16 tothe case when A and B contain a copy of Dn = ∞n as a unital C∗-subalgebra,and Φ: A → B is a unital completely contractive map. In this case, A and Bdecompose as n×n matrix algebras, with respect to the n canonical idempotentsin Dn. Also, Φ is a Dn-bimodule map if and only if we may write Φ = [ϕij ]. Inthis case we say again that Φ is corner-preserving. Again 1.3.12 implies that ifΦ is ‘the identity map’ on the copies of Dn, then Φ is corner-preserving.

2.7 DUAL OPERATOR ALGEBRAS

We shall now consider dual objects in the category of operator algebras, bothfrom the concrete and from the abstract point of view. By definition, a concretedual operator algebra is a w∗-closed subalgebra of B(H), for some Hilbert spaceH . As a consequence of 1.4.7, any concrete dual operator algebra is a dual oper-ator space. In the converse direction, let M be an operator algebra together witha given additional topology τ on M , and we suppose that τ is the w∗-topologygiven by some predual for M . We call such an algebra M , together with thetopology τ , an (abstract) dual operator algebra, if there exist a Hilbert spaceH and a w∗-continuous completely isometric homomorphism π : M → B(H).In this case, the range π(M) is w∗-closed by the Krein–Smulian theorem A.2.5,and π is a w∗-homeomorphism onto its range. Hence π(M) is a concrete dualoperator algebra acting on H , which we may identify with M . If M is a dualoperator algebra, then we say that the topology τ mentioned above, is a dualoperator algebra topology on M .

In the selfadjoint situation, we use the term ‘von Neumann algebra’ for aconcrete selfadjoint dual operator algebra, and the term ‘W ∗-algebra’ for anabstract selfadjoint dual operator algebra (see A.5.1 and A.5.3). By the famoustheorem of Sakai, a W ∗-algebra is exactly the same as a C∗-algebra with aBanach space predual; and in this case the Banach space predual is unique (see


[380, Sections 1.13 and 1.16]). We will see that in the nonselfadjoint context, somenew complications may occur. For example, in general the predual of an operatoralgebra need not be unique. That is, an operator algebra may possess more thanone dual operator algebra topology (see Corollary 2.7.8 and Proposition 2.7.16).However we will establish an abstract characterization of dual operator algebras(see Theorem 2.7.9) which is close to Sakai’s result.

We often identify any two dual operator algebras M and N which are w∗-homeomorphically and completely isometrically isomorphic. In this case we some-times say that M ∼= N as dual operator algebras.

2.7.1 If M is a dual operator algebra, then we will reserve the notation M∗ forany operator space predual of M which induces the given dual operator algebratopology on M . If M ⊂ B(H) is represented as a w∗-closed subalgebra (by amap π as above), then by 1.4.6 we may take M∗ = S1(H)/M⊥.

2.7.2 Unital dual operator algebras play a particularly prominent role in thesequel. A slight modification of the argument in 2.1.4 shows that any unital dualoperator algebra may be represented as a w∗-closed subalgebra of some B(H)containing IH . (Note that the map a → a|K in 2.1.4 is a w∗-continuous completeisometry into B(K), and is thus a w∗-homeomorphism by A.2.5.)

2.7.3 As a consequence of Corollary 2.5.6, the second dual of any operatoralgebra is a dual operator algebra, as one might hope.

The next result collects together several superficial facts about dual algebras:

Proposition 2.7.4 Let M be a (possibly nonunital) dual operator algebra.(1) The product on M is separately w∗-continuous.(2) For any a ∈ M and any ϕ ∈ M∗, the functionals ϕa and aϕ (using notation

from 2.5.1) both belong to M∗.(3) If M is approximately unital, then M is actually unital.(4) The w∗-closure of a subalgebra of M is a dual operator algebra.(5) The unitization (see Section 2.1 above) of M is also a dual operator algebra.(6) If M is nonunital, and if π : M → B(H) is a w∗-continuous contractive

homomorphism, then the canonical unital extension of π to the unitizationof M (see 2.1.13) is w∗-continuous.

Proof Item (1) is clear from the separate w∗-continuity of the product on B(H)(see A.1.2). Item (2) is a restatement of (1). For (3), assume that (et)t is a caifor M , and let e ∈ M be a w∗-cluster point of this net. For any a ∈ M and anyϕ ∈ M∗, and passing to a subnet, we have

〈ϕ, eta〉 = 〈aϕ, et〉 −→ 〈aϕ, e〉 = 〈ϕ, ea〉

by (2). Since eta → a, we deduce that 〈ϕ, ea〉 = 〈ϕ, a〉. Similarly, 〈ϕ, ae〉 = 〈ϕ, a〉.Since ϕ ∈ M∗ was arbitrary, this shows that e is a unit (of norm 1) for M .

For (4), let x, y belong to the w∗-closure of the subalgebra A in M , andlet (aα)α and (bβ)β be two nets of A converging to x and y respectively in


the w∗-topology. By (1), xbβ = w∗-limα aαbβ. Hence xbβ belongs to Aw∗

. Then

xy = w∗-limβ xbβ , and so xy ∈ Aw∗

.Finally, suppose that M is a w∗-closed nonunital subalgebra of B(K), and

write I for IK . Suppose that (xt)t and (λt)t are nets in M and C respectively,with (xt +λtI) converging in the w∗-topology. By the Hahn–Banach theorem, itis easy to see that (λt)t converges in C. It follows that (xt)t converges in M , inthe w∗-topology. From this, (5) and (6) follow easily.

2.7.5 (Examples) The following completes the list considered in Section 2.2.We refer the reader to the extensive literature on dual algebras for deeper andmore interesting examples (see the first paragraph of the Notes section for thischapter for a few references).

(1) As in 2.2.1, the ∞-direct sum of dual operator algebras is again a dualoperator algebra. We leave this fact as an exercise using the last facts in 1.4.13.In particular, if M is a dual operator algebra and I is a set, then ∞I (M) is adual operator algebra.

(2) Let M and N be two dual operator algebras. Then it follows from 2.2.2and 2.7.4 (4), that their normal spatial tensor product M ⊗N (defined in 1.6.5)is a dual operator algebra. We leave it as an exercise that if π, θ are w∗-continuouscompletely bounded representations of M and N , then the extension of π ⊗ θ toM ⊗N is again a w∗-continuous completely bounded homomorphism.

(3) The result in 2.2.4 extends as follows. Let (Ω, µ) be a σ-finite mea-sure space, let M be a dual operator algebra with separable predual, and re-call from 1.6.6 the w∗-continuous completely isometric isomorphism jM fromL∞(Ω)⊗M onto the dual operator space L∞(Ω; M) of w∗-measurable essen-tially bounded functions f : Ω → M . If f, g ∈ L∞(Ω; M) then the pointwiseproduct function (fg)(t) = f(t)g(t) is w∗-measurable and hence belongs toL∞(Ω; M). Moreover the resulting multiplication on L∞(Ω; M) is separatelyw∗-continuous. Indeed these results are established in [380, Theorem 1.22.13]in the case when M is a W ∗-algebra and the proofs given there extend almostverbatim to the nonselfadjoint case. Now consider L∞(Ω; M) as equipped withthis pointwise product. Then the last lines of 1.6.6 show that the restriction ofjM to L∞(Ω) ⊗ M is a homomorphism. Since the products on L∞(Ω)⊗M andL∞(Ω; M) are both separately w∗-continuous, we deduce that jM is a homomor-phism. Thus L∞(Ω; M) is a dual operator algebra for the pointwise multipli-cation and L∞(Ω)⊗M = L∞(Ω; M) ‘as dual operator algebras’. These resultsextend the well-known fact (e.g. see [380, Theorem 1.22.13]) that L∞(Ω; M) isa W ∗-algebra if M is a W ∗-algebra.

(4) The discussion in 2.2.6 and 2.2.7 can be continued as follows. LetH∞(D) ⊂ L∞(T) be the Hardy space of all f ∈ L∞(T) whose negative Fouriercoefficients vanish. Then H∞(D) is a w∗-closed subalgebra of L∞(T), and henceit is a unital dual operator algebra. Using Poisson integrals, H∞(D) can also bedescribed as the algebra of all bounded analytic functions from D into C. Moregenerally, if M is a dual operator algebra with separable predual, then the space


H∞(D; M) of all functions f ∈ L∞(T; M) whose negative Fourier coefficientsvanish is a dual operator algebra. Moreover H∞(D; M) can also be described asthe algebra of all bounded analytic functions from D into M and we have anidentification H∞(D; M) = H∞(D)⊗M ‘as dual operator algebras’. We omitthe proofs of these assertions, as they diverge from our main concerns.

(5) When M = MI , for a cardinal I, then (2) yields a dual operator algebrastructure on MI ⊗N . On the other hand, if N ⊂ B(H) is represented as a dualoperator algebra on H , then MI(N) is the w∗-closure of M

finI (N) in B(2

I(H))by Corollary 1.6.3 (3), hence is a dual operator algebra by 2.7.4 (4). From thisit is not hard to see that the identification MI ⊗N = MI(N) given by (1.54)holds ‘as dual operator algebras’. This is analoguous to (2.8) and is related tothe latter by the following proposition.

Proposition 2.7.6 Let A be an operator algebra, and I any cardinal. ThenKI(A)∗∗ ∼= MI(A∗∗) as dual operator algebras.

Proof The canonical embedding A ⊂ A∗∗ induces a completely isometric homo-morphism π : KI(A) → KI(A∗∗) ⊂ MI(A∗∗). By (1.62) the unique w∗-continuousextension π of π to KI(A)∗∗ is a completely isometric isomorphism. However by2.5.5, it is a homomorphism too.

We now return to the algebra U(X) constructed in 2.2.10.

Lemma 2.7.7 Let X be an operator space.

(1) If U(X) is a dual operator algebra, then X ⊂ U(X) is w∗-closed. Hence Xis a dual operator space.

(2) If X is a dual operator space, then U(X) is a dual operator algebra. IndeedU(X) has an operator space predual for which the canonical embedding fromX into U(X) is w∗-continuous.

Proof Assume that U(X) is a dual operator algebra. Let p be the projection

in U(X) defined by p =[

1 00 0

]. Then pU(X)(1 − p) is the subspace of U(X)

corresponding to X . It is a fairly obvious general principle that a corner pM(1−p)of a dual algebra M is w∗-closed. Indeed by 2.7.4 (1) and the fact that p is aprojection, it follows that if (yt) is a net in pM(1 − p) converging to b ∈ M say,then yt = pyt(1 − p) converges to pb(1− p). Thus b = pb(1− p) ∈ pU(X)(1 − p).

Assume conversely that X is a dual operator space. By 1.4.7 we may assumethat X ⊂ B(H) is a w∗-closed subspace of B(H) for some Hilbert space H .Then U(X), as defined by (2.10), is easily seen to be a w∗-closed subalgebra ofB(H(2)) and the result follows at once.

Corollary 2.7.8 There exist a unital operator algebra A with two distinct dualoperator algebra topologies. Equivalently, there exist two completely isometrichomomorphisms π1 : A → B(H1) and π2 : A → B(H2) such that M1 = π1(A)and M2 = π2(A) are w∗-closed but π∗

1(M1∗) = π∗2(M2∗).


Proof Let E be a Banach space with two distinct preduals Z1 and Z2. Moreprecisely, we think of these preduals as two distinct subspaces Z1 ⊂ E∗ andZ2 ⊂ E∗. We consider the operator space X = Min(E), and the unital operatoralgebra A = U(X) of 2.2.10. Let J : X → U(X) be the canonical embedding.Since X = Max(Z1)∗ = Max(Z2)∗, the second part of Lemma 2.7.7 ensuresthat there exist two completely isometric homomorphisms π1 : A → B(H1) andπ2 : A → B(H2) respectively, such that M1 = π1(A) and M2 = π2(A) are w∗-closed, and such that π1 J : X → M1 and π2 J : X → M2 are w∗-continuous.In the last statement X is regarded as the dual of Z1 and of Z2 respectively; thatis, (π1 J)∗(M1∗) = Z1 and (π2 J)∗(M2∗) = Z2. Thus if π∗

1(M1∗) and π∗2(M2∗)

were equal, we would have the contradiction that Z1 = Z2.

We have noticed that a dual operator algebra is a dual operator space. Themain result of this section is the following converse, which may be regarded as adual version of the BRS theorem, and also as a nonselfadjoint version of Sakai’stheorem mentioned at the start of this section.

Theorem 2.7.9 Let M be an operator algebra which is a dual operator space.Then the product on M is separately w∗-continuous, and M is a dual operatoralgebra. That is, there exists a Hilbert space H and a w∗-continuous completelyisometric homomorphism π : M → B(H).

The first assertion saying that the product on M is separately w∗-continuouswill only be proved in Chapter 4, in 4.7.2. The rest of the proof will requireseveral intermediate steps of independent interest, and will be completed in 2.7.13below. We start with a version of Proposition 2.4.20 (1), for unital dual operatoralgebras.

Theorem 2.7.10 Let M be a unital operator algebra which is also a dual opera-tor space. Let u : M → B(H) be a w∗-continuous completely contractive map, Hbeing a Hilbert space. Then there exist a Hilbert space K, two linear contractionsV : H → K, W : K → H, and a unital w∗-continuous completely contractivehomomorphism π : M → B(K), such that u(a) = Wπ(a)V for any a ∈ M .

Proof We first apply Proposition 2.4.20 (1) to u. We obtain a Hilbert spaceG, two linear contractions T : H → G and S : G → H , and a unital completelycontractive homomorphism θ : M → B(G) such that

u(a) = Sθ(a)T, a ∈ M. (2.25)

Moreover we may assume that G = [θ(M)T (H)]. Let K = [θ(M)∗S∗(H)] ⊂ G,and let p = PK be the projection onto K. If a, b, c ∈ M and ζ, η ∈ H then

〈u(acb)ζ, η〉 = 〈pθ(b)Tζ, θ(c)∗θ(a)∗S∗η〉 = 〈pθ(c)pθ(b)Tζ, θ(a)∗S∗η〉.Thus

u(acb) = Sθ(a)θ(c)pθ(b)T = Sθ(a)pθ(c)pθ(b)T. (2.26)

Setting a = b = 1 in (2.26) yields u = Wπ(·)V , where W = S|K , V = pTand π = pθ(·)|K . A well-known general principle states that if θ : A → B(H)


is a homomorphism and if K is a subspace of H such that θ(A)∗K ⊂ K (orequivalently, such that θ(A)K⊥ ⊂ K⊥, then π = PKθ(·)|K is a homomorphism.To see this, note that for any a, b ∈ A and any ζ, η ∈ K, we have

〈π(ab)ζ, η〉 = 〈θ(ab)ζ, η〉 = 〈θ(a)θ(b)ζ, η〉 = 〈θ(b)ζ, θ(a)∗η〉 = 〈π(b)ζ, θ(a)∗η〉.

On the other hand,

〈π(a)π(b)ζ, η〉 = 〈θ(a)π(b)ζ, η〉 = 〈π(b)ζ, θ(a)∗η〉.

Thus π is a unital completely contractive homomorphism. To show that π isw∗-continuous, we fix a, b ∈ M and ζ, η ∈ H . We noticed before Theorem 2.7.10that the product on M is separately w∗-continuous (by 4.7.2). Since u is w∗-continuous, the mapping c → u(acb) is therefore w∗-continuous from M intoB(H). Also observe that for any c ∈ M , we have using (2.26) that

〈u(acb)ζ, η〉 = 〈pθ(c)pθ(b)Tζ, θ(a)∗S∗η〉 = 〈π(c)pθ(b)Tζ, θ(a)∗S∗η〉.

If (ct)t is a bounded net in M converging in the w∗-topology to c ∈ M , then

〈π(ct)k1, k2〉 → 〈π(c)k1, k2〉 (2.27)

for k1 = pθ(b)Tζ, k2 = θ(a)∗S∗η. The span of the set of such k1 is dense in K,and similarly for the span of the set of such k2. Therefore, by a simple normdensity argument, (2.27) holds for all k1, k2 ∈ K. That is, π(ct) → π(c) in theWOT. Since this is a bounded net, it converges by A.1.4 in the w∗-topology.Thus π is w∗-continuous by A.2.5 (2).

We next give a dual version of Proposition 2.3.4. Recall from 1.4.4, that ifX = Z∗ is a dual operator space and if Y ⊂ X is w∗-closed, then the quotientspace X/Y is the dual operator space of Y⊥ ⊂ Z.

Proposition 2.7.11 Let M be a dual operator algebra and let J ⊂ M be aw∗-closed (two-sided) ideal. Then M/J is a dual operator algebra.

Proof We may assume that M ⊂ B(H) is a w∗-closed subalgebra of someB(H). Then by 2.7.4 (5) its unitization M 1 = SpanM, IH is also a w∗-closedsubalgebra, any ideal of M is still an ideal of M 1, and the canonical embed-ding M/J ⊂ M1/J is w∗-continuous. Replacing M by M 1 if necessary, we maytherefore assume that M is unital.

If M is unital, then N = M/J is unital, hence N is both a dual operator spaceand a unital operator algebra (by 2.3.4). By 1.4.7, there exists a w∗-continuouslinear complete isometry u : N → B(H), for some Hilbert space H . ChooseK, π, V and W as in Theorem 2.7.10, with u = Wπ(·)V . For any n ≥ 1 and anymatrix [aij ] ∈ Mn(N), we have∥∥[aij ]

∥∥n

=∥∥[u(aij)]

∥∥n≤ ‖W‖

∥∥[π(aij)]∥∥

n‖V ‖ ≤

∥∥[π(aij)]∥∥

n≤∥∥[aij ]

∥∥n.

Hence π is a complete isometry, so that N is a dual operator algebra.


Lemma 2.7.12 Let M be an Arens regular Banach algebra. Assume moreoverthat M = X∗ is a dual Banach space, and that the product on M is separatelyw∗-continuous. Let iX : X → X∗∗ be the canonical embedding. Then the adjointmapping Q = i∗X : M∗∗ → M is a homomorphism.

Proof Take η and ν in M∗∗ and consider two nets (at)t and (bs)s in M con-verging respectively to η and ν in the w∗-topology. Then it follows immediatelyfrom the separate w∗-continuity that

Q(ην) = w∗- limt

lims

Q(atbs) = w∗- limt

lims

atbs = w∗- limt

atQ(ν) = Q(η)Q(ν).

Thus Q is a homomorphism.

2.7.13 (Proof of Theorem 2.7.9) Assume that M is an operator algebra whichis also a dual operator space. We already emphasized that the product on M isseparately w∗-continuous (by 4.7.2). Let Q : M ∗∗ → M be defined as in the abovelemma. Combining the latter with 2.5.4, we obtain that Q is a homomorphism.Since iX is a complete isometry, Q also is a w∗-continuous complete quotient map,by 1.4.3. Thus Ker(Q) is a w∗-closed ideal of M∗∗. Factoring through its kernel,Q induces a w∗-continuous (by A.2.4) and completely isometric homomorphismfrom M∗∗/Ker(Q) onto M . In fact this map is a w∗-homeomorphism, by A.2.5.Since M∗∗ is a dual operator algebra, as we remarked in 2.7.3, Proposition 2.7.11shows that M∗∗/Ker(Q) is a dual operator algebra. Thus M is a dual operatoralgebra.

Corollary 2.7.14 Suppose that A is an operator algebra, and that m is anyproduct on A∗∗ extending the product on A, for which A∗∗ is (completely isomet-rically) an operator algebra. Then m must be the Arens product.

Proof By Theorem 2.7.9, m is automatically w∗-continuous. By 2.5.3, m is theArens product.

In the light of Sakai’s theorem (which we described at the start of this sec-tion), one might imagine that Theorem 2.7.9 remains valid if we replace theassumption that M is a dual operator space by the weaker assumption that Mis merely a dual Banach space. In fact this is not true, and our next goal is toshow this. We will use the following subtle result which clarifies some differencesbetween Banach space and operator space duality.

Lemma 2.7.15 There exists an operator space X with a unique Banach spacepredual Z, such that X possesses no operator space predual. Moreover, wheneverZ is equipped with an operator space structure, the canonical mapping betweenthe dual operator space Z∗ and X is not a complete isomorphism.

Proof Let B be an operator space that will be specified later on. From 1.4.13,

1(B∗)∗ = ∞(B∗∗) and c0(B)∗ = 1(B∗) (2.28)


completely isometrically. Thus c0(B)∗∗ = ∞(B∗∗). If η ∈ ∞(B∗∗), let η de-note its class in the quotient space ∞((B∗∗)op)/c0((B∗∗)op) (see 1.2.25 for thisnotation), and write η for the pair (η, η). Consider the operator space

X =η : η ∈ ∞(B∗∗)

⊂ ∞(B∗∗) ⊕∞ (

∞((B∗∗)op)/c0((B∗∗)op)). (2.29)

The map κ : X → ∞(B∗∗) taking η to η is a complete contraction, that is,

‖η‖Mn(∞(B∗∗)) ≤ ‖η‖Mn(X) (2.30)

for any n ≥ 1 and any η ∈ Mn⊗∞(B∗∗). Moreover since η → η is a contraction,κ is an isometric isomorphism. The main point to be kept in mind is the operatorspace structure on X induced by the right-hand side of (2.29).

Obviously, η = 0 for any η ∈ c0(B). Thus if η ∈ Mn ⊗ c0(B) then

‖η‖Mn(c0(B)) = ‖η‖Mn(X).

That is, the canonical map ι : c0(B) → X is a complete isometry.Fix z ∈ Mn ⊗ 1(B∗) for some arbitrary integer n ≥ 1, and let u be the asso-

ciated linear mapping from ∞(B∗∗) to Mn (as in (1.27), and using (2.28)). Notethat u is w∗-continuous, and therefore it is the unique w∗-continuous extensionto ∞(B∗∗) of the map u|c0(B) : c0(B) → Mn. Hence according to 1.4.8, and alsousing (1.27) and (2.28), we have∥∥u : ∞(B∗∗) −→ Mn

∥∥cb

=∥∥u|c0(B) : c0(B) −→ Mn

∥∥cb

= ‖z‖Mn(1(B∗)).

It follows from the last equations, and the fact that κ and ι are completelycontractive, and that u κ ι = u|c0(B), that we also have

‖z‖Mn(1(B∗)) =∥∥u κ : X −→ Mn

∥∥cb

. (2.31)

Being isometric to ∞(B∗∗), X is a dual Banach space by (2.28). We shall nowassume that B is a C∗-algebra. Then ∞(B∗∗) is a W ∗-algebra, and so 1(B∗)is the unique Banach space predual of X . Equation (2.31) may be rephrased asthe fact that if we view 1(B∗) ⊂ X∗ in the natural way, then the operator spacestructure induced on 1(B∗) from X∗, coincides with its usual one from 1.4.13.Let Z denote 1(B∗) equipped with another operator space structure for whichX = Z∗ completely isomorphically via the canonical map. Then Z = 1(B∗)completely isomorphically by our rephrased version of (2.31). Passing to operatorspace duals, we find that X = ∞(B∗∗) completely isomorphically as well. Thismeans that there is a constant c > 0 such that

‖η‖Mn(∞((B∗∗)op)/c0((B∗∗)op)) ≤ c‖η‖Mn(∞(B∗∗))

for any n ≥ 1 and any η ∈ Mn(∞(B∗∗)). Restricting to matrices η whose entriesare constant sequences valued in B, we deduce that

‖η‖Mn(Bop) ≤ c‖η‖Mn(B), n ≥ 1, η ∈ Mn(B).

Hence we obtain a contradiction by choosing our B so that the last inequality isfalse (take for example B to be B(H) or S∞(H), with dim(H) = ∞).


An example which also satisfies the last lemma, but which looks much simpler,is obtained by defining X to be the subspace of B(H) ⊕∞ (B(H)/S∞(H))op

consisting of pairs (T, T ) for T ∈ B(H). The arguments of the lemma go throughverbatim almost to the end. However the last three lines of the proof (in the casethat c > 1) then require nontrivial facts about the Calkin algebra B(H)/S∞(H).

Corollary 2.7.16 There exists a unital operator algebra A with a Banach spacepredual for which the product is separately w∗-continuous but for which there isno complete isometry J : A → B(H) which is w∗-continuous, or whose range isw∗-closed. Such an algebra A has no operator space predual.

Proof Let X be an operator space satisfying Lemma 2.7.15, and let Z denoteits Banach space predual. Consider the unital operator algebra A = U(X) definedby 2.2.10, and we will prove that it satisfies the requirements above. First notethat Min(X) = Max(Z)∗ is a dual operator space (see 1.4.12). Thus U(Min(X))is a dual operator algebra by Lemma 2.7.7 (2). Moreover A is isometricallyisomorphic to U(Min(X)) by Proposition 2.2.12. This proves that A has a Banachspace predual for which the product is separately w∗-continuous. Second, notethat if A were a dual operator space, then it would be a dual operator algebraby Theorem 2.7.9. By Lemma 2.7.7 (1), this would imply the contradiction thatX ⊂ A is a dual operator space. If there existed a w∗-continuous completeisometry J : A → B(H) then the range of J is w∗-closed by A.2.5, so that Awould be a dual operator space (using 1.4.6).


Nonselfadjoint operator algebras were first studied by Kadison and Singer [215],and a little later by Ringrose (see references in [364]). Shortly after this studycommenced, many major and foundational papers on the subject were writtenby Arveson—for example, see [20–23,25]. It was Arveson who defined completecontractions, and complete isometries, and recognized their importance to op-erator algebras and operator spaces. Around that time, motivated in large partby operator theory and the invariant subspace problem, dual operator algebrasbegan to be studied extensively using Banach space techniques. We will not at-tempt the long list of names that might be mentioned here, we would be sureto omit some. For some citations see [23,32,84,108,240,337,354], and referencestherein. Operator algebras occur quite naturally in mathematics, for example asthe ‘classifying object’ for many problems. In the 1970s also, Varopoulos initi-ated another direction in the study of operator algebras as Banach algebras. Forexample, see [417,116,124,123,411,82]; amongst other things, these authors gavesome characterizations, and produced many surprising examples of Banach alge-bras which are/are not bicontinuously isomorphic to operator algebras. We saya little more on this work (which greatly influenced the subject of our book) inChapter 5. We have included some results on Banach algebras in the Appendix;others may be found in the standard texts, for example [74, 106,297,298].


Again, we will not attempt to describe the huge literature on nonselfadjointoperator algebras from the 1980s and later. The deepest examples and specialclasses of operator algebras are out of the range of our book. We hope that atsome future point we might see a text entitled ‘Operator algebras by example’ tofollow [108,110]! General results, which apply to all operator algebras, seem to berather scarce in the literature. Our book of course only treats topics connectedwith operator spaces, and even then is incomplete due to limitations of space.We have omitted to address a few major topics which are extensively covered inother books [108,314,335,337], such as dilation theory.

2.1: The result 2.1.7 is from [46]. The unitization of approximately unital(matrix normed) operator algebras may first appear in [344]. Meyer’s results [277]suggest that perhaps other questions about the ball can be transformed intoquestions involving positivity. The diagonal ∆(A) has played a role since thebeginning of the subject. Arazy and Solel show in [14] that a surjective linearisometry T : A → B between unital operator algebras takes the diagonal ontothe diagonal. It then follows from a result of Kadison [212] that T (1) is a unitaryv in ∆(B), so that v∗T (·) is a unital isometry of A onto B. They also showthat for unital surjective isometries, T (xy + yx) = T (x)T (y) + T (y)T (x) for allx, y ∈ A. See also [196].

If A is a unital C∗-algebra, then ‖a‖ = sup |ϕ(a)| for a ∈ A+, the supremumover states ϕ of A. There is a result resembling this which is true if A is an oper-ator algebra. Namely, define a matrix state of A to be a completely contractiveunital ϕ : A → Mn. Then ‖a‖ = sup ‖ϕ(a)‖ for any a ∈ A, where the supremumis taken over matrix states of A. A similar formula holds for a ∈ Mn(A). Indeed,suppose that A ⊂ B(H), that a ∈ Mn(A), and consider a finite subset of vectorsin H on which a ‘achieves its norm’ within ε. Let K be the span of these vectors,let PK be the projection onto K, and let ϕ(x) = PKx|K . Since B(K) ∼= Mm forsome m, we may view ϕ as a matrix state on A. The result is now easy (see [69]and [394]).

If A is an operator algebra which is not approximately unital, we say thata homomorphism π : A → B(H) is ∗-nondegenerate if the span of terms of theform c1c2 · · · cnζ, for ζ ∈ H and ci ∈ π(A) ∪ π(A)∗, is dense in H . The universalrepresentation defined in 2.4.4 is a good example of a ∗-nondegenerate repre-sentation. For operator algebras with cai, one can show that ∗-nondegeneracycoincides with our earlier notion of nondegeneracy. Indeed, it is not hard to seeusing A.1.5 that an operator algebra A possesses a cai if and only if wheneverπ : A → B(H) is a ∗-nondegenerate completely contractive homomorphism, andwhenever x ∈ H , then x ∈ [π(A)x] (see also [72]).

Our book focuses on approximately unital operator algebras for the mostpart, however some analoguous theory of operator algebras with a one-sided caimay be found in [55] and [60].

2.2: The constructions in this section are well-known, with the exception of afew of the facts mentioned in 2.2.3. An early study of the spatial tensor productof nonselfadjoint algebras may be found in [316]. The constructions in 2.2.9 and


2.2.10 we have heard attributed to Arveson. The norm relation in 2.2.11 is well-known (e.g. see Chapter IV Section 2 in [159] for a generalization); its usage in2.2.11 is taken from [279]. A frequently used variant of the algebra in 2.2.10, isits subalgebra with repeated diagonal entries.

Simple modifications to 2.2.1 show that the ‘c0-sum’ (see 1.2.17) of operatoralgebras is an operator algebra. Many other general operator algebra construc-tions have been considered by others. As just one example, we mention the workof Gilfeather and Smith (see [170,385], and references therein; in the former pa-per the automorphisms of certain operator algebras are parametrized and studiedusing the weak* Haagerup tensor product of 1.6.9).

2.3: The BRS theorem is due to Blecher, Ruan, and Sinclair [69]. The nonuni-tal case appeared first in [373]. The proof given here is due to Le Merdy [246];we have presented this proof in part because of its capacity to generalize. Infact one need not assume associativity of the product in the BRS theorem, thisis automatic as proved in the original paper [69]. That the quotient Banach al-gebra of an operator algebra A by an ideal is isometrically isomorphic to anoperator algebra, was proved by Cole when A is a function algebra [424], andthen extended to general operator algebras by Bernard [34] and Lumer [259]in the early 1970s. A direct proof of this may also be found in [337], and it isused there to give an alternative proof of the BRS theorem. It is an interestingopen question as to whether every unital Banach algebra for which von Neu-mann’s inequality holds, is an operator algebra (e.g. see [125]). Interpolation ofoperator algebras considered as Banach algebras was briefly studied in [415].See [44, 62] for interpolation in the sense considered here. Nonselfadjoint directlimit operator algebras have been well studied; Power used the BRS theorem toconstruct such algebras in [350]. In [241] there is an explicit identification of the‘C∗-envelope’ (see Section 4.3) of an interesting class of direct limit algebras. Seealso [111,112,351] for example.

Matrix normed algebras were first mentioned in [42]; it was observed therethat the Fourier algebras of a locally compact group were examples of these. Theywere used extensively in the middle 1990s by Effros, Ruan, and Kraus, under thename completely contractive Banach algebra, in order to study Banach algebraswhich have an operator space structure of specific interest. For example, see [377]or Chapter 16 of [149], and references therein.

2.4: The name ‘C∗-cover’ is due to Muhly we believe, however the conceptitself was developed by Arveson. The maximal universal C∗-algebra C∗

max(A)of an operator algebra A was mentioned in [67, 64], and it (and the associated‘universal representation of A’) played a crucial role in [50, 52, 72]. The latterpaper treated the nonunital case first. Another construction of C∗

max(A), andExample 2.4.5, may be found in [50]. Duncan has established some functorialproperties of C∗

max(·), and computed it for certain interesting classes of algebras(e.g. see [129]).

Note that saying that one C∗-cover (B, j) is dominated by another (B′, j′)in the ordering defined in 2.4.1, is the same as saying that if one forms any poly-


nomial p in variables coming from A and the set a∗ : a ∈ A of formal adjoints,then ‖p‖B ≤ ‖p‖B′. In this case, the kernel I of the canonical ∗-homomorphismfrom B′ to B is called a boundary ideal of B′ for A. Note that then B′/I ∼= B asC∗-algebras, and indeed also as C∗-covers of A. We show in Theorem 4.3.1 thata minimal C∗-cover exists in this ordering. We will take this fact for granted inthe following discussion. Let A be an operator algebra, and consider the familyof all C∗-covers of A. This has two extremal elements, namely the maximal oneC∗

max(A) from 2.4.3, and a minimal one whose existence we are taking on faith,and which we denote by C∗

e (A). For any C∗-cover (B, i) of A, we define KB tobe the kernel of the ∗-homomorphism C∗

max(A) → B provided by 2.4.2. It is easyto see from the universal property in 2.4.2 that (B, i) ≤ (B ′, i′) if and only ifKB′ ⊂ KB. In particular (B, i) and (B′, i′) are equivalent with respect to thecanonical equivalence relation on C∗-covers, if and only if KB′ = KB. Next con-sider the ideal J defined to be KB in the particular case when B = C∗

e (A). Thenby the above universal property of these covers, it is easy to see that KB ⊂ Jfor any C∗-cover B of A. Hence such KB is an ideal of J . Conversely, for anyclosed ideal K of J , it is easy to see that C∗

max(A)/K is a C∗-cover of A. Puttingall the above together, we have that the correspondence B → KB is a bijectiveorder-reversing correspondence between the partially ordered set C(A) of equiv-alence classes of C∗-covers of A, and the set of closed ideals of J . The last setis a lattice. Indeed by [122] 3.2.2, this lattice is lattice isomorphic to the set ofopen sets in a topological space, namely the open sets in the spectrum J of theC∗-algebra J . We recall that the spectrum J is the set of the equivalence classesof irreducible representations of J .

Thus we have shown that if A is an operator algebra, then C(A) is a com-plete lattice, which is lattice anti-isomorphic to a certain topology. More pre-cisely, C(A) is lattice anti-isomorphic to the set of open sets in the generallynon-Hausdorff topological space J . In this canonical way, one can associate atopological space with any operator algebra A. It is an interesting exercise tocompute this topological space explicitly in the case that A = T 2, using 2.4.5.

The matrix factorization idea and several of the key examples in this sectionare from [67]. Some remarks on [67], and a list of other examples that may betreated in this framework, including the enveloping operator algebra discussedhere, may be found in [45]. In connection with 2.4.14, it should be noticed thatsince von Neumann’s inequality does not extend to the case of three commutingcontractions [416], the unital operator algebra O(N3

0) cannot be isometricallyidentified with the tridisc algebra A(D3) ⊂ C(T3). Pisier put the matrix fac-torization idea to excellent use in his notion of similarity degree—see [337] andreferences therein. He also has various extensions and complements to this circleof ideas in [337]. For example, the notion of the ‘universal operator algebra’ O(E)of an operator space E is thoroughly developed there, together with its relationsto the topics above. See [250] for some versions appropriate to the w∗-topology.Paulsen has also further exploited such factorization formulae in a series of re-cent papers. For example, he cleverly uses such formulae to recover and extend


a result of Agler related to interpolation. See [314, Chapter 18], [312, 313], andreferences therein.

One may define the noncommutative maximal ideal space of any Banach al-gebra which is also an operator space, to be the C∗-algebra C∗

max(O(B)). Thisis easily seen to have the following universal property: if π : B → D is anycompletely contractive homomorphism into a C∗-algebra D, then there exists a(necessarily unique) ∗-homomorphism π : C∗

max(O(B)) → D such that π j = π.A variant of 2.4.20 (2) and 2.4.21, in the C∗-algebra case, was first proved

by Christensen, Effros, and Sinclair [92]. The version presented here is due toPisier [329]. The ‘unital free product operator algebra’ discussed in this sec-tion is ‘amalgamated over the scalars’. If one does not amalgamate, one obtainsanother free product, or ‘coproduct’, of operator algebras. See [337] and ref-erences therein. We have also not discussed ‘reduced free products’ here (e.g.see [418]). It is also shown in [67] that the free product operator algebra con-struction is injective. That is, if Ai is a unital-subalgebra of Bi for i = 1, 2, thenA1∗A2 ⊂ B1∗B2 completely isometrically. It is fairly clear that if G1 and G2 aretwo discrete semigroups, then the universal properties in 2.4.9 and 2.4.18 showthat O(G1) ∗ O(G2) = O(G1 ∗ G2).

2.5: Most Banach algebras are not Arens regular; for example, see [106]and [130]. The latter is a good survey of the Arens product and the basic featuresof Arens regularity. For instance if G is an infinite locally compact Hausdorffgroup, then the group algebra L1(G) is not Arens regular [438]. See [107] (andreferences therein) for more on this theme. A version of the result 2.5.6 appearedin [144] (one also needs 1.4.11 to make the results match precisely). The fact2.5.8 has perhaps been known since [97]. The principle 2.5.10 is from [65], witha different proof. It is easy to see that if A is an Arens regular matrix normedalgebra, then A∗∗ is a matrix normed algebra too.

2.6: The two-sided multiplier algebra of an operator algebra was first studiedby Poon and Ruan [344] in terms of the double centralizer algebra. This objectmay be defined just as in the Banach algebra case (see [297, 106]). Poon andRuan check that with natural matrix norms, the double centralizer algebra isa unital operator algebra containing A completely isometrically, and that it isisomorphic to the appropriate subalgebras of A∗∗ and B(H) (for nondegeneratecompletely isometric representations of A on H). One-sided and two-sided multi-plier algebras of operator algebras were developed from a different angle aroundthe same period by Blecher, Muhly, and Paulsen as a necessary tool for thetheory of nonselfadjoint operator algebra Morita equivalence (see [65, 46, 309]).Proposition 2.6.12 is taken from [46]. Kraus and Ruan discuss multiplier algebrasfor a matrix normed algebra in [236]. Item 2.6.17 is well known.

We leave it as an exercise (using 2.1.6, say) that if A is an approximatelyunital operator algebra, then its diagonal (2.1.2) satisfies

∆(A) = A ∩ ∆(M(A)) = A ∩ ∆(LM(A)).

As another exercise, the reader might check that for a unital operator algebra A


and a cardinal I, we have LM(KI(A)) = MwrI (A) (see 2.2.3 for this notation).

Note that this implies that M(KI(A)) = MwrI (A) ∩ M

wcI (A) for such A. If A is

nonunital then these do not hold necessarily; in this case we can add to the list ofinfinite matrix operator algebras in 2.2.3, the algebras LM(KI(A)), RM(KI(A)),and also the subalgebras Cw

I (A)RI(A) and CI(A)RwI (A).

2.7: These results are due to Le Merdy [248, 249], with the following excep-tions. The separate w∗-continuity assertion in Theorem 2.7.9 follows immediatelyfrom the notion of operator space multipliers and Magajna’s result 4.7.1. It hadbeen first established in the unital case by Blecher in [54]. Corollary 2.7.16 isalso from the latter paper. The proof of Theorem 2.7.9 shows that an operatoralgebra which is a dual Banach space is isometrically w∗-homeomorphic to a dualoperator algebra if and only if the product is separately w∗-continuous. Ruanfirst used the canonical projection from M ∗∗ to M to study dual operator alge-bras [372, Lemma 2.2]. The fact that the predual of a dual operator algebra neednot be unique is probably quite old (we heard it first mentioned by Westwood,using the U(X) trick too), and was further investigated by Ruan [372]. Gode-froy’s work on unique preduals (e.g. see the survey [171]) may also be combinedwith the U(X) trick. The first example of an operator space which is a dual Ba-nach space but not a dual operator space was found by Le Merdy in [244]. Thelater example described in Lemma 2.7.15 is a variation on examples found inde-pendently by Peters and Wittstock [323], and Effros, Ozawa, and Ruan [141]. Weare grateful to Neufang for communicating the example of Peters and Wittstockto us. Proposition 2.7.11 was independently obtained by Arias and Popescu [16].Observation 2.7.14 is new, and is related to [172, Theorem II.1].

By an almost identical argument to that of 2.4.2, if M is a dual operatoralgebra then one may define a maximal W ∗-algebra W ∗(M). There is a w∗-continuous completely isometric homomorphism j : M → W ∗(M) whose rangegenerates W ∗(M) as a W ∗-algebra, and which possesses the following universalproperty: given any w∗-continuous representation π : M → B(H), there exists aw∗-continuous ∗-representation of W ∗(M) on H extending π. See [72].

General dual function algebras are discussed in [257,443,83], for example. In-deed by Theorem 2.7.9, the objects called uniform dual algebras by those authors,are exactly the function algebras with a Banach space predual. For a sample ofother recent results on general dual operator algebras the reader might con-sult [54], [55] (for dual algebras with a one-sided identity), [113] (and referencestherein), or [352, Theorem 4]. The ideas in the proof of 2.7.9 break down for gen-eral Banach algebras. Indeed, the related question for Banach algebras may bephrased: Does a unital Arens regular Banach algebra with a predual, necessarilyhave a separately w∗-continuous product? We are grateful to Lau for showing usa counterexample.

3

Basic theory of operator modules

3.1 INTRODUCTION TO OPERATOR MODULES

Operator modules are one of the main themes of our book. One is presentedimmediately with operator modules when one first meets operator algebras, inthe form of the Hilbert spaces on which they act. Then as one gets deeper into thetheory of operator algebras one finds that they arise naturally in many places, andprovide an important tool for the analysis of operator algebras. Indeed they forma class that is large enough to include the Hilbert modules just mentioned, theoperator algebras themselves, and important subclasses such as the C∗-modules;and yet is not so large that one loses too much contact with the underlyingHilbert space geometry. For example, we shall see in Chapter 8 that most basicC∗-module constructions, such as their usual tensor products, are in essenceoperator module constructions, but are not Banach module constructions.

In this chapter we treat the basics of the ‘completely isometric’ theory ofmodules. Thus in this chapter if X and Y are (left, say) A-modules over analgebra A which are also operator spaces, then we consider these modules to be‘the same’ if X and Y are completely isometrically isomorphic via an A-modulemap. In this case we say that X and Y are completely isometrically A-isomorphic,and we write X ∼= Y completely A-isometrically. A similar terminology will beused for bimodules. We shall not discuss the ‘completely isomorphic’ theory ofmodules here; this will be touched on in Section 5.2.

Unless stated otherwise, throughout the first section of this chapter, the sym-bols A, B are reserved for algebras which are at least Banach algebras with anoperator space structure. They do not necessarily have any kind of identity orapproximate identity. This class includes of course all operator algebras.

3.1.1 (Operator modules) A concrete left operator A-module is a linear sub-space X ⊂ B(K, H), which we take to be norm closed as always, together witha completely contractive homomorphism θ : A → B(H) for which θ(A)X ⊂ X .Such an X is a left A-module via θ. An (abstract) left operator A-module is anoperator space X which is also a left A-module, such that X is completely iso-metrically A-isomorphic to a concrete left operator A-module. This is equivalentto saying that there is a complete isometry Φ: X → B(K, H), for some Hilbertspaces H, K, and a completely contractive homomorphism θ : A → B(H) suchthat Φ(ax) = θ(a)Φ(x) for any a ∈ A and x ∈ X .

Basic theory of operator modules 103

Similar definitions hold for right modules and bimodules. Thus a concreteright operator B-module is a subspace X ⊂ B(K, H) with Xπ(B) ⊂ X for acompletely contractive homomorphism π : B → B(K). For a concrete operatorA-B-bimodule we have both θ(A)X ⊂ X and Xπ(B) ⊂ X . In fact it is oftenconvenient to treat one-sided modules as bimodules. This is done by simplydefining another action of the scalars C on X in the obvious way.

3.1.2 (Examples of operator modules)(1) If H, K are Hilbert spaces, and if θ : A → B(H) and π : B → B(K) are

completely contractive homomorphisms, then B(K, H) is a concrete operatorA-B-bimodule (with the canonical module actions).

(2) Submodules of operator modules are clearly operator modules.(3) Any operator space X is an operator C-C-bimodule.(4) Any operator algebra A is an operator A-A-bimodule.(5) If A is an approximately unital operator algebra and if p, q are pro-

jections in A (or projections in the multiplier algebra M(A)), then qAp is anoperator (qAq)-(pAp)-bimodule. If q = 1− p, then we say as in 2.6.14, that qApis a corner of A. We shall in fact see in 3.3.4 that these are essentially the onlyexamples of operator bimodules over approximately unital operator algebras. IfA is a C∗-algebra, then the operator bimodule qAp is an example of a ‘rightC∗-module’ over pAp (see Chapter 8), and conversely we shall see in 8.1.19 thatevery right C∗-module is a corner of a C∗-algebra.

(6) A closed left or right ideal J in a C∗-algebra A is an operator bimoduleover JJ and JJ . Here J denotes the adjoint space of J (see 1.2.25). Moregenerally, let Z be a norm closed linear subspace of B(H), or of B(K, H) fortwo Hilbert spaces H and K, such that ZZZ ⊂ Z. We say that such Z is aternary ring of operators, or TRO. Such a space Z is an operator bimodule overthe C∗-algebras ZZ and ZZ. This example will be important in Sections 4.4and 8.3.

3.1.3 (h-modules) Let X be an operator space which is also a left A-module.We say that X is a (left) h-module over A if the module action on X , viewed asa linear mapping A ⊗ X → X , extends to a complete contraction

A ⊗h X −→ X . (3.1)

Since the Haagerup tensor product linearizes completely bounded bilinear maps(see 1.5.4), the last line is also equivalent to saying that the spaces Mn(X) areleft Banach Mn(A)-modules in the canonical way, for every n ∈ N. That is,

‖ax‖n ≤ ‖a‖n‖x‖n, n ≥ 1, a ∈ Mn(A), x ∈ Mn(X). (3.2)

Similar definitions hold for right modules. Then an h-bimodule over A and B isby definition an A-B-bimodule which is both a left h-module over A and a righth-module over B.

Any left operator A-module is an h-module over A. Indeed suppose that X isa θ(A)-submodule of B(K, H), for a completely contractive θ as in 3.1.1. Then for

104 Introduction to operator modules

a ∈ Mn(A) and x ∈ Mn(X), we have ‖θn(a)x‖n ≤ ‖θn(a)‖n‖x‖n ≤ ‖a‖n‖x‖n.Thus we have verified (3.2). Of course a similar result holds for right modules, andhence for bimodules. In Section 3.3 and in 4.6.7, we will prove that conversely, ifA and B are approximately unital and if X is a nondegenerate A-B-bimodule,then X is an operator A-B-bimodule provided that the actions A ⊗h X → Xand X ⊗h B → X are both completely contractive.

3.1.4 (Matrix normed modules) It is natural in view of the discussion in A.6.1to consider the A-modules that correspond to completely contractive homomor-phisms A → CB(X), for a operator space X . We shall call these matrix normedmodules. It is easy to see that these are exactly the left A-modules X , such thatX is also an operator space, and such that also

‖[aijxkl]‖nm ≤ ‖[aij ]‖n‖[xkl]‖m (3.3)

for all [aij ] ∈ Mn(A), [xkl] ∈ Mm(X), and all m, n ∈ N. By 1.5.11, a reformula-tion of (3.3) is that the module action on X extends to a complete contraction

A⊗ X −→ X. (3.4)

Similar definitions hold for right matrix normed modules. Then a matrix normedA-B-bimodule is by definition an A-B-bimodule which is both a (left) matrixnormed A-module and a (right) matrix normed B-module.

3.1.5 (Examples of matrix normed modules)(1) Every h-module, and hence every operator module is a matrix normed

module. To see this we simply compare (3.1) and (3.4), and we apply 1.5.13.Similar assertions hold for bimodules.

(2) If X is a matrix normed A-B-bimodule, then X∗ is a matrix normedB-A-bimodule for the dual actions defined by

〈bϕ, x〉 = 〈xb, ϕ〉 and 〈ϕa, x〉 = 〈ϕ, ax〉

for a ∈ A, b ∈ B, x ∈ X, ϕ ∈ X∗. That this is a matrix normed bimodule is asimple exercise using the definitions (see 1.2.20).

(3) Any matrix normed algebra A (see 2.3.9) is a matrix normed A-moduleover itself. More generally, if X is an operator space, then A

⊗X is a left matrix

normed A-module. To see this we appeal to (3.4). That the obvious left A-moduleaction on A

⊗X satisfies (3.4) follows from the ‘associativity’ and ‘functoriality’

of the operator space projective tensor product (see 1.5.11):

A⊗ (A

⊗ X) ∼= (A

⊗ A)

⊗ X

m⊗IX−→ A⊗ X

where m is the product mapping on A. A similar argument shows that if Y is anyleft matrix normed A-module, then Y

⊗ X is a left matrix normed A-module.

In the case that X = S1n, the dual of Mn, then we write S1

n[A] for A⊗ S1

n. This


is a basic ‘building block’ in the category of matrix normed modules over A, aswe will see in 3.4.7.

(4) Suppose that X is a Banach A-module over a Banach algebra A. In 2.3.9we showed that Max(A) is a matrix normed algebra. By a similar argument(using 1.5.12 (2) and (3.4)) it is easy to see that Max(X) is a matrix normedMax(A)-module. Thus the class of matrix normed modules contains the class ofBanach A-modules in some sense. This shows that one usually cannot expectvery strong general theorems to hold for all matrix normed modules.

3.1.6 (Hilbert modules) These constitute one of the simplest and most impor-tant classes of operator modules. We define a Hilbert A-module to be a Hilbertspace H which is a left A-module whose associated homomorphism θ : A → B(H)is completely contractive. If A is a C∗-algebra then the last condition is equiva-lent to saying that θ is a ∗-homomorphism, as we noted in 1.2.4. We sometimeswrite such a Hilbert module as Hθ. Some authors call this a completely con-tractive Hilbert module, and are also interested in the case when θ is merelybounded or completely bounded. We will be more restrictive. Unless otherwisestated we will assume that our Hilbert modules are nondegenerate, which by2.1.5 is equivalent to saying that θ is nondegenerate. Moreover a Hilbert moduleH will always be viewed as an operator space by assigning it the ‘column Hilbertspace structure’ Hc discussed in 1.2.23. It is easy to see that Hilbert modulesare exactly the left operator modules from Example 3.1.2 (1) in the case thatK = C. An alternative, and often useful, identification of Hilbert modules withcertain concrete operator modules proceeds as follows. Let θ : A → B(H) be acompletely contractive homomorphism, let η be a fixed unit vector in a Hilbertspace K, and let H ⊗ η be defined as in 1.2.23, namely, the set of rank oneoperators ζ⊗η ∈ B(K, H) : ζ ∈ H. Being left invariant under θ(A), H ⊗η is aconcrete left operator A-module. By 1.2.23 this module is completely isometricto Hc, and it is clear that this isometry is also an A-isomorphism.

The next result is a useful characterization of Hilbert modules.

Proposition 3.1.7 Let H be a Hilbert space, let θ : A → B(H) be a nondegener-ate bounded homomorphism, and consider Hc as a left A-module in the canonicalway (via θ). The following are equivalent:

(i) θ is completely contractive,(ii) Hc is a left operator A-module,(iii) Hc is a left matrix normed A-module.

Proof That (i) implies (ii) follows from Example 3.1.2 (1) as we noted in 3.1.6.That (ii) implies (iii) follows from 3.1.5 (1). That (iii) implies (i) follows fromthe fact that B(H) = CB(Hc) completely isometrically (see (1.14)), and fromthe first few lines of 3.1.4.

3.1.8 (The ∞-direct sum of operator modules) For a family Xi : i ∈ Iof operator A-B-bimodules, the ∞-direct sum ⊕i Xi discussed in 1.2.17 is anoperator A-B-bimodule. This is easily seen by an argument analoguous to the


one in 2.2.1, using, for example, a property of the ∞-sum mentioned in the lastparagraph of 1.4.13. Similarly, ⊕i Xi is an operator bimodule over ⊕i Ai and⊕i Bi, if each Xi is an operator Ai-Bi-bimodule.

3.1.9 (The 1-direct sum of matrix normed modules) Let Xi : i ∈ I be afamily of left matrix normed A-modules say. The 1-sum ⊕1

i Xi defined in 1.4.13,with the obvious left A-module action, is a left matrix normed A-module. Oneway to see this is to note that the canonical complete contractions A

⊗Xi → Xi

give, via the universal property of the 1-direct sum (see 1.4.13), a completecontraction ⊕1

i (A⊗ Xi) → ⊕1

i Xi. Putting this together with (1.52) yields acanonical complete contraction

A⊗ (⊕1

i Xi) −→ ⊕1i Xi.

Appealing to (3.4) we see that ⊕1i Xi is a left matrix normed A-module.

3.1.10 (Quotient modules) Let X be an operator space, and suppose that X isalso an A-B-bimodule. If Y is an A-B-submodule of X (as always, assumed to beclosed), then the quotient operator space (see 1.2.14) X/Y is an A-B-bimodule inthe canonical way. This is simple algebra, indeed if q : X → X/Y is the canonicalquotient map then q is an A-B-bimodule map too. It is also easy to see that if inaddition A and B act nondegenerately on X , then they also act nondegeneratelyon X/Y . This follows from the simple fact that if V is a dense subspace of Xthen q(V ) is dense in X/Y . If X is a matrix normed A-B-bimodule, then weclaim that X/Y is a matrix normed A-B-bimodule too. To see this suppose thatz ∈ Mm(X/Y ) and a ∈ Mn(A), both with norm < 1. Then there is a matrixx ∈ Mm(X) of norm < 1 which is mapped to z by the quotient map. Then[aijxkl] has norm < 1 in Mnm(X), so that [aijzkl] = qnm([aijxkl]) has norm< 1 in Mnm(X/Y ). Thus X/Y is a matrix normed A-module, and similarly itis a matrix normed B-module. A similar proof shows that the quotient of anh-bimodule is an h-bimodule.

3.1.11 (Multiplier actions) There is a useful way to make a nondegeneratemodule over an approximately unital algebra, into a module over a unital algebra.For example, suppose that X is a nondegenerate left Banach module over anapproximately unital Banach algebra A. As at the beginning of Section 2.6, wedefine LM(A) = BA(A), and we define a left action of LM(A) on X by settingηx = η(a)x′ if η ∈ LM(A) and x = ax′ with a ∈ A, x′ ∈ X . Notice that

ηx = η(a)x′ = limt

η(eta)x′ = limt

η(et)x, (3.5)

if (et)t is any cai for A. This together with Cohen’s factorization theorem A.6.2shows that this action of LM(A) on X is well defined. Indeed from (3.5) it iseasy to check that X is a nondegenerate left Banach LM(A)-module. An almostidentical argument shows that if X is a nondegenerate left matrix normed module


over an approximately unital matrix normed algebra A, then the action describedin (3.5) makes X into a nondegenerate left matrix normed LM(A)-module.

Assume now that in addition X is a nondegenerate left operator A-module.We may assume that X is a θ(A)-submodule of B(K, H) for some Hilbert spacesK, H and a completely contractive homomorphism θ : A → B(H). By replacingH by its subspace [XK], which is θ(A)-invariant, we may assume that θ isnondegenerate. By definition, Hc is a Hilbert A-module with A-module actiondefined via θ. Hence Hc is a matrix normed A-module by 3.1.7. By the lastparagraph, Hc is canonically a matrix normed LM(A)-module. Using 3.1.7 again,we see that we have extended θ to a completely contractive unital homomorphismθ : LM(A) → B(H). Thus B(K, H) becomes a left operator LM(A)-module.With notation as in (3.5) we have

ηx = θ(η(a))x′ = θ(ηa)x′ = θ(η)θ(a)x′ = θ(η)x.

Thus θ(LM(A))X ⊂ X and the resulting action coincides with the one definedin (3.5). Thus X is actually a left operator LM(A)-submodule of B(K, H).

In particular, a nondegenerate left operator A-module is also a left operatormodule over M(A) or over the unitization A1, if A is an operator algebra, say.

3.1.12 (Prolongations) Let A, B be Banach algebras and assume that X isa left Banach A-module and that θ : B → A is a contractive homomorphism.If we define bx = θ(b)x for x ∈ X and b ∈ B, then X becomes a left BanachB-module. This B-module is called the prolongation of X by θ, and is sometimeswritten as θX . It is easy to see that if A, B, X are also operator spaces and if θ iscompletely contractive, then θX is an operator B-module (resp. matrix normedB-module) if X is an operator A-module (resp. matrix normed A-module).

Assume now that A, B are approximately unital matrix normed algebrasand that X is a nondegenerate left operator A-module. If θ : B → LM(A) is acompletely contractive homomorphism, then using 3.1.11 and the considerationsin the last paragraph, we see that X becomes a left operator B-module. If θis a left multiplier-nondegenerate morphism (in the sense of 2.6.11) from B toLM(A), then X is easily seen to be a nondegenerate left operator B-module. Acommonly encountered particular case of this, occurs when A is a left ideal in B.Since B is a matrix normed algebra, the canonical map from B into CBA(A) isa completely contractive homomorphism. Hence X becomes an operator moduleover B, which is nondegenerate if B is approximately unital.

3.1.13 (Matrix spaces) If X is a left operator module or matrix normed module,and if I, J are sets, then there are two obvious ways to attempt to make thematrix space MI,J(X) discussed in 1.2.26 into a module, namely via the formulae:

a[xij ] = [axij ] or [aij ][xij ] =[∑

k

aikxkj

],

assuming that these are defined. Similarly for right modules or bimodules. It isthese actions that are referred to in the next result.


Proposition 3.1.14 If X is an operator (resp. matrix normed) A-B-bimodule,and if I, J are sets, then KI,J(X) and MI,J(X) are operator (resp. matrixnormed) A-B-bimodules. If A and B are operator algebras, and if X is an oper-ator A-B-bimodule, then the following also hold:(1) KI,J(X) and MI,J(X) are operator KI(A)-KJ(B)-bimodules,(2) MI,J(X) is an operator M

wcI (A)-Mwr

J (B)-bimodule (see 2.2.3).(3) KI,J(X) and MI,J(X) are operator MI-MJ -bimodules.

Proof We treat only the left module action, the right being similar. We firstprove (2). Suppose that X is an operator A-B-bimodule. We may assume that Xis a θ(A)-submodule of B(K, H), where θ : A → B(H) is a completely contractivehomomorphism. Using (1.19), we will identify MI,J(B(K, H)) and MI(B(H))with B(K(J), H(I)) and B(H(I)) respectively. Thus we have a completely iso-metric embedding MI,J(X) ⊂ B(K(J), H(I)). Consider the ‘amplification’ θI , amap from M

wcI (A) to B(H(I)), this is completely contractive (see 1.2.26). It is

an easy exercise to check that θI is a homomorphism. By (1.21) it is clear thatMI,J(X) is a M

wcI (A)-submodule of B(K(J), H(I)). We have now established (2).

The second assertion in (1) follows from (2), since, for example, KI(A) is asubalgebra of M

wcI (A). To see the other assertion in (1), note that the product of

two ‘finitely supported’ matrices in MI(A) and MI,J(X) respectively, is clearlyfinitely supported, and thus is in KI,J(X). By density considerations we musthave KI(A) KI,J(X) ⊂ KI,J(X). Taking A = B = C in (2) gives the secondassertion in (3). One way to see the first part of (3) is to first note that KI,J(X)is an operator KI -KJ -bimodule, by the first part of (1), and then to use 3.1.11.

We finally prove the first part of the proposition. Here we only assume thatA, B are Banach algebras with an operator space structure. If X is an operatorA-B-bimodule then a variant of the proof of (2) above shows that MI,J(X) isan operator A-B-bimodule, and KI,J(X) is clearly an A-B-submodule. Considernow the case that X is merely a matrix normed A-B-bimodule. If I and Jare finite integers then it is quite obvious from a formula such as (1.5) thatMI,J(X) also is a matrix normed A-B-bimodule. In the case that I, J are infinitesets, we observe that since the norm in Mnm(MI,J(X)) ∼= MI,J(Mnm(X)) isthe supremum of the norm of finite submatrices, the calculation boils down tothe previous case. Thus MI,J(X) is a matrix normed A-B-bimodule. As before,KI,J(X) is a submodule, and hence is also a matrix normed A-B-module.

Corollary 3.1.15 If A is an approximately unital operator algebra, then CI(A)is a nondegenerate operator KI(A)-A-bimodule, and RI(A) is a nondegenerateoperator A-KI(A)-bimodule.

3.1.16 (Opposite and adjoint modules) These constructions allow one to makea left module into a right module, and vice versa. We will need the notation in1.2.25 and 2.2.8. Assume for simplicity that A is an operator algebra. If X is aleft operator A-module (resp. matrix normed A-module), then Xop is canonicallya right operator module (resp. matrix normed module) over Aop. Similarly, the


adjoint operator space X is a right operator module (resp. matrix normed mod-ule) over the adjoint operator algebra A. We leave these assertions as exercisesfor the interested reader.

3.2 HILBERT MODULES

We characterized Hilbert modules in 3.1.6 and the proposition following it. Inthis section we say a little more about their theory and applications.

3.2.1 (Submodules and quotients of Hilbert modules) It is clear that a sub-module (as always, assumed to be closed) of a Hilbert A-module is a HilbertA-module. Also, the quotient of a Hilbert A-module by a closed submodule isa Hilbert A-module too. This may be seen by combining 3.1.7 with the factthat the classes of nondegenerate matrix normed A-modules and Hilbert columnspaces are both closed under quotients (see 3.1.10 and 1.2.23). However there isanother way to see this fact which is more informative. Of course a submodule Kof a Hilbert A-module H is nothing more than a θ(A)-invariant subspace, whereθ : A → B(H) is the associated homomorphism. Let PK⊥ denote the projectiononto K⊥ = H K. Then the completely contractive map π = PK⊥θ(·)|K⊥ fromA into B(H K) is a homomorphism. Indeed this follows from the ‘generalprinciple’ proved in the lines following (2.26) in 2.7.10. This makes H K intoa Hilbert A-module and it is easy to check that this A-module is completelyA-isometrically isomorphic to the quotient module H/K discussed in 3.1.10.

If θ is a representation of A on a Hilbert space H , then a closed subspace Lof H is called a semi-invariant subspace for this representation, if PLθ(·)|L is ahomomorphism from A into B(L). Here PL is the projection onto L.

Proposition 3.2.2 (Sarason) Let θ : A → B(H) be a representation. A sub-space L of H is a semi-invariant subspace for θ if and only if there exist twoA-submodules H2 ⊂ H1 ⊂ H such that L = H1 H2.

Proof The ‘if part’ follows from the above discussion, with H and K replacedby H1 and H2 respectively. Conversely, if L is a semi-invariant subspace of H ,let H1 be the closure of the span of L and θ(A)L, and let H2 = H1 L. ThenL = H1 H2. Let P be the projection of H onto L. For a, b ∈ A and ζ ∈ L,

Pθ(a)(I − P )θ(b)ζ = Pθ(a)θ(b)ζ − Pθ(a)Pθ(b)ζ = 0

by hypothesis. Clearly Pθ(a)(I − P )L = 0. Thus by linearity and continuity,Pθ(a)(I − P )H1 = 0. Hence Pθ(a)H2 = 0, so that θ(A)H2 ⊂ L⊥. Since H1

is θ(A)-invariant, we see now that H2 is θ(A)-invariant.

3.2.3 (Reducing submodules) Let H be a Hilbert A-module with associ-ated homomorphism θ : A → B(H)), and let K be an A-submodule, that is,a closed θ(A)-invariant subspace. We say that K is a reducing submodule (or isA-reducing) if its orthogonal complement K⊥ also is θ(A)-invariant. We also usethe term summand for such a K. It is easy to see that K is reducing if and only

110 Hilbert modules

if θ(a) commutes with PK⊥ for any a ∈ A. In that case, the restriction of θ(·)to K⊥ maps into K⊥, and the resulting homomorphism mapping into B(K⊥)coincides with the homomorphism π = PK⊥θ(·)|K⊥ discussed in 3.2.1. Hence thesubmodule K⊥ coincides with the quotient module H/K.

If A is a C∗-algebra, then so is θ(A) by A.5.8. Thus in this case, every closedA-submodule of H is A-reducing.

3.2.4 (Direct sums of Hilbert A-modules) It is clear that the usual Hilbertspace direct sum ⊕i∈I Hi of a family Hi : i ∈ I of Hilbert A-modules, is againa Hilbert A-module. This simply corresponds to the usual ‘direct sum’ of theassociated representations. If each Hi is a copy of a single Hilbert module H , thenaccording to 1.1.4 we write H (γ) for ⊕i∈I Hi, if γ is the cardinal associated withI. We call this a multiple of H . If θ : A → B(H) is the associated representation,then we write θ(γ) for the associated representation of A on H (γ), and call thisa multiple of θ. We will need the simple fact that

θ(γ)(A)w∗

=(θ(A)

w∗)(γ). (3.6)

Indeed, suppose that T is in the right-hand set, that is, there exists S ∈ θ(A)w∗

with S(γ) = T . If (St) is a net in θ(A) converging to S, then S(γ)t → T , and so

T is in the left-hand set. Conversely, if (St) is a net in θ(A) such that S(γ)t → T ,

then to see that T is in the right-hand set, note that if Pi is the projectiononto the ith copy of H , then PiS

(γ)t Pj → PiTPj in the w∗-topology, for all i, j.

However PiS(γ)t Pj = 0 if i = j, and PiS

(γ)t Pi = St.

3.2.5 (The category HMOD) By 1.2.23, linear maps T between Hilbert mod-ules are bounded (resp. isometric) if and only if they are completely bounded(resp. completely isometric) with respect to the usual operator space structureon the Hilbert modules (that is, their ‘column space’ structure). Indeed for suchT , we have ‖T ‖ = ‖T ‖cb.

We define AHMOD to be the category of Hilbert A-modules, with boundedA-module maps as the morphisms. Sometimes these morphisms are called ‘in-tertwining maps’, since they ‘intertwine’ the associated representations. We saythat two Hilbert A-modules are spatially equivalent, and write H ∼= K, if theyare isometrically A-isomorphic, that is if there exists a unitary A-module mapfrom H onto K. We say that representations π, θ of A are quasi-equivalent ifthere is a multiple of π which is spatially equivalent to a multiple of θ. Thustwo Hilbert A-modules H and K are quasi-equivalent if and only if there arecardinals α and β (which by basic set theory we may assume to be equal) suchthat H(α) ∼= K(β). If A is a C∗-algebra this coincides with the usual definitionof quasi-equivalence found in C∗-algebra texts (e.g. see [122, 5.3.1]).

3.2.6 (Hilbert modules and noncommutative ring theory) The transplantingof important ideas from the purely algebraic theory of rings and modules (eg.see [8, 368]), into the theory of operator algebras, usually happens in two main


ways, at least from the standpoint of this book. In both, the ring is of coursereplaced by an operator algebra A. In the first method, one takes the A-modulesto be operator modules or matrix normed modules, including A itself. This isthe main method the reader will encounter in this book. In the second methodone takes the modules to be Hilbert modules. Although there are some strongrelations between these two methods, they are usually quite distinct. For exam-ple, the notion of ‘projective module’ means quite different things in the twocategories. As examples of the importance of the second method we cite thework of Rieffel and Connes on Hilbert space modules, which developed into thetheory of correspondences (discussed briefly in Section 8.5 below), the work ofDouglas and coauthors on Hilbert modules over function modules (see Douglasand Paulsen’s text [127], or [126] and references therein, for recent progress),and the work of Muhly and Solel on Hilbert modules over operator algebras(e.g. see [281,279]). We are unfortunately not able here to touch on the deep andimportant ideas developed by these authors. Instead we shall give a simpler illus-tration of the transplanting of ring theoretic ideas via Hilbert modules. Namely,we devote most of the rest of this section to a discussion of topics connected tothe algebraic concept of a ‘generator of a category’, leading to a generalization ofvon Neumann’s double commutant theorem to nonselfadjoint operator algebras.

Henceforth in this section we assume that A is an approximately unital oper-ator algebra, and we recall that all Hilbert modules are assumed nondegenerate.This is simply to make things clearer, the general case of the topics below isdiscussed in [72].

3.2.7 (Universal Hilbert modules and generators) A Hilbert A-module H (orequivalently, the associated nondegenerate representation of A on H) is said tobe A-universal, if every Hilbert A-module K is isometrically A-isomorphic (thatis, spatially equivalent) to an A-reducing submodule of a direct sum of copiesof H . We say that a module H in AHMOD is a generator (resp. cogenerator)for AHMOD if for every nonzero morphism R : K → L of AHMOD, thereexists a morphism T : H → K (resp. T : L → H) of AHMOD with RT = 0(resp. TR = 0). As in pure algebra (e.g. see [8]), there is a host of equivalentformulation of the notions of generator and cogenerator (see [72]).

3.2.8 (Examples of generators) We shall not use this, but if A is a C∗-algebra,then it is easy to see that a Hilbert A-module H is A-universal if and only if His a generator for AHMOD, and if and only if the unique w∗-continuous mapfrom A∗∗ to B(H) extending the representation of A on H , is one-to-one. Theseassertions are easy to deduce from, for example, the later results 3.2.11 (6), 3.2.12,3.8.5, and 3.8.6 and the remark following it. The situation is a little different forgeneral nonselfadjoint algebras. One can see that A-universal representationsare generically ‘very large’. However generators and cogenerators may be quite‘small’, as we see next.

3.2.9 (A generator for T 2) To illustrate the definition, we show that the directsum of the usual two-dimensional representation of T 2, the upper triangular 2×2

112 Hilbert modules

matrices, with the one-dimensional representation of ‘evaluation at the 1-1 entry’,is a generator. There are more sophisticated ways of obtaining this fact, but to seeit directly from the definition given here, suppose that R : K → L is a nonzeroT 2-module map between Hilbert T 2-modules. We may write K = K1⊕K2, withthe action of T 2 on K given by the expression[

a b0 c

] [ζη

]=

[aζ + bSη

cη

],

for a linear map S : K2 → K1. Here a, b, c ∈ C, ζ ∈ K1, η ∈ K2. Since R = 0we have R|K1 = 0 or R|K2 = 0. First suppose that R|K1 = 0. Then the actionof T 2 on K1 is just multiplication by the scalar in the 1-1 entry. Choose ζ inK1 (K1∩Ker(R)), ζ = 0, and take a nonzero map T : C

3 → Cζ which is zeroexcept on the third coordinate. It is easy to see that T is a module map intoK1, and RT = 0. Thus we may assume that R|K1 = 0 and R(η0) = 0 for someη0 ∈ K2. Define a map T : C

3 → K1 ⊕Cη0 taking e1 to Sη0, e2 to η0, and e3 to0. One easily checks that T is a module map into K, and RT = 0.

3.2.10 (A connection with the C∗-theory) Let A be an approximately unitaloperator algebra, and let C∗

max(A) be its ‘maximal C∗-algebra’ (see 2.4.3). From2.4.2 it follows that may regard Hilbert A-modules as Hilbert C∗

max(A)-modules,and vice versa, in a canonical way.

Suppose that H, K are Hilbert A-modules, that θ and π are the representa-tions of A on H and K respectively, and that T : H → K is a bounded A-modulemap. Then H , K are Hilbert modules over the adjoint operator algebra A aswell (see the last part of 2.2.8). If a ∈ A then π(a)T = Tθ(a), and so

T ∗π(a∗) = T ∗π(a)∗ = θ(a)∗T ∗ = θ(a∗)T ∗.

Hence T ∗ : K → H is an A-module map. Conversely, by symmetry, if T ∗ is anA-module map then T is an A-module map.

We call a bounded A-module map T between Hilbert A-modules adjointableif T ∗ is also an A-module map. From the last two paragraphs we deduce thatT is adjointable if and only if T is a C∗

max(A)-module map. We shall not needadjointable maps very much, except in the special case that i is an isometric mapbetween Hilbert A-modules, such that i and i∗ are A-module maps. In this caseit follows from the last paragraph that i and i∗ are C∗

max(A)-module maps. Inparticular, we deduce that unitary morphisms, that is, unitary A-module maps,are C∗

max(A)-module maps. Thus two Hilbert A-modules are spatially equivalentas A-modules if and only if they are spatially equivalent as C∗

max(A)-modules.From the above, it also follows that the set of A-reducing submodules of a HilbertA-module H is the same as the set of closed C∗

max(A)-submodules of H . IndeedHilbert A-module direct sums (resp. summands) of Hilbert A-modules are thesame as Hilbert C∗

max(A)-module direct sums (resp. summands).We collect together several simple deductions:


Proposition 3.2.11 Let A be an approximately unital operator algebra.

(1) A Hilbert A-module H is A-universal if and only if H is C∗max(A)-universal.

(2) The universal representation of A considered in 2.4.4 is A-universal.(3) Two Hilbert A-modules are quasi-equivalent as Hilbert A-modules if and only

if they are quasi-equivalent as Hilbert C∗max(A)-modules.

(4) Any two A-universal representations are quasi-equivalent.(5) If θ is a representation of A which is quasi-equivalent to an A-universal

representation, then θ is an A-universal representation.(6) Any A-universal module is a generator and a cogenerator for AHMOD.(7) H is a generator for AHMOD if and only if H cogenerates AHMOD.(8) If π and θ are quasi-equivalent representations of A, then there exists a

(necessarily unique) w∗-homeomorphic completely isometric isomorphism

ρ : π(A)w∗

−→ θ(A)w∗

such that ρ π = θ.

Proof (1) and (3) are obvious from the discussion above. Item (2) follows fromthe discussion above too, together with the fact that the universal representationof a C∗-algebra C is C-universal in the sense above (see the last fact in A.5.5).Item (5) is clear by the definitions, or is an easy exercise.

Assertion (4) is a simple application of set theory, and the well-known ‘Eilen-berg swindle’. If H and K are two A-universal representations, then there existcardinals α and β, and Hilbert A-modules M and N , such that H ⊕ M ∼= K(α)

and K ⊕ N ∼= H(β). By adding on extra multiples of H or K to the last twoequations, we may suppose that α = β. We may also assume (by replacing by alarger cardinal if necessary) that α · α equals α. Then

K(α) ∼= K(α) ⊕ K(α) ⊕ · · · ∼= H ⊕ M ⊕ H ⊕ M ⊕ · · ·∼= H ⊕ K(α) ⊕ K(α) ⊕ · · · ∼= H ⊕ K(α)

by associativity. Since α · α = α, a multiple of the last equation yields thatK(α) ∼= H(α) ⊕ K(α). Similarly, H(α) ∼= K(α) ⊕ H(α), hence H(α) ∼= K(α).

Item (7) follows from the definitions of generators and cogenerators, and thefirst two paragraphs of the discussion in 3.2.10. To prove (6), suppose that H isA-universal, and that R : K → L is a nonzero bounded A-module map. There is acardinal α such that K may be identified with an A-reducing submodule of H (α).Let Q be the associated projection onto K from H (α). Let εi be the inclusionmap of H into H(α) as its ith summand H . If every map R Q εi is zero, thenR Q and R are zero, which is a contradiction. Thus H is a generator. Using (1)we have that H is A-universal. By the above and (7), H is a cogenerator.

If π and θ in (8), are spatially equivalent, then (8) is clear. However this case,together with (3.6), easily implies the general case.

Thus the A-universal representations comprise one entire equivalence classfor the equivalence relation of quasi-equivalence.

114 Hilbert modules

3.2.12 (A property of universal representations) If a nondegenerate representa-tion π : A → B(H) is A-universal, then we claim that its w∗-continuous extension

π : A∗∗ → B(H) (see 2.5.5) is a complete isometry. Hence π(A)w∗ ∼= A∗∗. Indeed,

if π = πu is the universal representation of A (see 2.4.4), then the claim followsimmediately from the proof of 2.5.6. For a general A-universal representation θof A, we use (4) and (8) of 3.2.11. By those results, there exists a w∗-continuous

completely isometric isomorphism ρ : πu(A)w∗

→ θ(A)w∗

such that ρ πu = θ.By w∗-continuity, we deduce that ρ πu = θ. Thus θ is a complete isometry.

3.2.13 (The double commutant property) We say that a subspace Z of B(H)has the double commutant property if its double commutant equals its w∗-closure.In this case, it also equals the WOT-closure, since Z

w∗⊂ Z

WOT ⊂ Z ′′. We saythat a nondegenerate completely contractive representation π of an algebra A,or the associated Hilbert module Hπ, has the double commutant property if the

latter property holds for π(A). That is, if π(A)′′ = π(A)w∗

in B(H).We make several simple observations. If Z ⊂ B(H) then, by the w∗-continuity

of the involution (see A.1.2), we have Zw∗= (Z

w∗). It is easy algebra to check

that (Z)′′ = (Z ′′). Hence Z has the double commutant property if and only ifZ has the double commutant property. If K is another Hilbert space, then it isalso easy algebra to see that (Z ⊗ IK)′′ = Z ′′ ⊗ IK in B(H ⊗ K). In particular,and using also (3.6), if π : A → B(H) is a nondegenerate completely contractiverepresentation and if α is a cardinal, then π has the double commutant propertyif and only if π(α) has the double commutant property.

Theorem 3.2.14 Let A be an approximately unital operator algebra. If H isa generator or cogenerator for AHMOD, then H has the double commutantproperty. In particular, for any A-universal representation π we have

π(A)′′ = π(A)w∗ ∼= A∗∗, completely isometrically.

Proof The last part follows from the preceding assertions, Proposition 3.2.11(6), and 3.2.12. The ‘cogenerator case’ follows from the ‘generator case’ by Propo-sition 3.2.11 (7) and an assertion in 3.2.13. Thus we may suppose that H is agenerator for AHMOD, with associated nondegenerate representation π. LetL = H(∞) be the countably infinite multiple of H and let θ = π(∞) be its associ-ated representation. It is easy to see that L is a generator for AHMOD too. Fix anonzero ζ ∈ L, and consider K = [θ(A)ζ] ⊂ L. By 2.1.5 we have ζ ∈ K. Supposethat T ∈ θ(A)′′. If V : L → K is a bounded A-module map then V (regarded asa map into L) is in θ(A)′. Thus TV (L) = V T (L) ⊂ K. Consequently, T mapsthe range of V into K. Claim: T (K) ⊂ K. To see this, let W be the closure in Kof the span of the ranges of all such V as above. Clearly W is an A-submoduleof K, and by the above, T (W ) ⊂ K. If W = K, consider the quotient HilbertA-module K/W , and the canonical quotient map q : K → K/W . Since L is agenerator, there exists a bounded A-module map V : L → K such that qV = 0.This contradicts the definition of W , and proves our claim.


For ζ ∈ L, the Claim above shows that Tζ ∈ [θ(A)ζ] ⊂ [θ(A)w∗

ζ]. By A.1.5,

the set θ(A)w∗

, which coincides by (3.6) with π(A)w∗

⊗I∞, is ‘reflexive’. Putting

these facts together, T ∈ θ(A)w∗

. Thus we have shown that θ(A)′′ is contained

in θ(A)w∗

. Since the other inclusion trivially holds, θ has the double commutantproperty. By 3.2.13, π also has this property.

Thus we have a formal generalization of von Neumann’s double commutanttheorem. Indeed assume that A is a C∗-algebra and let π : A → B(H) be a

nondegenerate ∗-representation. To deduce the identity π(A)′′ = π(A)w∗

fromthe above proof, just observe that the submodule K ⊂ L automatically reducesA, as we noted in the last line of 3.2.3. Hence the orthogonal projection V fromL onto K is an A-module map. Thus Tζ = TV ζ = V Tζ ∈ K for ζ ∈ L, which iswhat is needed to obtain the desired conclusion.

A similar proof yields an analogue of 3.2.14 above, for any unital dual op-erator algebra M . In particular, there exist w∗-continuous completely isometricrepresentations π of M with π(M)′′ = π(M). See [72] for details.

3.3 OPERATOR MODULES OVER OPERATOR ALGEBRAS

The next theorem, the main result in this section, is a variation on a theoremof Christensen, Effros, and Sinclair. We will refer to it as the ‘CES theorem’.It gives a complete characterization of nondegenerate h-bimodules over a pairof approximately unital operator algebras. A generalization to bimodules overBanach algebras which are also operator spaces will be given in 4.6.7. Recall from3.1.3 that if A, B are operator algebras, say, then an A-B-bimodule X is called anh-bimodule, if the module actions extend to complete contractions A⊗h X → Xand X ⊗h B → X . Call these maps u and v respectively. Thus we obtain fromthe functoriality and associativity of the Haagerup tensor product (see 1.5.5) asequence of complete contractions

A ⊗h X ⊗h Bu⊗IB−→ X ⊗h B

v−→ X.

The composition of these maps is a complete contraction A ⊗h X ⊗h B → X ,taking a ⊗ x ⊗ b to axb for a ∈ A, b ∈ B and x ∈ X . It is often convenient toformulate the bimodule action as a single map from A ⊗h X ⊗h B to X .

Theorem 3.3.1 (CES theorem) Let A and B be approximately unital operatoralgebras, and let X be an operator space which is a nondegenerate h-bimoduleover A and B. Then there exist Hilbert spaces H and K, a completely isometriclinear map Φ: X → B(K, H), and completely contractive nondegenerate repre-sentations θ of A on H, and π of B on K, such that

θ(a)Φ(x) = Φ(ax) and Φ(x)π(b) = Φ(xb), (3.7)

for all a ∈ A, b ∈ B and x ∈ X. Thus X is completely A-B-isometric to theconcrete operator A-B-bimodule Φ(X). Moreover, Φ, θ, π may chosen to all becompletely isometric, and such that H = K.

116 Operator modules over operator algebras

If A = B then one may choose, in addition to all the above, π = θ.

Proof Suppose that X ⊂ B(L), for a Hilbert space L, and apply 1.5.7 (2) (andthe remark after it) to the map from A⊗h X⊗h B to X ⊂ B(L) described above3.3.1. We obtain

axb = Rθ1(a)Ψ(x)π1(b)S

for a ∈ A, b ∈ B and x ∈ X . Here Ψ: X → B(K1, H1) is a linear completecontraction, S : L → K1 and R : H1 → L are contractions; and θ1 and π1 are,respectively, the restrictions to A and B of ∗-representations of C∗-covers, onthe Hilbert spaces H1 and K1. Thus θ1 and π1 are completely contractive ho-momorphisms. We may assume, using 2.1.10 if necessary, that θ1 and π1 arenondegenerate representations.

Let K be the subspace [π1(B)SL] of K1, let P be the projection of K1

onto K, and let π = π1(·)|K = Pπ1(·)|K . View π as a completely contractivenondegenerate representation of B on K. Let H be the subspace [θ1(A)∗R∗L]of H1, let Q be the projection of H1 onto H , and let θ = Q θ1(·)|H . Note thatθ1(A)∗H ⊂ H , so that H⊥ is θ1(A)-invariant. Hence by 3.2.1, θ is a completelycontractive representation of A on H , and it is easy to see using 2.1.6 that θ isnondegenerate. Let Φ = QΨ(·)|K : X → B(K, H). For any a ∈ A, b ∈ B, π1(b)Smaps into K, whereas Rθ1(a) = Rθ1(a)Q. Hence for any x ∈ X , we have

axb = Rθ1(a)Φ(x)π1(b)S. (3.8)

One consequence of the last relation is that

‖Φ(x)‖ ≥ ‖Rθ1(et)Φ(x)π1(fs)S‖ = ‖etxfs‖,where (et)t and (fs)s are cai’s for A and B respectively. From this it is clear thatΦ is an isometry. Similarly, Φ is a complete isometry.

Consider a, a2 ∈ A, b, b2 ∈ B, x ∈ X , and ζ, η ∈ L. By (3.8),

〈(a2axbb2)ζ, η〉 = 〈Rθ1(a2)Φ(axb)π1(b2)Sζ, η〉= 〈Φ(axb)π1(b2)Sζ, θ1(a2)∗R∗η〉.

On the other hand, by (3.8) again this same quantity also equals

〈Rθ1(a2a)Φ(x)π1(bb2)Sζ, η〉 = 〈θ1(a)Φ(x)π1(b)π1(b2)Sζ, θ1(a2)∗R∗η〉= 〈θ(a)Φ(x)π(b)π1(b2)Sζ, θ1(a2)∗R∗η〉.

Putting together, this shows that

〈Φ(axb)π1(b2)Sζ, θ1(a2)∗R∗η〉 = 〈θ(a)Φ(x)π(b)π1(b2)Sζ, θ1(a2)∗R∗η〉.Now using linearity and density considerations, one sees that

θ(a)Φ(x)π(b) = Φ(axb).

If A and B are unital, we clearly have (3.7). In the contrary case, we note that asimplification of the argument above can be used to prove each of the relationsin (3.7) separately. This proves the first assertion or two of the theorem.


Next, we take a sufficiently large multiple of all our maps. More specifically,for some cardinal γ we know that H (γ) = K(γ) unitarily. We have

θ(a)γ Φ(x)γ π(b)γ = Φ(axb)γ .

Take V : H(γ) → K(γ) to be a unitary, and define Φ′ = V Φ(·)γ , θ = V θ(·)γ V ∗,and π = π(·)γ . Then it is easy to see that θ′ and π′ are nondegenerate com-pletely contractive representations on K (γ). Thus we may assume that H = K.By a similar argument we may also assume that there exist completely isometricnondegenerate representations ρ and σ, of A and B respectively, on H . Moreexplicitly, take two more Hilbert spaces on which there do exist such represen-tations and take large enough multiples of all our Hilbert spaces so that thesemultiples are unitarily equivalent.

To adjust things so that all maps are completely isometric, we replace H byH(2), and we modify θ, π, Φ by considering instead the representations θ ⊕ ρ,σ ⊕ π, and the 2 × 2 operator matrix which is zero except for a Φ(·) as the 1-2entry. This gives the ‘Moreover’ assertion of the theorem.

If A = B, take Φ, θ, π as in the ‘Moreover’ assertion of the theorem. Wereplace H by H(2), and we modify θ, π, Φ by considering instead the representa-tions θ ⊕ π, π ⊕ θ, and the 2× 2 operator matrix which is zero except in the 1-2entry; which entry is Φ(x). This gives the final assertion.

3.3.2 (Quotients of operator modules) Suppose that X is a nondegenerateoperator A-B-bimodule over a pair of approximately unital operator algebras. IfY ⊂ X is a closed A-B-submodule, then the quotient X/Y is nondegenerate andis an h-bimodule by the last sentence in 3.1.10. Hence the CES theorem ensuresthat X/Y is an operator A-B-bimodule.

3.3.3 (CES-representations) Let A and B be Banach algebras which are alsooperator spaces, and let X be an A-B-bimodule which is also an operator space.To say that X is an operator A-B-bimodule is equivalent to saying that Xpossesses a CES-representation. By the latter term we mean a triple (Φ, θ, π)consisting of a linear complete isometry Φ: X → B(K, H), and completely con-tractive representations θ of A on H , and π of B on K, satisfying (3.7). Indeed ifthere exists such a triple, then X is completely A-B-isometric to the concrete op-erator A-B-bimodule Φ(X). If A and B are operator algebras and if θ and π arecompletely isometric then we say that (Φ, θ, π) is a faithful CES-representation.

Similar definitions pertain to one-sided modules, except that in this case theCES-representation will be a pair as opposed to a triple.

If (Φ, θ, π) is a CES-representation of an A-B-bimodule X , with Φ(X) asubspace of B(K, H), then one may adjust the representation to obtain certainadditional properties. For example, it is always possible to replace the CES-representation by one for which H = K, by the trick in the third last paragraphof the proof of 3.3.1. Or, if we replace H by H ′ = [Φ(X)K], and θ by θ′ = θ(·)|H′

viewed as a map from A to B(H ′), then we obtain a new CES-representationof X having the additional properties that Φ(X)K is now dense in H , and

118 Operator modules over operator algebras

also that θ′ is a nondegenerate representation (even if θ was degenerate). Notehowever that θ′ need not be isometric even if the original θ was isometric. Finally,replace both H by H ′ above, and K by [Φ(X)H ]; and define Φ′ = Φ(·)|K′ ,viewed as a map from X into B(K ′, H ′), and define θ′ as above, and defineπ′ = PK′π(·)|K′ , viewed as a map from B into B(K ′). We claim that (Φ′, θ′, π′)is a CES representation too, but now θ′ and π′ are both nondegenerate. Theproof of this follows from routine arguments of the type we have seen in 3.3.1,for example, so that we will only sketch part of it. For x ∈ X, ζ ∈ K, η ∈ H wehave

〈Φ(x)ζ, η〉 = 〈ζ, Φ(x)∗η〉 = 〈PK′ζ, Φ(x)∗η〉 = 〈Φ′(x)PK′ζ, η〉.Thus Φ = Φ′(·)PK′ , and it follows that the complete contraction Φ′ is a completeisometry. A similar argument shows that (3.7) holds for the adjusted maps. Itis easy to see that θ′ is a nondegenerate homomorphism. To see that π′ is ahomomorphism one may use the argument in 3.3.1 showing that θ was a homo-morphism there. To see that π′ is nondegenerate, use 2.1.6 together with the factthat Φ(x)π(et) = Φ(xet) → Φ(x), for a cai (et)t for B, and x ∈ X .

3.3.4 (The algebra of a bimodule) We consider a ‘bimodule variant’ of the U(X)construction of 2.2.10. Suppose first that X is an A-B-bimodule over algebras Aand B. Consider the algebra D of 2 × 2 matrices[

a x0 b

]for a ∈ A, b ∈ B, x ∈ X . The product here is the formal product of 2 × 2matrices, implemented using the module actions and algebra multiplications.Assume that A and B are unital and that X is nondegenerate. Let p and q

be the idempotents[

1A 00 0

]and

[0 00 1B

]. Then pDp ∼= A and qDq ∼= B as

algebras, and pDq ∼= X as bimodules. Conversely, given any unital algebra D,and idempotents p, q ∈ D with p + q = 1D, then pDq is a pDp-qDq-bimodule.This characterizes nondegenerate bimodules in terms of unital algebras.

For a nondegenerate operator A-B-bimodule X over approximately unitaloperator algebras A and B, we may form the 2×2 matrix algebra D as above. If(Φ, θ, π) is a faithful and nondegenerate CES-representation on Hilbert spaces Hand K as provided by Theorem 3.3.1, then we may identify D with the concreteoperator algebra C of matrices [

θ(a) Φ(x)0 π(b)

],

which should be viewed as a nondegenerate subalgebra of B(H⊕K). This algebrais approximately unital, and by elementary facts from Section 2.6, the canonicaldiagonal projections p = IH ⊕ 0 and q = 0⊕ IK are in M(C). Again we see thatpCp ∼= A and qCq ∼= B as algebras, and pCq ∼= X as bimodules, but now these


isomorphisms are also completely isometric. The converse of this is true too, aswe noted in 3.1.2 (5). This characterizes nondegenerate operator bimodules overapproximately unital operator algebras as ‘corners’ pC(1 − p) of approximatelyunital operator algebras C (in the sense of 2.6.14).

Lemma 3.3.5 Suppose that A, B, C are approximately unital operator algebras,that X is a nondegenerate operator A-B-bimodule, and that Y is a nondegener-ate operator B-C-bimodule. Then there exists two faithful CES-representations(Φ, π, θ) and (Ψ, θ, σ) for X and Y respectively, where all representations are ona single Hilbert space H. Thus, up to appropriate identifications, we may viewA, B, C as subalgebras of B(H), and X, Y as subbimodules of B(H). If B isunital, then we may assume in addition that θ is a unital map too.

Proof By 3.3.1 there exist faithful CES-representations (Φ, π, θ) and (Ψ, ρ, σ)of the two bimodules. We may suppose that the representations in the first tripleare on a Hilbert space H1, and that the representations in the second triple areon a Hilbert space H2. Let H = H1 ⊕ H2, and replace θ and ρ with θ(b) ⊕ ρ(b).Then replace Φ with Φ⊕ 0, Ψ with 0⊕Ψ, π with π⊕ 0, and σ with 0⊕σ.

3.4 TWO MODULE TENSOR PRODUCTS

3.4.1 (The algebraic module tensor product) Suppose that X and Y are,respectively, right and left modules over an algebra A. If Z is a vector space,then bilinear map u : X × Y → Z is said to be balanced (or A-balanced) ifu(xa, y) = u(x, ay) for all x ∈ X, y ∈ Y, a ∈ A. Given such X and Y , we definethe algebraic module tensor product X ⊗A Y to be the quotient of X ⊗ Y by thesubspace spanned by terms of the form xa⊗ y− x⊗ ay, for x ∈ X, y ∈ Y, a ∈ A.It is clear that X ⊗A Y has the following universal property: given any vectorspace Z, and any balanced bilinear u : X × Y → Z, then there exists a uniquelinear u : X ⊗A Y → Z mapping the class of x⊗ y to u(x, y) for all x ∈ X, y ∈ Y .Thus the module tensor product ‘linearizes balanced bilinear maps’.

3.4.2 (Two operator module tensor products) We now introduce two operatorspace analogues of the algebraic module tensor product construction, the mod-ule Haagerup tensor product, and the module operator space projective tensorproduct. Let A be an algebra, and let X and Y be operator spaces which are,respectively, right and left A-modules. We define X ⊗hA Y (resp. X

⊗A Y ) to be

the quotient in the sense of 1.2.14, of X ⊗h Y (resp. X⊗ Y ) by the closure of

the subspace spanned by terms of the form xa ⊗ y − x ⊗ ay. For the purpose ofthis paragraph and the next one only we write x ⊗A y for the equivalence classof x⊗ y in either X ⊗hA Y or X

⊗A Y . Later on, as is customary in algebra, we

will continue to write x ⊗ y for this equivalence class. Such expressions are theelementary tensors. We will also write ⊗A for the canonical bilinear completelycontractive map X × Y → X ⊗hA Y (resp. jointly completely contractive mapX × Y → X

⊗A Y ). In particular, the norm of x ⊗A y in either module tensor

120 Two module tensor products

product is dominated by ‖x‖‖y‖ for x ∈ X, y ∈ Y ; and a similar assertion holdsfor matrices. The elements in the linear span of the range of ⊗A we will refer toas finite rank tensors, they are dense in X ⊗hA Y (resp. X

⊗A Y ).

It is clear that the space W = X ⊗hA Y with the map µ = ⊗A, possesses thefollowing universal property:

Given any operator space Z and bilinear completely contractive balancedu : X × Y → Z, then there exists a (necessarily unique) linear completelycontractive u : W → Z such that u µ = u.

Thus X⊗hA Y ‘linearizes’ balanced completely bounded maps. In fact this prop-erty characterizes X ⊗hA Y , analoguously to the fact at the end of 1.5.4.

A similar universal property characterizes X⊗A Y , namely the property of

‘linearizing’ balanced jointly completely bounded maps.

3.4.3 (An alternate construction) There is an alternate way of defining the twomodule tensor products of 3.4.2, which provides some extra information quiteeasily. We treat the module Haagerup tensor product, the other is similar.

Begin with a right A-module X and a left A-module Y as before and formtheir algebraic module tensor product X ⊗A Y (see 3.4.1). On X ⊗A Y we definea sequence of matrix seminorms ‖ · ‖n∞n=1 by the formula

‖z‖n = inf‖x‖ ‖y‖ : z = x A y, (3.9)

where x ranges over Mn,p(X), y ranges over Mp,n(Y ). Here p is variable, z is inMn(X⊗AY ), and xAy is defined to be [

∑k xik⊗Aykj ]. Following verbatim the

usual proof that the Haagerup tensor product is an operator space (see 1.5.4),one sees that the sequence of seminorms just defined is an operator seminormstructure in the sense of 1.2.16. By 1.2.16, the completion of the quotient ofX ⊗A Y by the null space N of ‖ · ‖1 is an operator space. It is straightforwardto check that this completed quotient has the universal property described in3.4.2, and hence is completely isometrically isomorphic to X ⊗hA Y . Note thatthe isomorphism takes an ‘elementary tensor’ x⊗ y+N in the quotient space, tothe equivalence class x ⊗A y in the quotient space of X ⊗h Y considered in ourfirst construction of the module Haagerup tensor product. Indeed the followingdiagram of canonical maps commutes:

X ⊗ Y −→ X ⊗A Y( (X ⊗h Y −→ X ⊗hA Y

One useful and immediate consequence of this is a useful norm formula. Ifz ∈ Mm,n(X ⊗ Y ), then the norm in Mm,n(X ⊗hA Y ) of its equivalence classis precisely the infimum considered in (3.9), now viewing z ∈ Mm,n(X ⊗A Y ).There is a similar formula valid for tensors which are not finite rank:


Lemma 3.4.4 Let A be an algebra, and let X and Y be operator spaces whichare, respectively, right and left A-modules. If z ∈ X ⊗hA Y and if ε > 0 is given,then there exists x = [x1 x2 · · · ] ∈ R(X) and y = [y1 y2 · · · ]t ∈ C(X) such thatz equals the norm convergent sum

∑∞k=1 xk ⊗ yk in X ⊗hA Y . Moreover this can

be done with ‖x‖R(X) = ‖y‖C(Y ) <√‖z‖+ ε. If z is finite rank, then the x and

y above may be chosen in Rn(X) and Cn(X) respectively, for some n ∈ N.

Proof The last assertion was mentioned in the last paragraph of 3.4.3. Therest follows immediately from 1.5.6 and the basic properties of a Banach spacequotient map (in this case, the canonical map X ⊗h Y → X ⊗hA Y ).

Lemma 3.4.5 (Functoriality of the module tensor product) Let B be an al-gebra, let X1 and X2 be operator spaces which are right B-modules, and letY1 and Y2 be operator spaces which are left B-modules. If u : X1 → X2 andv : Y1 → Y2 are completely bounded B-module maps, then the map u⊗ v extendsuniquely to a well defined linear completely bounded map (which we also writeas u ⊗ v) from X1 ⊗hB Y1 to X2 ⊗hB Y2 (resp. X1

⊗B Y1 to X2

⊗B Y2). Indeed,

‖u ⊗ v‖cb ≤ ‖u‖cb‖v‖cb.

Proof We just prove this for the Haagerup tensor product; the other is identi-cal. By the functoriality of the Haagerup tensor product 1.5.5, we obtain a com-pletely bounded linear map X1 ⊗h Y1 → X2 ⊗h Y2 taking x ⊗ y to u(x) ⊗ v(y).Composing this map with the quotient map X2 ⊗h Y2 → X2 ⊗hB Y2 we obtaina map X1 ⊗h Y1 → X2 ⊗hB Y2. It is easy to see that the kernel of the last mapcontains all terms of form xa ⊗ y − x ⊗ ay, with a ∈ B, x ∈ X1, y ∈ Y1, so thatwe obtain a map X1 ⊗hB Y1 → X2 ⊗hB Y2 with the required properties.

Lemma 3.4.6 Let A be an approximately unital Banach algebra, which is alsoan operator space. If X is a left h-module (resp. matrix normed module) overA, then the operator space A ⊗hA X (resp. A

⊗A Y ) is canonically completely

isometrically isomorphic to the essential part of X (see A.6.4).

Proof We prove the case involving A⊗hAX , the other is identical. By definition(see 3.1.3), the module action A × X → X is a completely contractive balancedbilinear map. By 3.4.2, it induces a completely contractive map m from A⊗hA Xto X . Conversely, if (et)t is a cai for A, let φt : X → A ⊗hA X be the mapx → et ⊗ x, which is easily seen to be completely contractive. We have

φt(m(a ⊗ x)) = eta ⊗ x −→ a ⊗ x, a ∈ A, x ∈ X.

Hence by linearity and density considerations, φt(m(z)) → z for all z ∈ A⊗hAX .Thus for zij ∈ A ⊗hA X ,

‖[zij ]‖n = limt

‖[φt(m(zij))]‖n ≤ ‖[m(zij)]‖n.

Thus m is a complete isometry. It is easily seen that m maps onto the essentialpart of X .

122 Two module tensor products

The following result is often useful (although we shall not need it later), sinceit can help reduce certain calculations involving matrix normed modules to thesimple case of S1

n[A] (see 3.1.5 (3)).

Proposition 3.4.7 Let X be a nondegenerate left matrix normed module overan approximately unital matrix normed algebra A. Then there exists a set I,natural numbers ni for i ∈ I, and a complete quotient map u : ⊕1

i S1ni

[A] → X,such that u is also a left A-module map. Hence X is completely A-isometric toa quotient A-module of ⊕1

i S1ni

[A].

Proof As noted in 1.4.13 there is a complete quotient map from ⊕1i S1

nionto

X . By the projectivity of⊗ (mentioned in 1.5.11) we obtain a complete quotient

map from A⊗ (⊕1

i S1ni

) onto A⊗ X . By (1.52), the definition of

⊗A, and 3.4.6,

we obtain a complete quotient map ⊕1i S1

ni[A] → A

⊗A X ∼= X.

3.4.8 (Threefold tensor products) Let B, C be algebras, and suppose thatX, Y, Z are operator spaces. Suppose that X is also a right B-module, that Y isa B-C-bimodule, and that Z is a left C-module. We define X ⊗hB Y ⊗hC Z tobe the quotient of X ⊗h Y ⊗h Z by the closure of the linear span of terms of theform xb⊗ y ⊗ z − x⊗ by⊗ z and x⊗ yc⊗ z − x⊗ y ⊗ cz. Then exactly as above,but using the universal property of the threefold Haagerup tensor product (see1.5.5), one sees that X ⊗hB Y ⊗hC Z has the following universal property: Ifu : X × Y × Z → W is a completely contractive, balanced (in the sense thatu(xb, y, z) = u(x, by, z) and u(x, yc, z) = u(x, y, cz)) trilinear map, then there isa unique completely contractive linear map u : X ⊗hB Y ⊗hC Z → W such thatu(x ⊗ y ⊗ z) = u(x, y, z) for all x ∈ X, y ∈ Y, z ∈ Z. This is a universal propertyin the same sense as the one discussed at the end of 3.4.2.

Similarly there is a threefold tensor product X⊗B Y

⊗C Z linearizing jointly

completely bounded balanced trilinear maps. Indeed for n ≥ 4 there are n-foldvariants of both module tensor products, which are defined in an analoguousway.

3.4.9 (Tensor products as bimodules) Let A and C be Banach algebras whichare also operator spaces and let B be an algebra. Let X, Y be operator spaceswhich are respectively an A-B-bimodule and a B-C-bimodule. Suppose that theleft action makes X a matrix normed A-module. Then X

⊗ Y is canonically a

matrix normed A-module, as noted in 3.1.5 (3). Using 3.1.10, we see that X⊗B Y

is also a left matrix normed A-module. If A acts nondegenerately on X , then it iseasy to see using 3.1.10 that A also acts nondegenerately on X

⊗B Y . Likewise

if Y is a (resp. nondegenerate) right matrix normed C-module, then X⊗B Y

is a (resp. nondegenerate) right matrix normed C-module as well. Moreover ifboth hold, then the left and right actions on X

⊗B Y commute hence X

⊗B Y

is actually a matrix normed A-C-bimodule.An almost identical argument shows that if X is a left h-module over A and

Y is a right h-module over C, then X ⊗hB Y is an h-bimodule over A and C.


Furthermore, this bimodule is nondegenerate if X and Y are nondegenerate.If we appeal to the CES theorem (or its extension to be proved in 4.6.7) wefind that the module Haagerup tensor product of left and right nondegenerateoperator modules is a nondegenerate operator bimodule.

We also remark that in Lemma 3.4.5, if further X1 and X2 are left matrixnormed modules (resp. h-modules) over A, and if further Y1 and Y2 are rightmatrix normed modules (resp. h-bimodules) over C, and if u, v are bimodulemaps, then u ⊗ v is a A-C-bimodule map on X

⊗B Y (resp. X ⊗hB Y ).

Theorem 3.4.10 (Associativity of the module tensor product) Let A, B, C, Dbe Banach algebras which are also operator spaces. Assume that X is a matrixnormed A-B-bimodule, that Y is a matrix normed B-C-bimodule, and that Z isa matrix normed C-D-bimodule. Then

X⊗B (Y

⊗C Z) ∼= (X

⊗B Y )

⊗C Z ∼= X

⊗B Y

⊗C Z

completely isometrically (via an A-D-bimodule map). Similarly, if X, Y, Z areh-bimodules, then

X ⊗hB (Y ⊗hC Z) ∼= (X ⊗hB Y ) ⊗hC Z ∼= X ⊗hB Y ⊗hC Z

completely isometrically (via an A-D-bimodule map).

One way to prove the last result (see [65, Theorem 2.6]) is to show that theiterated module tensor products have the universal property of the threefoldmodule tensor product (see 3.4.8).

3.4.11 Suppose that A is an algebra, and that X and Y are operator spaceswhich are, respectively, right and left A-modules. Then it is not hard to extend1.5.14 (7) to show that there is a canonical completely isometric isomorphism

Cm(X) ⊗hA Rn(Y ) ∼= Mm,n(X ⊗hA Y )

taking [x1 x2 · · ·xm]t ⊗ [y1 y2 · · · yn] to the matrix [xi ⊗ yj ]. Indeed a variant of3.4.10 and 1.5.14 (6) show that

Cm(X) ⊗hA Rn(Y ) ∼= (Cm ⊗h X) ⊗hA (Y ⊗h Rn)∼= Cm ⊗h (X ⊗hA Y ) ⊗h Rn

∼= Mm,n(X ⊗hA Y ).

A ‘minimal’ tensor product of operator modules is discussed in the Notes.

3.5 MODULE MAPS

In this section we again use the convention from 3.1 that A, B, C are at leastBanach algebras with an operator space structure.

124 Module maps

3.5.1 (The category OMOD) If X and Y are left matrix normed A-modules,then we write ACB(X, Y ) for the set of completely bounded A-module mapsfrom X to Y . Evidently ACB(X, Y ) is an operator space, namely a subspaceof the operator space CB(X, Y ) (see 1.2.19). Its matrix norms are specified bythe canonical identification Mn(ACB(X, Y )) ∼= ACB(X, Mn(Y )). Similar nota-tions will apply for right B-modules and A-B-bimodules, except that we writeCBB(X, Y ) (resp. ACBB(X, Y )) for the completely bounded right B-modulemaps (resp. the A-B-bimodule maps).

We write AOMOD for the category of nondegenerate left operator A-modules,with morphisms the completely bounded left A-module maps. We write OMODB

for the category of nondegenerate right operator B-modules and AOMODB forthe category of nondegenerate operator A-B-bimodules. It is usually convenientfor us to view the category AHMOD of Hilbert A-modules (see 3.2.5) as a sub-category of AOMOD. To do this, note that by 3.1.7 every Hilbert A-module is aleft operator A-module (with its ‘Hilbert column space’ structure), and by 3.2.5the morphisms are the same.

3.5.2 (Mapping spaces as modules) Suppose that X and Y are matrix normedbimodules over A and B, and A and C, repectively. Then ACB(X, Y ) becomesa B-C-bimodule with the ‘interior’ left action (bu)(x) = u(xb), and the ‘ex-terior’ right action defined by (uc)(x) = u(x)c. Here b ∈ B, x ∈ X, c ∈ Cand u ∈ ACB(X, Y ). We say that such a map bu is left B-essential. We writeACBess(X, Y ) for the subset of ACB(X, Y ) consisting of such maps bu, for b ∈ Band u ∈ ACB(X, Y ). It is easy to see that ACB(X, Y ) and ACBess(X, Y ) aboveare matrix normed B-C-bimodules.

Similarly, CBB(X, W ) is a matrix normed D-A-bimodule for any matrixnormed D-B-bimodule W , with the exterior left D-module action, and the inte-rior right A-module action (ua)(x) = u(ax). We write CBess

B (X, W ) for the setof such right A-essential maps ua.

3.5.3 (Transplanting relations from algebra) In 3.2.6 we have briefly discussedthe process of transferring basic concepts and relations from algebra (of the typefound in basic algebra texts such as [8, 368]), into the setting of modules overoperator algebras. For some results or notions the transferal is routine; othersrequire a great deal of analysis, or have no satisfactory variant. Also, quite oftenan algebraic notion has several interesting analogues in the operator framework.Such results, while sometimes initially looking a little ‘formal’, are nonethelessthe ‘right way’ to approach certain topics, as is testified to, for example, inthe literature on Hilbert modules cited in 3.2.6. In the present text we will notattempt to systematically collect such ‘transferals from pure algebra’, partlybecause we will not be able to reach the more sophisticated topics that requiremany such ‘transplanted relations’. We will however give a few simple and usefulresults of this sort in this section. Indeed we will treat some properties of the‘Hom(−,−) functor’ in our category; namely the module mapping spaces of thetype ACB or ACBess defined above.


Lemma 3.5.4 Let X be a nondegenerate left matrix normed module over anapproximately unital matrix normed algebra A. Then(1) X ∼= ACBess(A, X) ⊂ ACB(A, X) completely isometrically.(2) If A is unital, or if X is a dual space and the map x → ax on X is w∗-

continuous for all a ∈ A, then X ∼= ACB(A, X) in (1).

Proof Let (et)t be a cai for A. For each x ∈ X define rx(a) = ax for a ∈ A.It is easy to see that the map x → rx is completely contractive. In fact it iscompletely isometric, as may be seen from the relation

‖[xij ]‖n = limt

‖[etxij ]‖n, [xij ] ∈ Mn(X).

By Cohen’s factorization theorem A.6.2, we may write any x ∈ X as x = a1x1

with a1 ∈ A, x1 ∈ X . Therefore rx(a) = rx1(aa1) for a ∈ A. Thus rx = a1rx1 ,a map in ACBess(A, X). Conversely, if u ∈ ACB(A, X) and if a1 ∈ A, then(a1u)(a) = u(aa1) = au(a1) for a ∈ A. Hence a1u = ru(a1). This yields (1).

Assertion (2) follows from (1). This is obvious if A is unital. In the other case,consider u ∈ ACB(A, X), and let x be a w∗-cluster point in X of the boundednet (u(et))t. Then u(a) = limt au(et) = ax for any a ∈ A.

If X is a Hilbert A-module, or more generally if X is reflexive, then thehypothesis in (2) of the above lemma clearly holds, so that X ∼= ACB(A, X).

3.5.5 (Module maps and multipliers) Suppose that X and Y are left matrixnormed modules over an approximately unital matrix normed algebra A, andthat the action on X is nondegenerate. If u ∈ ACB(X, Y ) and x ∈ X thenby Cohen’s theorem A.6.2 we may write x = ax′ for an a ∈ A, x′ ∈ X . Thusu(x) = au(x′). Therefore u maps into the essential part (see A.6.4) Y ′ of Y , andso ACB(X, Y ) = ACB(X, Y ′). If, in addition, Y is nondegenerate then by 3.1.11we may view both X and Y as left modules over LM(A), If η ∈ LM(A) and ifu is as above, then by 3.1.11 we have

u(ηx) = limt

u(η(et)x) = limt

η(et)u(x) = ηu(x).

That is, u is a LM(A)-module map. If A is in addition an operator algebra, thenit follows that u is also an A1-module map.

Likewise if H is a Hilbert A-module and K is any Hilbert space, then B(K, H)is canonically a left operator LM(A)-module (see 3.1.11) and any A-module mapu : X → B(K, H) is an LM(A)-module map.

3.5.6 (Module maps and matrix spaces) We next consider some frequentlyencountered module map variants of the matrix space relations in 1.2.28 and1.2.29. For example, if X and Y are matrix normed A-B-bimodules then wehave ACBB(X, MI,J(Y )) ∼= MI,J(ACBB(X, Y )) completely isometrically. Thisis easily seen from 1.2.29. One needs to check that the isomorphisms in that prooftake matrices of bimodule maps to bimodule maps, but this is fairly evident.

126 Module maps

By the idea of the proof as 1.2.28 and the proof of (1.14) that we sketchedin the Notes to Section 1.2, one may prove the following commonly encounteredidentities. We omit the tedious details. In this result, the expression CBess

B refersto the space of right B-essential maps (see 3.5.2); which coincides, for examplein the CBess

B (Cn(B), X) case, with the set of right Mn(B)-essential maps. Ofcourse if B is unital, then one may delete the symbol ‘ess’ below.

Proposition 3.5.7 Let X be a right operator B-module over an approximatelyunital matrix normed algebra B. Let I, J be cardinals, and m, n ∈ N. Then wehave the following canonical completely isometric isomorphisms:

(1) CBB(CJ (B), CI(X)) ∼= MI,J(CBB(B, X)), and(2) CBess

B (Cn(B), Cm(X)) ∼= Mm,n(X).

Also, there are canonical complete isometries

CBessB (CI(B), X) → Rw

I (X) → CBB(CI(B), X).

If B is unital then (1) becomes CBB(CJ (B), CI(X)) ∼= MI,J(X).

3.5.8 (The role of complete boundedness) The canonical isomorphisms in theresult above are perhaps most quickly grasped by looking at the special case of(2) when n = m and X = B. We see that if B is a unital operator algebra, then

Mn(B) ∼= CBB(Cn(B)) completely isometrically.

This isomorphism is the same as that in the fundamental fact from ring theorythat Mn(A) ∼= HomA(A(n)) (which in turn is a trivial generalization of the ele-mentary linear algebra fact that Mn(C) is isomorphic to the set of linear mapson C

n). Namely, a matrix b ∈ Mn(B) is taken by this isomorphism to the mapon Cn(B) given by left matrix multiplication by b. It should be carefully notedthat for a general operator algebra B, the operator algebra Mn(B) is not iso-metrically isomorphic to BB(Cn(B)) via the canonical map (e.g. see [182]). Thisis an illustration of the importance of the operator space approach to operatoralgebras. The failure of this elementary relation at the isometric level is the firstindication of the complete breakdown further down the road if one attempts togeneralize certain facts from purely algebraic module theory without using thematrix norms of operator space theory (see, for example, [46]).

Proposition 3.5.9 Let A, B and C be Banach algebras which are also operatorspaces, suppose that X is a matrix normed A-B-bimodule, that Y is a matrixnormed B-C-bimodule, and that Z is a matrix normed A-C-bimodule. Then thereare canonical completely isometric isomorphisms

ACBC(X⊗B Y, Z) ∼= BCBC(Y,A CB(X, Z)) ∼= ACBB(X, CBC(Y, Z)).


Proof We prove just the first isomorphism, the other is similar. By definition,X

⊗B Y is a complete quotient of X

⊗ Y , hence we have complete isometries

CB(X⊗B Y, Z) −→ CB(X

⊗ Y, Z) ∼= CB(Y, CB(X, Z)).

The last relation is simply (1.50), and the first ‘arrow’ is ‘composition with q’,where q : X

⊗ Y → X

⊗B Y is the canonical quotient map. Let r be the

composition of the two maps in the sequence. If u ∈ CB(X⊗B Y, Z) then

(r(u)(y))(x) = u(x ⊗ y), x ∈ X, y ∈ Y. (3.10)

From this it is clear that if u is an A-C-bimodule map, then r(u)(y) ∈ ACB(X, Z)for any y ∈ Y , and r(u) ∈ BCBC(Y,A CB(X, Z)). Thus r is a complete isometryfrom ACBC(X

⊗BY, Z) to BCBC(Y,A CB(X, Z)). If Ψ ∈ BCBC(Y,A CB(X, Z)),

then in particular, Ψ ∈ CB(Y, CB(X, Z)). Thus there is by (1.50), a map v inCB(X

⊗ Y, Z) with v(x ⊗ y) = Ψ(y)(x) for x ∈ X and y ∈ Y . Then

v(xb ⊗ y) = Ψ(y)(xb) = (bΨ(y))(x) = Ψ(by)(x) = v(x ⊗ by).

Using 3.4.2, we obtain a map u ∈ CB(X⊗B Y, Z) with v = uq. Then Ψ = r(u),

and it is simple algebra using (3.10) to check that u is an A-C-bimodule map.Thus r maps ACBC(X

⊗B Y, Z) onto BCBC(Y,A CB(X, Z)).

The following is a special case corresponding to Z = A = C = C.

Corollary 3.5.10 Let B be a Banach algebra which is also an operator space,and suppose that X and Y are respectively right and left matrix normed modulesover B. Then there are canonical completely isometric isomorphisms

(X⊗B Y )∗ ∼= BCB(Y, X∗) ∼= CBB(X, Y ∗).

In particular, BCB(Y, X∗) is a w∗-closed subspace of CB(Y, X∗).

Corollary 3.5.11 If H and K are Hilbert modules over A and B respectively,and if X is a matrix normed A-B-bimodule, then ACBB(X, B(K, H)) is a dualoperator space with predual

Hr⊗A X

⊗B Kc = Hr ⊗hA X ⊗hB Kc. (3.11)

Proof By 1.5.14 (1) we have Hr⊗hX = Hr⊗X , so that Hr⊗hAX = Hr

⊗AX .

Similarly, we have (Hr ⊗hA X)⊗hB Kc = (Hr⊗A X)

⊗B Kc. Then (3.11) follows

from the associativity of the two tensor norms. From 3.5.10, (1.15), and 3.5.9,we have

(Hr⊗A (X

⊗B Kc))∗ ∼= ACB(X

⊗B Kc, Hc) ∼= ACBB(X, CB(Kc, Hc)),

and the latter space is just ACBB(X, B(K, H)), by (1.14).

128 Module map extension theorems

3.6 MODULE MAP EXTENSION THEOREMS

In this section all algebras are C∗-algebras, and all operator modules are assumednondegenerate for simplicity.

3.6.1 (The bimodule Paulsen system) Let X be an operator A-B-bimoduleover unital C∗-algebras A and B. By choosing a nondegenerate faithful CES-representation, we may assume that A and B are unital-subalgebras of B(H)and B(K) respectively, and that X is an A-B-submodule of B(K, H). Accordingto 3.1.16, we regard X ⊂ B(K, H) as a B-A-bimodule. We define the bimodulePaulsen system to be the set S of formal matrices[

a xy∗ b

]for a ∈ A, b ∈ B, x, y ∈ X . Clearly this may be viewed as a unital selfadjoint(A ⊕ B)-(A ⊕ B)-submodule of B(H ⊕ K).

Next suppose that we have unital ∗-representations θ : A → B(H1) andπ : B → B(K1), and suppose that u : X → B(K1, H1) is a completely contractiveA-B-bimodule map. That is, u(axb) = θ(a)u(x)π(b) for a ∈ A, b ∈ B, x ∈ X . Afairly literal modification of the proof of Lemma 1.3.15 shows that the map

Θ :[

a xy∗ b

]→

[θ(a) u(x)u(y)∗ π(b)

](3.12)

from S to B(H1 ⊕ K1) is completely positive.By 1.3.3, we have that Θ is completely contractive. By symmetry, if in ad-

dition u is a complete isometry and θ and π are one-to-one, then we see thatΘ is a unital complete order isomorphism, and a complete isometry. This showsthat the bimodule Paulsen system may be defined as an operator system in-dependently of the particular nondegenerate faithful CES-representation of thebimodule chosen in the beginning of this section.

Theorem 3.6.2 (Wittstock) Let Y be an operator A-B-bimodule over unitalC∗-algebras A and B, and suppose that X is an A-B-submodule of Y . We supposethat θ : A → C and π : B → C are unital ∗-homomorphisms, where C is aninjective unital C∗-algebra. Suppose that u : X → C is a completely contractiveA-B-bimodule map (i.e. u(axb) = θ(a)u(x)π(b) for a ∈ A, b ∈ B, x ∈ X). Thenthere exists a completely contractive A-B-bimodule map u : Y → C extending u.

Proof Suppose that C is represented as a unital-subalgebra of B(L) for aHilbert space L. Since C is injective, there is a unital completely contractive pro-jection from B(L) onto C, which is a θ(A)-π(B)-bimodule map by 1.3.12. Thus itis clear that we may assume that C = B(L). On the other hand we may assumethat A and B act faithfully and nondegenerately on some Hilbert spaces H and Krespectively, that X ⊂ B(K, H) is an A-B-submodule, and that Y = B(K, H).Let S ⊂ B(H ⊕K) be the bimodule Paulsen system considered above. By 3.6.1,


we obtain a completely positive Θ: S → M2(B(L)) = B(L ⊕ L) defined by(3.12). By Arveson’s extension theorem 1.3.5, Θ extends to a completely posi-tive map Θ: B(H ⊕K) → B(L⊕L). Clearly this map is a ∗-homomorphism onthe diagonal C∗-algebra CIH ⊕ CIK . Thus by 1.3.12 we have that Θ is corner-preserving in the sense of 2.6.15 and 2.6.16. Indeed since Θ is a ∗-homomorphismon the diagonal A⊕B, we have by 1.3.12 that Θ is an (A⊕B)-bimodule map. Ifu : B(K, H) → B(L) is the 1-2-corner of Θ (see 2.6.16), then u clearly extendsu and is completely contractive. Since Θ is a (A ⊕ B)-bimodule map it followsthat θ(a)u(y)π(b) = u(ayb) for all a ∈ A, b ∈ B, y ∈ B(K, H).

Corollary 3.6.3 Let A and B be C∗-algebras, and let H and K be Hilbert mod-ules over A and B respectively. Suppose also that Y is an operator A-B-bimodule,and that X is a closed A-B-submodule of Y . Given any completely contractiveA-B-bimodule map u : X → B(K, H), then there exists a completely contractiveA-B-bimodule map u : Y → B(K, H) extending u.

Proof We let θ : A → B(H) and π : B → B(K) be the nondegenerate contrac-tive homomorphisms (and hence ∗-homomorphisms by A.5.8) associated to theHilbert modules. Case 1: H = K, A and B are unital. Here π and θ are unital,and the result follows by applying Theorem 3.6.2 with C = B(H).

Case 2: H = K, and π and θ are possibly nonunital. In this case, let A1 andB1 be the unitizations of A and B. By 3.1.11 we have that Y is an operatorA1-B1-bimodule, and X is an A1-B1-submodule. We may extend θ and π tounital ∗-representations of A1 and B1 respectively on H and K. By 3.5.5, u isan A1-B1-bimodule map. Thus by Case 1, there exists a completely contractiveA1-B1-bimodule map from Y into B(H) extending u.

Case 3: H = K. Choose a cardinal γ and a unitary V as in the proof of 3.3.1.Let θ′ = V θ(·)γV ∗ and u′ = V u(·)γ . Note that u′ is a completely contractiveA-B-bimodule map from X into B(K(γ)). By Case 2, there is a completelycontractive A-B-bimodule map from Y into B(K (γ)) extending u′. Multiply thismap by V ∗. Pre- (resp. post-) multiplying the resulting map by the canonicalinclusion (resp. projection) map from H → H (γ) (resp. K(γ) → K) gives acompletely contractive bimodule map from Y into B(K, H) extending u.

3.6.4 (Injective bimodules) Assume for simplicity that A and B are unitalC∗-algebras. One may rephrase the last result as the statement that for Hilbertmodules H and K over A and B respectively, the space B(K, H) is injective inthe category AOMODB . As a special case (taking B = K = C) we see that anyHilbert module H over a C∗-algebra A is injective in the category AOMOD.

Theorem 3.6.5 Suppose that A is a C∗-algebra, and that X (resp. Y ) is anA-submodule of a right (resp. left) operator A-module W (resp. Z).(1) If L is a Hilbert space and if u : X × Y → B(L) is a completely contractive

A-balanced bilinear map, then u has a completely contractive A-balancedbilinear extension u : W × Z → B(L).

(2) The canonical map X ⊗hA Y → W ⊗hA Z is a complete isometry.

130 Module map extension theorems

Proof (1) By the last remark in 3.1.11 we may assume that A is unital (byreplacing A by A1 if necessary). By Lemma 3.3.5 we may assume that there is asingle C∗-algebra B = B(H) containing A as a unital C∗-subalgebra and W andZ as A-A-submodules. It clearly suffices to extend u to a completely contractiveA-balanced bilinear map B ×B → B(L). We will use the well-known techniquesseen in the proof of 3.3.1. By 1.5.7 (2) we may write u(x, y) = Rθ(x)Sπ(y)T forHilbert spaces H1, K1, ∗-representations π and θ of B on K1 and H1 respectively,and contractions R, S, T between these Hilbert spaces. Let K2 = [π(Y )TL], andlet P be the projection of K1 onto K2. Since AY ⊂ Y , the space K2 is obviouslyπ(A)-invariant. Let H2 = [θ(X)∗R∗L] and let Q be the projection of H1 onto H2.Since XA ⊂ X , we have [θ(A)∗H2] = [θ(A)∗θ(X)∗R∗L] = [θ(XA)∗R∗L] ⊂ H2.Since θ is a ∗-representation, H2 is θ(A)-invariant. Let Θ(x) = Rθ(x)|H2 and letΨ(y) = Pπ(y)T = π(y)T . Let S ′ = QS|K2 viewed as a map from K2 to H2.Then by standard arguments of the type found in 3.3.1 we have

u(x, y) = Rθ(x)Sπ(y)T = Rθ(x)QSΨ(y) = Θ(x)S ′Ψ(y).

Let θ′ : A → B(H2) (resp. π′ : A → B(K2)) be defined by restricting θ(A) (resp.π(a)) to its invariant subspace H2 (resp. K2) for any a ∈ A. Then

〈θ′(a)S′π(y)Tζ, θ(x)∗R∗η〉 = 〈Rθ(xa)Sπ(y)Tζ, η〉 = 〈u(xa, y)ζ, η〉,

for ζ, η ∈ L and a ∈ A. A similar calculation shows that

〈S′π′(a)π(y)Tζ, θ(x)∗R∗η〉 = 〈u(x, ay)ζ, η〉.

Since u is balanced these are equal. By linearity and density considerations,

θ′(a)S′ = S′π′(a), a ∈ A. (3.13)

For a ∈ A, x ∈ X , and ζ ∈ H2, η ∈ L we have

〈Θ(xa)ζ, η〉 = 〈Rθ(x)θ(a)ζ, η〉 = 〈Θ(x)θ′(a)ζ, η〉,

so that Θ: X → B(H2, L) is an A-module map. A similar but easier calculationshows that Ψ: Y → B(L, K2) is an A-module map. Thus by 3.6.3, Ψ and Θhave completely contractive A-module map extensions Ψ′ : B → B(L, K2) andΘ′ : B → B(H2, L). Then u(b1, b2) = Θ′(b1)S′Ψ′(b2) is a completely contractivebilinear map extending u, and we have using (3.13) that for a ∈ A, b1, b2 ∈ B,

u(b1a, b2) = Θ′(b1)θ′(a)S′Ψ′(b2) = Θ′(b1)S′π′(a)Ψ′(b2) = u(b1, ab2).

(2) Let V be the closure in W ⊗hA Z of the span of the finite rank tensorsx ⊗ y for x ∈ X, y ∈ Y . It is straightforward to show using (1) that V has theuniversal property discussed at the end of 3.4.2. Thus V is completely isometricto X ⊗hA Y .


3.6.6 (Extensions of multilinear module maps) Theorem 3.6.5 says that themodule Haagerup tensor product over C∗-algebras is injective. From this wemay make the following deduction (generalizing 3.6.5 (1)). Suppose in additionto the hypotheses of 3.6.5 (1), that we have two other C∗-algebras B and Cwhich W and Z are respectively left and right operator modules over, and L isboth a Hilbert B-module and a Hilbert C-module. Suppose that X and Y aresubbimodules of W and Z respectively, and suppose that u : X × Y → B(L) isas in 3.6.5 (1) but with the further property that u(bx, yc) = bu(x, y)c for allx, y, b, c. Then we may choose the extension u to also have this further property.This may be proved by using (2) of 3.6.5 to view X ⊗hA Y as a subbimoduleof W ⊗hA Z, by noting that the latter is an operator bimodule by 3.4.9, andthen using 3.6.3 to extend the associated bimodule map X ⊗hA Y → B(L) to abimodule map W ⊗hA Z → B(L). We leave it to the reader to make this precise.

With the last paragraph in hand, together with the associativity of the mod-ule Haagerup tensor product, it follows by induction that there are similar exten-sion theorems for multilinear completely contractive ‘N -balanced’ maps Φ froma product X1 × X2 × · · · × XN of N bimodules, into B(L).

The last fact, namely that multilinear completely bounded module maps intoB(H) may be extended, is a key tool in completely bounded cohomology theory(see references in the Notes to Section 3.6).

3.7 FUNCTION MODULES

Function modules resemble operator modules, except that we consider subspacesof a C(K)-space rather than subspaces of B(K, H). They form a good exampleof operator modules, but are less interesting as a class. Some of the most impor-tant modules over uniform algebras, such as those coming from representationson Hilbert space, are not function modules but operator modules. On the otherhand, in this section we shall demonstrate a link between this class of opera-tor modules and the classical theory of ‘M -structure’, that will simultaneouslyfacilitate a deeper understanding of function modules, and point us toward themethods in Chapter 4 which give similar insights into the structure of generaloperator modules.

Throughout this section E is a Banach space, with E = 0. We will alsoconsistently write KE for the topological space of extreme points of Ball(E∗)equipped with the w∗-topology. We write Cb(KE) for the commutative C∗-algebra of bounded continuous functions on KE , and j : E → Cb(KE) for thecanonical map given by j(x)(ψ) = 〈ψ, x〉 for x ∈ E and ψ ∈ KE. This is anisometry by the Krein–Milman theorem. We will use several times the obviousfact that ‖ψ‖ = 1 for any ψ ∈ KE.

3.7.1 (The function multiplier algebra) This is defined to be, for a Banachspace E, the closed unital algebra M(E) = f ∈ Cb(KE) : fj(E) ⊂ j(E). Thecentralizer algebra of E is Z(E) = f ∈ Cb(KE) : f, f ∈ M(E).

132 Function modules

Note that Z(E) is a commutative unital C∗-algebra, and M(E) is a uniformalgebra. Every Banach space E is a Banach M(E)-module. Indeed there is acanonical homomorphism π : M(E) → B(E), defined by

π(f)(x) = j−1(fj(x)), x ∈ E, f ∈ M(E).

This unital homomorphism is easily seen to be contractive and in fact it isisometric. To see this note that

‖π(f)‖ = sup‖fj(x)‖ : x ∈ Ball(E)= sup|f(ψ)||ψ(x)| : x ∈ Ball(E), ψ ∈ KE= sup|f(ψ)|‖ψ‖ : ψ ∈ KE = ‖f‖.

The isometry π above maps Z(E) onto a closed unital-subalgebra of B(E). Wewill usually identify M(E) and Z(E) with their images in B(E) under thishomomorphism.

One of the main results of this section, is that just as Banach A-moduleactions on a Banach space E correspond to contractive homomorphisms fromA into B(E) (see A.6.1), function module actions on E are in a canonical one-to-one correspondence with contractive homomorphisms into M(E). If A is aC∗-algebra then the correspondence is with ∗-homomorphisms into Z(E).

Theorem 3.7.2 Let E be a Banach space and recall that KE = ext(Ball(E∗)).If T ∈ B(E), then the following are equivalent:

(i) T ∈ M(E).(ii) There is a nonnegative constant C such that

|ψ(T (x))| ≤ C|ψ(x)|, x ∈ E, ψ ∈ KE.

(iii) ψ is an eigenvector for T ∗ for all ψ ∈ KE.(iv) There is a compact space Ω, a linear isometry σ : E → C(Ω), and an

f ∈ C(Ω), such that σ(Tx) = fσ(x), for all x ∈ E.The least C in (ii), and least ‖f‖∞ in (iv), coincides with the usual norm of T .

Proof Given (iii), let aT (ψ) be the eigenvalue associated with ψ. Then

|ψ(T (x))| = |aT (ψ)||ψ(x)|, x ∈ E.

Taking the supremum over x ∈ Ball(E) gives |aT (ψ)| = ‖ψ T ‖ ≤ ‖T ‖. Fromthe last two equations, (ii) is obvious. On the other hand, if (ii) holds then thekernel of ψ T contains the kernel of ψ. Hence by linear algebra, ψ T is a scalarmultiple of ψ. Thus (ii) implies (iii).

Suppose again that (iii) holds, and define aT as above. We noticed thataT : KE → C is bounded. Let (ψi)i be a net in KE converging in the w∗-topology to some ψ ∈ KE. Choose x ∈ E with ψ(x) = 0. Then ψi(x) = 0 forlarge enough i, and


aT (ψi) =ψi(T (x))

ψi(x)−→ ψ(T (x))

ψ(x)= aT (ψ).

Thus aT belongs to Cb(KE). Since aT j(x) = j(T (x)), it is clear that (i) holds.Given (i), we appeal to Gelfand theory (see A.5.4) to see that Cb(KE) ∼= C(Ω)

for some compact Ω. This yields (iv).That (iv) implies (iii) follows by the well-known consequence of the Krein–

Milman theorem, that extreme points of Ball(σ(E)∗) extend to extreme pointsof Ball(C(Ω)∗) (e.g. see the proof of Lemma 4.1.2 below), and the fact that thelatter extreme points are the obvious ones, namely point evaluations multipliedby a unimodular constant. If ψ ∈ KE = ext(Ball(E∗)), then ψ σ−1 is anextreme point of Ball(σ(E)∗), so that ψ(·) = ασ(·)(w) for some w ∈ Ω, andsome α ∈ C, |α| = 1. This gives (iii), since for any x ∈ E,

ψ(T (x)) = ασ(T (x))(w) = αf(w)σ(x)(w) = f(w)ψ(x).

Thus (i)–(iv) are equivalent. If T ∈ M(E), with associated function g on K,and if f is obtained as in the proof of ‘(i) ⇒ (iv)’, then ‖f‖∞ = ‖g‖∞ = ‖T ‖.With this same f , and tracing through the argument that (iv) implied (ii), wesee that C in (ii) may be chosen to be ‖T ‖. If C satisfies (ii), then taking thesupremum over ψ ∈ ext(Ball(E∗)) and x ∈ Ball(E), we see that ‖T ‖ ≤ C. Thus‖T ‖ is the least constant possible in (ii). Similarly, if f is as in (iv), then it isclear that ‖f‖∞ ≥ ‖T ‖ since σ is an isometry.

Since Z(E) ⊂ M(E), by the theorem above we may associate to any central-izer T ∈ Z(E) and ψ ∈ ext(Ball(E∗)), an eigenvalue aT (ψ) for T ∗.

Theorem 3.7.3 For T ∈ B(E) the following are equivalent:(i) T ∈ Ball(Z(E)) and aT ≥ 0.(ii) There is a compact space Ω and an f ∈ C(Ω) with 0 ≤ f ≤ 1, and a linear

isometry σ : E → C(Ω) such that σ(Tx) = fσ(x) for all x ∈ E.(iii) ‖T (x) + y − T (y)‖ ≤ max‖x‖, ‖y‖ for all x, y ∈ E.

Proof That (i) implies (ii) is exactly as in the proof that (i) implied (iv) in3.7.2. Assuming (ii), we have

‖T (x) + y − T (y)‖ = ‖σ(Tx) + σ(y) − σ(T (y))‖ = ‖fσ(x) + (1 − f)σ(y)‖.

Pointwise, fσ(x) + (1 − f)σ(y) is a convex combination of σ(x) and σ(y). Thus

‖T (x) + y − T (y)‖ ≤ max‖σ(x)‖, ‖σ(y)‖ = max‖x‖, ‖y‖,

which is (iii).Assume (iii). That is, the map ν : (x, y) → T (x) + y − T (y) is a contraction

from E ⊕∞ E → E. Hence ν∗ is a contraction. Thus if ψ ∈ ext(Ball(E∗)) then

‖T ∗(ψ)‖ + ‖ψ − T ∗(ψ)‖ ≤ 1.

134 Function modules

This implies that

1 = ‖ψ‖ = ‖T ∗(ψ) + (ψ − T ∗(ψ))‖ ≤ ‖T ∗(ψ)‖ + ‖ψ − T ∗(ψ)‖ ≤ 1. (3.14)

If T ∗(ψ) is neither 0 nor ψ, (3.14) yields a convex combination

ψ = ‖T ∗(ψ)‖ T ∗(ψ)‖T ∗(ψ)‖ + ‖ψ − T ∗(ψ)‖ ψ − T ∗(ψ)

‖ψ − T ∗(ψ)‖ .

Since ψ is an extreme point, we obtain T ∗(ψ) = ‖T ∗(ψ)‖ψ and of course thisequality holds as well if T ∗(ψ) is 0 or ψ. Thus T satisfies (iii) of 3.7.2, withaT (ψ) = ‖T ∗(ψ)‖. This yields (i).

3.7.4 (Centralizers and M -projections) An M -projection on a Banach spaceE is an idempotent linear map P : E → E such that

‖P (x) + y − P (y)‖ = max‖P (x)‖, ‖(I − P )(y)‖, x, y ∈ E. (3.15)

This is simply saying that E = P (E) ⊕∞ (I − P )(E). These M -projections arerelated to the important notion of M -ideals that we shall meet again in Section4.8. If P is an M -projection then setting x = y shows that P and I − P arecontractive, from which it is clear that 3.7.3 (iii) holds. Hence P is an idempotentin the centralizer algebra Z(E). Conversely, any idempotent P in Z(E) is an M -projection, as may be seen by the following argument. An idempotent P in Z(E)is contractive, as is I − P , since Z(E) is a function algebra. Applying P to thequantity P (x) + y − P (y) shows that ‖P (x)‖ ≤ ‖P (x) + y − P (y)‖. Similarly,‖y − P (y)‖ ≤ ‖P (x) + y − P (y)‖, and now we have one direction of (3.15). Toobtain the other, replace x by Px and y by y − Py in 3.7.3 (iii).

From this we see that the set of M -projections on E is exactly the lattice ofprojections in the centralizer algebra Z(E).

The following is the main result of this section. We will not prove all theimplications, since the purpose of this section is mostly to illustrate the connec-tions between the main ideas. However we remark that the ‘missing proof’ alsofollows easily from a formulation of (v) given in 3.7.7, together with results inChapter 4 (for example, from 4.6.2 in conjunction with 4.5.10).

Theorem 3.7.5 Let A be a unital Banach algebra and let E be a nondegenerateleft Banach A-module. The following are equivalent:

(i) There exists a compact space Ω, a contractive and unital homomorphismθ : A → C(Ω), and an isometric linear map Φ: E → C(Ω), such thatΦ(ax) = θ(a)Φ(x) for all a ∈ A, x ∈ E.

(ii) There is a contractive unital homomorphism θ : A → Cb(KE) such thatj(ax) = θ(a)j(x) for all a ∈ A, x ∈ E (see 3.7.1).

(iii) The canonical homomorphism A → B(E) given by the module action, mapsinto M(E) (indeed into Z(E) if A is a C∗-algebra).


(iv) The module action A×E → E extends to a contraction A⊗E → E, where⊗ is the injective Banach space tensor product (see A.3.1 and 1.5.3).

(v) The module action A × E → E extends to a contraction A ⊗g2 E → E,where g2 is the tensor norm in (A.6).

Proof That (iii) implies (ii) is clear from the discussion of multipliers in 3.7.1.Since Cb(KE) is a C(Ω) for some compact space Ω, we see that (ii) implies (i).If (i) holds, and if a1, . . . , an ∈ A, x1, . . . , xn ∈ E, then∥∥∥∑

k

akxk

∥∥∥ =∥∥∥Φ

(∑k

akxk

)∥∥∥ =∥∥∥∑

k

θ(ak)Φ(xk)∥∥∥. (3.16)

If ω ∈ Ω then θ(·)(ω) and Φ(·)(ω) are contractive functionals on A and E re-spectively. By the definition in A.3.1 it follows that∣∣∣∑

k

θ(ak)(ω) Φ(xk)(ω)∣∣∣ ≤

∥∥∥∑k

ak ⊗ xk

∥∥∥,

the last norm taken in A⊗E. Together with (3.16), this yields (iv). Since g2

dominates the injective tensor norm, (iv) implies (v).We omit the proof that (v) implies (iii), instead referring the reader to [63] for

the argument, which is a reprise of an earlier argument of Tonge [411,120], anduses the Pietsch factorization theorem and a little measure theory. For the last as-sertion in (iii): if A is a C∗-algebra then since the canonical homomorphism fromA to B(E) is a contractive homomorphism into M(E), it is a ∗-homomorphismby the last paragraph in 2.1.2, and maps into Z(E).

3.7.6 (Characterization of function modules) A module E satisfying one of theequivalent conditions of Theorem 3.7.5, will be called a function module over A,or a function A-module. The most interesting point, perhaps, is the following.For any Banach space E, first, E is a function M(E)-module, and second, by3.7.5 (iii), any function A-module action on E is a prolongation (see 3.1.12) ofthe M(E) action. We have also established the correspondences mentioned inthe last paragraph of 3.7.1.

3.7.7 Let A be a unital Banach algebra equipped with an operator space struc-ture, and let E be a nondegenerate left Banach A-module. Claim: E is a functionA-module if and only if Min(E) is an operator A-module. The ‘only if’ asser-tion is clear from condition (i) in 3.7.5. Conversely, if Min(E) is an operatorA-module, then it is an h-module (see 3.1.3). Hence the module action induces acomplete contraction from Max(A)⊗h Min(E) to Min(E). But this is equivalentto condition (v) in 3.7.5 by (1.46) and (1.10). Hence E is a function A-module.

3.7.8 (Examples)(1) We show, as asserted at the start of this section, that there are no interest-

ing function module representations on a Hilbert space H . Indeed ext(Ball(H∗))is precisely the unit sphere of H∗. Thus if T is a multiplier of H then ev-ery nonzero vector is an eigenvector for T ∗. If ζ, η is an orthonormal set

136 Dual operator modules

in H , and if λ, ν are the corresponding eigenvalues, then for some µ we haveT ∗(ζ + η) = µ(ζ + η) = λζ + νη, showing that λ = µ = ν. Therefore T ∗ and Tare scalar multiples of the identity operator. Thus H has no nontrivial multipli-ers. Hence there are no nontrivial function module actions on H .

(2) In contrast, one can show that if A is a uniform algebra then M(A) ∼= Aisometrically isomorphically (as Banach algebras). We omit the details sincethese facts are generalized in 4.5.11.

(3) If E is a closed subspace of C(Ω) and A is a uniform algebra on Ω,then AE is a function A-module. Thus there are plenty of examples of nontrivialfunction modules.

Corollary 3.7.9 (Tonge–Kaijser) If A is a unital Banach algebra, with multi-plication m viewed as a map A ⊗ A → A, then the following are equivalent:

(i) A is isometrically isomorphic to a uniform algebra,(ii) the mapping m : A⊗A −→ A is contractive,(iii) the mapping m : A ⊗g2 A −→ A is contractive.

Proof These follow by applying 3.7.5 with E = A regarded as an A-module.Indeed if this module satisfies condition (i) in 3.7.5 then we have

‖a‖ = ‖Φ(a)‖ = ‖θ(a)Φ(1A)‖ ≤ ‖θ(a)‖‖Φ(1A)‖ ≤ ‖θ(a)‖

for any a ∈ A, hence the contractive homomorphism θ : A → C(Ω) is an isometry.Thus A is (isometrically isomorphic to) a uniform algebra.

Corollary 3.7.10 If A is a unital operator algebra, then A is completely isomet-rically isomorphic to a uniform algebra if and only if A is a ‘minimal operatorspace’ (see 1.2.21).

Proof The one direction is obvious. Conversely, assume that A is a ‘minimaloperator space’ and an operator algebra. Then the product is contractive as amap from Min(A) ⊗h Min(A) to A, by 2.3.2. Hence it is contractive as a mapfrom Max(A) ⊗h Min(A) to A. Now use (1.46) and 3.7.9 (iii).

An alternate proof of this result will be given in the Notes to Section 4.6.

3.8 DUAL OPERATOR MODULES

In this section, M and N are at least unital Banach algebras which are alsodual operator spaces. We discuss M -N -bimodules below, this contains the one-sided module case by taking either M or N equal to C. Many of the resultsin this section are due to Effros and Ruan in the case that M and N arevon Neumann algebras.

3.8.1 (Dual operator bimodules) A concrete dual operator M -N -bimodule is aw∗-closed subspace X of B(K, H) such that θ(M)Xπ(N) ⊂ X , where θ and πare unital w∗-continuous completely contractive representations of M and N on


H and K respectively. An (abstract) dual operator M -N -bimodule is defined tobe an operator M -N -bimodule X , which is also a dual operator space, and whichpossesses a W ∗-representation. By the latter term we mean a CES-representation(Φ, θ, π) in the sense of 3.3.3, with Φ, θ and π w∗-continuous, and with θ, π unital.By A.2.5, if there exists such a triple, then Φ is a w∗-homeomorphism onto itsw∗-closed range, and indeed Φ(X) is a concrete dual operator M -N -bimodule.

If M and N are dual operator algebras, a W ∗-representation is called faithfulif θ and π are completely isometric. We remark that one may always choose aW ∗-representation of a dual operator bimodule acting on a single Hilbert space.This may be accomplished, for example, by taking a large enough direct sum ofthe maps involved, as in the third last paragraph of the proof of 3.3.1.

3.8.2 (Normal dual modules) An M -N -bimodule X which is also a dual opera-tor space is called a normal dual bimodule if the trilinear mapping (a, x, b) → axbfrom M ×X ×N to X is separately w∗-continuous. Since the product on B(H)is separately w∗-continuous, it is clear that every dual operator bimodule is anormal dual bimodule. The converse of this is true too (see 4.7.7 and 4.7.6). Fornow we just state a preliminary form of this converse in the important operatoralgebra case.

Theorem 3.8.3 Let M and N be unital dual operator algebras, and let X be anondegenerate operator bimodule (or, equivalently, an h-bimodule) over M andN . Assume that X is a normal dual M -N -bimodule.(1) X has a faithful W ∗-representation (Φ, θ, π) on a single Hilbert space H.

Thus X is a dual operator M -N -bimodule.(2) If M = N , then the representation in (1) may be accomplished with π = θ.

Proof We follow the proof of 3.3.1, except that we suppose that X ⊂ B(L) isa w∗-closed subspace. As in that proof we may write axb as a product of maps.Indeed there are three completely contractive maps α : M → B(H0, L), andv : X → B(K0, H0), and β : N → B(L, K0), such that axb = α(a)v(x)β(b) fora ∈ M, b ∈ N and x ∈ X . Since (a, x, b) → axb is assumed to be separately w∗-continuous, it follows as in 1.6.10 that the three maps α, v, β can be chosen to bew∗-continuous. By a slight modification of 2.7.10, we can then factor α and β asfollows. There exist a Hilbert space H1, a w∗-continuous unital homomorphismθ1 : M → B(H1) and two contractions T1 : H0 → H1 and R : H1 → L suchthat α(a) = Rθ1(a)T1. Likewise, there exist a Hilbert space K1, a w∗-continuousunital homomorphism π1 : N → B(K1), and two contractions S : L → K1 andT2 : K1 → K0 such that β(b) = T2π1(b)S. We define Ψ: X → B(K1, H1) byΨ(x) = T1v(x)T2. Then Ψ is w∗-continuous. Moreover,

axb = Rθ1(a)Ψ(x)π1(b)S, a ∈ M, x ∈ X, b ∈ N.

The rest of the proof is identical to that of 3.3.1, except that we need to checkthat all maps are w∗-continuous. This is clear by the principle mentioned in thesecond paragraph of 1.6.10.


3.8.4 (Examples) All of the examples in 3.1.2 have simple dual bimodulevariants. For instance, a unital dual operator algebra M is canonically a dualoperator M -M -bimodule. Since a w∗-closed M -N -submodule of a dual operatorM -N -bimodule is clearly a dual operator M -N -bimodule, it is easy to see thatif M is a unital dual operator algebra, and if p and q are projections in M , thenqMp is a dual operator (qMq)-(pMp)-bimodule. As in 3.3.4, one may view dualoperator bimodules over unital dual operator algebras, as the corners pR(1− p)in a unital dual operator algebra R. We leave the details as an exercise.

Another important class of dual bimodules are the ‘normal rigged bimodules’which we shall meet in Section 8.5.

3.8.5 (Normal Hilbert modules) If π : M → B(H) is a w∗-continuous unitalcompletely contractive representation, then H (with its Hilbert column spacestructure) is a left operator M -module, by 3.1.6. Clearly H is a dual operatorM -module. We call such H a normal Hilbert module. The category MNHMODof normal Hilbert M -modules is a subcategory of MHMOD. An obvious (butimportant) fact is that MNHMOD is closed under Hilbert space direct sums:this is just the fact that a direct sum of w∗-continuous representations of Mis a w∗-continuous representation. If M is a W ∗-algebra, then there is a ‘mostimportant’ module in MNHMOD, namely the standard form L2(M). We shallnot really need this module in this book, although it is mentioned briefly in8.5.39. It is worth pointing out that if M is a von Neumann algebra acting ona separable Hilbert space H , then H is (up to spatial equivalence) the standardform if and only if M has a ‘separating cyclic vector’ ξ ∈ H (that is, H = [Mξ],and the map x → xξ on M is one-to-one). See [175] and [408, Chapter IX].

It follows from 2.5.9 that there is a bijective correspondence between non-degenerate completely contractive representations of an approximately unitaloperator algebra A, and w∗-continuous completely contractive unital represen-tations of A∗∗. Thus as objects, AHMOD = A∗∗NHMOD. If T : H1 → H2 isa morphism in AHMOD, then by the separate w∗-continuity of operator multi-plication, T is also a morphism in A∗∗NHMOD. The converse is even easier, sothat we see that AHMOD and A∗∗NHMOD are equal as categories. Thus, forexample, H is a generator (see 3.2.7) for AHMOD if and only if H (consideredas an A∗∗-module in this canonical way) is a generator for A∗∗NHMOD.

Proposition 3.8.6 Let ρ : M → B(H) be a normal representation of a W ∗-algebra M , and view H ∈ MNHMOD. Then ρ is faithful if and only if H is agenerator for MNHMOD (and if and only if H is a cogenerator).

Proof Suppose that H is a generator, and that x ∈ M with ρ(x) = 0. Supposethat M is a von Neumann algebra in B(K), say. For any T ∈ MB(H, K), wehave xT = Tρ(x) = 0. Hence x = 0 on the closure L of the joint span of theranges of all such maps T . Clearly L is an M -submodule of K, and hence so isL⊥. Hence L⊥ = (0), by definition of L and of ‘generator’. Thus x = 0, and soρ is faithful.


For the converse, we may suppose that M is a von Neumann algebra in B(H).To show that H is a generator, it suffices to show that MB(H, K) = (0), for anynontrivial K in MNHMOD (since Ker(S)⊥ is an M -module for any morphismS of MNHMOD). Indeed, it suffices to show that there exists a nonzero mapT in MB(H(∞), K), where H(∞) is the countably infinite multiple of H . For ifthis were the case, and if εi is the inclusion of H into H(∞) as the ith summand,then T εi = 0 for some i.

Fix ξ ∈ K, ‖ξ‖ = 1, and let π be the representation of M on K. Then〈π(·)ξ, ξ〉 is a normal state on M . By the proof of 3.6.6 in [320], there existsη ∈ H(∞) such that 〈π(x)ξ, ξ〉 = 〈xη, η〉, for all x ∈ M . Define T : Mη → K byT (xη) = π(x)ξ. By the above, T is isometric, and hence extends to an isometricM -module map from [Mη] onto [π(M)ξ]. Let P be the projection onto [Mη],this is also an M -module map by 3.2.3. Then 0 = TP ∈ MB(H(∞), K).

The cogenerator assertion follows as in 3.2.11 (7).

We remark in passing that a simple Zorn’s lemma argument applied to thespaces [π(M)ξ] in the last proof, will show that any K in MNHMOD is spatiallyequivalent to a direct sum of submodules of H (∞). It follows that K is spatiallyequivalent to a submodule of a multiple of H . This shows that every generatorH for MNHMOD is ‘universal for MNHMOD’. Conversely, the proof of 3.2.11(6) shows that any ‘universal for MNHMOD’ is a generator. From this, and theremarks above 3.8.6, it is an easy exercise to prove the matching facts in 3.2.8.

3.8.7 (Direct sums of dual modules) For a family Xi : i ∈ I of dual oper-ator Mi-Ni-bimodules, the direct sum ⊕∞

i Xi is a dual operator bimodule overthe algebras ⊕∞

i Mi and ⊕∞i Ni. To see this, suppose that (Φi, θi, πi) are W ∗-

representations of Xi, for each i, and suppose (as we may) that Φi, θi, πi mapinto Bi = B(Hi). Then by the last paragraph in 1.4.13, we obtain w∗-continuouscomplete contractions Φ = (Φi), θ = (θi), and π = (πi), from ⊕∞

i Xi,⊕∞i Mi,

and ⊕∞i Ni, respectively, into ⊕∞

i Bi ⊂ B(⊕iHi). Moreover Φ is completely iso-metric. It is easy to check that θ and π are homomorphisms, and that (Φ, θ, π)is a W ∗-representation for the bimodule ⊕∞

i Xi. This yields the desired result.

3.8.8 (Quotients of dual modules) Let M and N be unital dual operator al-gebras for simplicity. Suppose that X is a dual operator M -N -bimodule andthat Y is a w∗-closed M -N -submodule. In 3.1.10 and 3.3.2 we said that X/Y isa nondegenerate operator M -N -bimodule, and we said that the canonical quo-tient map q : X → X/Y is an M -N -bimodule map. By the duality of subspacesand quotients (see 1.4.4), X/Y is the dual operator space of Y⊥, and q is w∗-continuous. From this and the fact that X is a normal bimodule, it is easy to seethat the bimodule actions on X/Y are separately w∗-continuous. That is, X/Yis a dual operator M -N -bimodule by Theorem 3.8.3.

3.8.9 (The second dual of a bimodule) Suppose that A and B are approx-imately unital operator algebras. Then A∗∗ and B∗∗ are unital dual operatoralgebras by 2.5.8 and 2.7.3. If X is a nondegenerate operator A-B-bimodule,


then X∗∗ is a dual operator A∗∗-B∗∗-bimodule in a canonical way. To see this,first notice that by 1.6.7, there is a unique separately w∗-continuous extensionfrom A∗∗ × X∗∗ to X∗∗, of the A-module action on X . Similarly we obtain aseparately w∗-continuous map from X∗∗ × B∗∗ to X∗∗. Using the separate w∗-continuities and routine w∗-approximation arguments, it is easy to check thatthese are nondegenerate module actions, and that they commute. Thus X∗∗ isan A∗∗-B∗∗-bimodule for these canonical second dual actions. Since these actionsA∗∗ ×X∗∗ → X∗∗ and X∗∗ ×B∗∗ → X∗∗ are completely contractive (see 1.6.7),X∗∗ is an h-bimodule. It therefore follows from Theorem 3.8.3 that X∗∗ is a dualoperator A∗∗-B∗∗-bimodule.

3.8.10 (W ∗-converging infinite sums) Let H be a Hilbert space, let I be acardinal, and suppose that a = (ai)i ∈ Rω

I (B(H)) and b = (bi)i ∈ CωI (B(H)).

According to (1.19), we may identify RωI (B(H)) and Cω

I (B(H)) with B(H(I), H)and B(H, H(I)) respectively. Let S : H(I) → H and T : H → H(I) correspond toa and b in these identifications. For each finite subset I0 of I consider the finitesum

∑i∈I0

aibi ∈ B(H). We obtain a bounded net indexed by such I0 and it iseasy to check that this net converges in the w∗-topology of B(H) to ST . In thesequel, we will denote this w∗-limit by

∑i∈I aibi ∈ B(H).

Proposition 3.8.11 (Dual matrix modules) If X is a dual operator bimoduleover M and N , and if I, J are sets, then MI,J(X) is also a dual operator bimoduleover M and N . If M and N are unital dual operator algebras then MI,J(X) isa dual operator MI(M)-MJ(N)-bimodule.

Proof Fix a W ∗-representation (Φ, θ, π) of X on a Hilbert space H , and fixcardinals I and J . Then by 1.6.3 (2), the amplification

Ψ = ΦI,J : MI,J(X) −→ MI,J(B(H)) ∼= B(H(J), H(I))

is a w∗-continuous complete isometry. Also, it is easy to see that the multiplesθ(I) : M → ∞I (B(H)) ⊂ B(H(I)) and π(J) : N → ∞J (B(H)) ⊂ B(H(J)) arew∗-continuous completely contractive homomorphisms. Since (Ψ, θ(I), π(J)) isevidently a W ∗-representation of the M -N -bimodule MI,J(X), we obtain thefirst assertion.

For the second, we may assume by Theorem 3.8.3 that M, N are unital w∗-closed subalgebras of B(H), and that X ⊂ B(H) is a w∗-closed submodule ofB(H). Using (1.19), we see that MI,J(B(H)) ∼= B(H(J), H(I)) is a dual op-erator MI(B(H))-MJ(B(H))-bimodule, and hence it is clearly a dual operatorMI(M)-MJ(N)-bimodule. Thus we only need to check that the w∗-closed sub-space MI,J(X) ⊂ MI,J(B(H)) is a submodule. This follows from the first para-graph of 3.8.10. Indeed if a = [aik] ∈ MI(M) and x = [xkj ] ∈ MI,J(X), thenthe i-j entry of ax ∈ MI,J(B(H)) is the (infinite) sum

∑k∈I aikxkj . The latter

is the w∗-limit of finite sums∑

k∈I0aikxkj of terms in X . Since X is w∗-closed,∑

k∈I aikxkj ∈ X . The argument for the right action is the same.

Henceforth in this section M and N are unital dual operator algebras.


3.8.12 Let X and Y be dual operator M -N -bimodules, and suppose thatu : X → Y is a w∗-continuous completely bounded M -N -bimodule map. Thenit is not hard to deduce from the proof of 3.8.11 that the w∗-continuous com-pletely bounded map uI,J : MI,J(X) → MI,J(Y ) is a MI(M)-MJ (N)-bimodulemap. In fact a surprising stronger statement holds in the case that M and N areselfadjoint. We assume that I = J (the contrary case is derivable from this one):

Theorem 3.8.13 (Effros and Ruan) Let X and Y be dual M -N -bimodulesover W ∗-algebras M and N , and suppose that u : X → Y is a given completelybounded M -N -bimodule map (we are not assuming that u is w∗-continuous). LetI be a cardinal. Then the map uI : MI(X) → MI(Y ) is a completely boundedMI(M)-MI(N)-bimodule map.

Proof By 3.8.3, we may assume that X ⊂ B(K, H) and Y ⊂ B(L, G) areconcrete dual operator M -N -bimodules, with M faithfully represented as a uni-tal von Neumann algebra on both H and on G, and N faithfully representedas a unital von Neumann algebra on both K and on L. Form the bimodulePaulsen systems S1 ⊂ B(H ⊕ K) and S2 ⊂ B(G ⊕ L) (see 3.6.1) of X and Yrespectively. Then the two systems MI(S1) ⊂ MI(B(H ⊕K)) = B(H(I) ⊕K(I))and MI(S2) ⊂ B(G(I) ⊕ L(I)), may respectively be thought of as the bimod-ule Paulsen systems of the MI(M)-MI(N)-bimodules MI(X) ⊂ B(K(I), H(I))and MI(Y ) ⊂ B(L(I), G(I)) (see 3.8.11 and its proof). By 3.6.1, we obtain acompletely positive unital map Θ: S1 → S2 taking[

a xy∗ b

]−→

[a u(x)

u(y)∗ b

], a ∈ M, b ∈ N, x, y ∈ X.

Then Ψ = ΘI : MI(S2) → MI(S2) is completely contractive, unital, and com-pletely positive by 1.3.3. Since Ψ is a ∗-homomorphism on MI(M) ⊕ MI(N),it is a bimodule map by 1.3.12. Thus the ‘1-2-corner’ of Ψ, which is uI , is anMI(M)-MI(N)-bimodule map.

3.8.14 (A module version of the weak* Haagerup tensor product) We will bevery brief, and omit proofs, since we are leaving the realm of ‘basic theory’ here.If X and Y are respectively right and left dual operator M -modules, then onemay define the weak* module Haagerup tensor product X ⊗w∗hM Y , to be thequotient of the weak* Haagerup tensor product X ⊗w∗h Y (see 1.6.9) by thesmallest norm closed subspace containing terms of form xr y − x ry. Herex ∈ Rw

I (X), y ∈ CwI (Y ), and r ∈ MI(M). Also, the expression x y means the

sum∑

i∈I xi ⊗ yi in X ⊗w∗h Y , in the sense of 1.6.9. The expressions such as xrabove make sense by virtue of 3.8.11. We continue to write

∑i∈I xi⊗yi and xy

for the equivalence class in the quotient X ⊗w∗hM Y . Every w ∈ X ⊗w∗hM Ymay be written in such a way, and we call the expression

∑i∈I xi ⊗ yi a weak

representation for w.It is not too difficult to check that some of the properties of the module

tensor product that we saw in Section 3.4, have appropriate versions for this


new tensor product. For example, this tensor product is functorial. Indeed, onecan show easily that if u : X1 → X2 and v : Y1 → Y2 are completely boundedM -module maps between dual operator M -modules, and if either (a) M is aW ∗-algebra, or (b) u and v are w∗-continuous, then there is a well defined linearcompletely bounded map u⊗v from X1⊗w∗hMY1 to X2⊗w∗hMY2 taking any weakrepresentation

∑i xi⊗yi to

∑i u(xi)⊗v(yi). Also, ‖u⊗v‖cb ≤ ‖u‖cb‖v‖cb. It is

also easy to see that if X is a left dual operator M -module then M⊗w∗hM X ∼= Xcompletely isometrically. Other, more sophisticated, properties of the moduleweak* Haagerup tensor product, such as associativity (in the sense of 3.4.10),are much harder to prove. Indeed, for these it is preferable to use equivalentformulations of this tensor product, which have been introduced and studied byMagajna in the case that M is a W ∗-algebra (see the Notes section).

One attractive feature of this tensor product, is that it can be quite useful inthe manipulation of certain dual space identities. To illustrate this, suppose thatM is a W ∗-algebra, and that X is a dual operator M -module. One can showusing (1.59), and the just mentioned associativity, that

CwI (M) ⊗w∗hM X ∼= CI ⊗w∗h M ⊗w∗hM X ∼= CI ⊗w∗h X ∼= Cw

I (X).

Setting X = RwJ (M) and using (1.20) yields

CwI (M) ⊗w∗hM Rw

J (M) ∼= CwI (Rw

J (M)) ∼= MI,J(M).


Linear spaces of Hilbert space operators which are bimodules over operator al-gebras have been studied for decades (e.g. see [108]). As usual we have restrictedour attention to topics particularly connected with operator spaces.

Operator modules over C∗-algebras were first defined by Wittstock around1980 in [432], where amongst other things he gave module map extension the-orems of the type in 3.6.2. See also [431, 273]. Other early theory of operatormodules and their tensor products appears in papers of Effros, alone or withcoauthors Kishimoto and Exel (e.g. see [135] and [138]). These were partly in-spired by some work of Haagerup [176, 177]. Christensen, Effros, and Sinclaircharacterized operator modules over unital C∗-algebras in [92], and used this tosolve some questions about the cohomology of operator algebras. See also [385],and references therein, for more on bounded cohomology of operator algebras;there is also the school of Helemskii (e.g. see [199,200]). Operator modules werealso studied in the 1980s (under that name) by Effros and Ruan (see [143]). How-ever later they changed notation and refer to what we call ‘matrix normed mod-ules’ by the name ‘operator module’. The objects which we call ‘matrix normedmodules’ appear first in Ruan’s work on operator amenability (see [149,377], andreferences therein), and later played a role in [236, 374, 375], for example. Thedevelopment of operator modules was of course influenced by important earlierwork on Hilbert modules (see references in 3.2.6). Another early source for op-erator modules and their tensor products are the papers of Smith with various


coauthors (e.g. see [391, 85]), and of Ara and Mathieu [12]. These date to thefirst few years of the 1990s decade, and contain many very useful and powerfultechniques that were beyond the scope of our book. Much of this work was on the‘central Haagerup tensor product’, that is the module Haagerup tensor productR ⊗hZ R of a von Neumann algebra R over its center Z. This is related to theconcept of ‘elementary operators’; for example the operators on a von Neumannalgebra R of the form x →

∑k akxbk. Here ak, bk ∈ R. See also [13]. The asso-

ciated map R ⊗Z R → CB(R) whose range is the set of such operators extendsto a map R ⊗hZ R → CB(R). It is shown in [85] that this map is an isome-try, generalizing an older result of Haagerup [176]. The memoir [65] of Blecher,Muhly, and Paulsen (circulating since the first half of the 1990s) is a standardreference for operator modules and the module Haagerup tensor product. Theideas in Section 3.4 (and many of the ideas in other sections) may be found here.Some of these facts were found independently in [261].

Some of the algebraic aspects of the theory have been inspired by Muhly’sconsistent advocation (for example, in his CBMS lectures of May 1990, or [279])of the transferal of the classical algebraic theory of rings and modules to theframework of modules over general operator algebras. Of course there are severalnatural obstacles, for example the dual of an operator module is usually not anoperator module (although it is a matrix normed module), and direct sums canbe problematic. Nonetheless this is a very important and useful perspective.

3.1: Many of the basic constructions with operator modules discussed inSection 3.1 are well-known (e.g. see [65]). Although we will not take the time to doso here, one may define, for example, limsup’s, direct limits, and ultraproducts,of operator bimodules. See 2.3.3, 2.3.8, 1.2.31, and 2.2.13 for the operator algebraversions of these constructions which are easily adapted. We are not sure whomto attribute 3.1.7 to—perhaps [65]. The Banach algebra version of 3.1.11 is dueto Johnson (see [106, 2.9.50]).

3.2: For historical sources of Hilbert modules, we refer to [21,22] in additionto those cited in 3.2.6. Item 3.2.2 is from [381]. The second half of the sectioncomes from [72], and is also valid for contractive as opposed to completely con-tractive representations. We also study there a sufficient condition for the doublecommutant property to hold which is more general than the notion of ‘genera-tor’. See that paper for more information. The double commutant theorem hereis an important tool, for example in the study of Morita equivalence of oper-ator algebras. The notions of A-universal, generator, cogenerator, in 3.2.7 arefunctorial. The facts in 3.2.8 are from [361].

The most common tensor product for Hilbert modules is given by the pro-cedure involving π ⊗ θ in 2.2.2. Another very important tensor product is theso-called ‘fusion’ or ‘relative tensor product’ of Connes’ bimodules (the ‘corre-spondences’ briefly discussed in 8.5.39). E.g. see [101, Section V.B] or [408, IX.3].

3.3: Theorem 3.3.1 is a remarkable result due to Christensen–Effros–Sinclair[92]. Its proof still works if A and B are not necessarily approximately unital,and without any nondegeneracy condition, if X is merely ‘normed by A and B’.


By this we mean

‖[xij ]‖n = sup‖[axijb]‖n : a ∈ Ball(A), b ∈ Ball(B), [xij ] ∈ Mn(X), n ∈ N.

Magajna, Johnson, and Pop have given isometric characterizations of Banachbimodules which admit an operator module structure (e.g. see [267,345]).

3.4. See references at the beginning of this Notes section. Some other factsmentioned in this section are from [19]. It is easy to see by an argument almostidentical to the proof of the projectivity of the Haagerup tensor product in 1.5.5,that the module Haagerup, and module projective, tensor products, are alsoprojective in an appropriate sense (for complete quotient module maps).

If X and Y are left operator modules over operator algebras A and B re-spectively, then the minimal tensor product X ⊗min Y is an operator moduleover A ⊗min B, with the ‘canonical action’ (a ⊗ b)(x ⊗ y) = (ax) ⊗ (by). Aneasy way to see that this is an operator module is to use 3.3.1 to choose twoHilbert spaces H and K with X → B(H) and Y → B(K), and completelycontractive representations θ and π of A and B on H and K respectively, suchthat θ(A)X ⊂ X and π(B)Y ⊂ Y . Then X ⊗min Y may be identified with asubspace of B(H ⊗ K), and by 2.2.2 there is a completely contractive homo-morphism θ ⊗ π : A ⊗min B → B(H ⊗ K). Clearly X ⊗min Y is a concrete leftoperator (A ⊗min B)-module for the action induced by θ ⊗ π. It is easy to seethat this tensor product is ‘functorial’ (in a sense similar to 3.4.5) for completelybounded module maps. Similarly one may define a weak* version X⊗Y of theminimal tensor product of dual operator bimodules using 1.6.5 and 3.8.3 (see8.2.17 or [48] for a special case of this). Another kind of module tensor productA⊗Z B of C∗-algebras over a commutative C∗-algebra Z may be found discussedin [37, 262].

3.5: Again [65] is a basic reference here. The space ACB(A, X) considered in3.5.4 sometimes appears in the literature (e.g. see [19,51,132]) as a useful moduleversion of the multiplier algebra of X . See also the last paragraph of the Notesto Section 8.1. One can show quite easily that if X is a left operator A-module,then so is ACB(A, X). Variants of 3.5.11 appear in [135,288]. Some other ideasin 3.5 (in some form) appear in e.g. [46–48,51,52]. Some of the module mappingspace results in this section have been observed by others too, for example byAristov [19], Peters [323], and Neufang (private communication). Magajna hassome results, in [265,268] for example, for the module Haagerup tensor productcorresponding to 3.5.9 and some of the relations in 1.5.4, 1.5.5, 1.5.7, 1.6.9.

A Hilbert module H over A is called locally cyclic if any finite subset of H iscontained in [Aξ], for some vector ξ ∈ H . If H and K are locally cyclic Hilbertmodules over C∗-algebras A and B respectively, and if X is a given operatorA-B-bimodule, then Smith has shown that ‖T ‖ = ‖T ‖cb for any A-B-bimodulemap T : X → B(K, H). There are also versions of this result valid for bimodulemaps. See [391, Section 2], [385, Section 1.6], [7].

There are some relations between modules over an approximately unital op-erator algebra A, and modules over a C∗-cover B of A, which are often useful.


It is shown in [50], using A.5.9, that a Banach A-module E has at most oneBanach B-module action extending the A-module action. Thus if T : E1 → E2

is a surjective isometric A-isomorphism between two Banach D-modules, thenT is a B-isomorphism too. It follows from this that the category of Banachmodules over B is a subcategory (via the ‘forgetful functor’) of the category ofBanach modules over A. Similarly, BOMOD is a subcategory of AOMOD, andBHMOD is a subcategory of AHMOD. This is not as obvious as it looks atfirst sight.

There is a natural variant for modules of the notion of C∗-cover from Chapter2. This is the dilation or adjunct, and was studied in [288, 280, 50]. It may bedefined to be B ⊗hA X , for an operator A-module X and a C∗-cover B of A. Inmodule theory this is called a change of rings. This construction has many usefulproperties which are noted in the cited references. In particular, its universalproperty provides another way to transfer certain problems into the C∗-algebraworld where they may be solved (e.g. see [52]). In the language of algebra, thisdilation, and the forgetful functor discussed in the last paragraph, form a pairof adjoint functors between BOMOD and AOMOD.

3.6: After Wittstock’s work on module extensions cited above, Suen gavein [402] the pretty proof of 3.6.2. Muhly and Solel consider various kinds ofinjectivity of Hilbert modules in [281] (e.g. see Theorem 3.1 there). Anotherimportant reference is Paulsen’s work on injectivity in various module categories,and relations to cohomology (e.g. see [311,163,217,314]). See also, for example,[68, 50], and papers of Christensen, Sinclair, and Smith cited here. For 3.6.5 (i)see [386, Theorem 3.1]; the main idea of the proof is essentially a remark in [318].The injectivity of the module Haagerup tensor product over C∗-algebras appearsin [7]. None of the results in Section 3.6 are valid in general with A replacedby a nonselfadjoint operator algebra. See, for example, Proposition 7.2.11, andExample 3.5 in [391].

If X is a nondegenerate operator module over a C∗-algebra, and Φ is a linearcontractive projection from X onto a closed submodule, then Φ is a module map.This is proved in the paper in preparation referred to in the Notes to 4.7.

3.7: Multipliers of Banach spaces were first considered by Cunningham [104],but Alfsen and Effros’ paper [4] is the definitive source for most of the basic ideasof this theory. Behrend’s contributions are also quite significant [31]. See [195] formore detailed references, and an excellent survey of Banach space multipliers,including several other important characterizations of multipliers and central-izers. For example, Z(A) for a C∗-algebra A is the center of M(A). This oldfact essentially dates back to [4]; it generalizes to approximately unital operatoralgebras but now one needs the diagonal of the center of M(A) (see [73, Section7]). Jarosz proves in [204] that a Banach space E not containing a isometric copyof c0 (and in particularly any reflexive Banach space) has Z(E) = M(E) ∼= ∞nfor some n ∈ N. Hence the function A-module actions on such E may be com-pletely characterized: the n minimal projections in Z(E) give an isomorphismE ∼= ⊕n

k=1Ek, and each Ek has only a ‘scalar’ A-module action.


Many of the results in this section are taken from [195], the main exceptionbeing the main theorem, which is a slightly weaker version of a result in [63],which in turn relies mainly on Tonge’s result from [411]. Corollary 3.7.9 is from[216,411], whereas 3.7.10 is from [40]. Some applications of 3.7.5 are given in [63],and in Part A of [53].

3.8: The material in this section developed out of Effros and Ruan’s excel-lent paper [143], treating dual operator modules over von Neumann algebras.They proved many of the results here (for example, 3.8.3 and 3.8.11) in thatcase. The case of 3.8.3 for A and B dual operator algebras is from [63]. Moregeneral results will be met later at the end of 4.7. For W ∗-algebras, the categoryNHMOD was studied under the name Normod by Rieffel [361]. Item 3.8.6, andthe remark after it, are from that paper, as are observations such as the factthat AHMOD = A∗∗NHMOD. For a discussion of the second dual of Banachbimodules, e.g. see 2.6.15 and A.3.51–54 in [106]; these results extend easily tomatrix normed modules. Many of the earlier results in this section have simple‘normal matrix normed module’ versions. The result 3.8.9 (or a small variantof it) was proved in [311]. Second duals have natural w∗-continuous extensionproperties. For example, let X be an operator bimodule over approximately uni-tal operator algebras A and B, and let Y ∗ be a dual matrix normed bimodule(in the sense of 3.1.5(2)) over A and B. Then if u : X → Y ∗ is a completelybounded A-B-bimodule map, its w∗-continuous extension u : X∗∗ → Y ∗ is easilyseen to be an A-B-bimodule map as well. Thus in analogy with (1.28), we havean isometric identity ACBB(X, Y ∗) = w∗-ACBB(X∗∗, Y ∗).

A special case of the result 3.8.13 was first proved by May [272], the generalcase is from [143] (see also [273]). The module weak* Haagerup tensor productwas mentioned in [139], and comprehensively explored in [263]. It is also knownas the extended module Haagerup tensor product. It is not obvious that the defi-nitions from those two papers coincide; Magajna has pointed out to us that oneneeds to use Lemma 3.2 (and Remark 3.7) from [263]. Magajna treats the moregeneral case of ‘strong operator modules’ over W ∗-algebras; unfortunately weare not able to develop these interesting modules here. Projection techniques inthe W ∗-algebras allow Magajna to show that the module weak* Haagerup ten-sor product behaves very similarly to the usual weak* Haagerup tensor product.In [48] this theory was applied to selfdual C∗-modules (see 8.5.40). For morerecent results on this tensor product, see [268] and references therein.

A sampling of other interesting papers on operator modules not mentionedabove, or elsewhere in our text, includes: [7, 18, 95, 202, 253, 288, 292, 345]. Morereferences may be found within these papers. The reader is also directed to thelong series of important papers by Magajna on operator modules, we have onlycited a sample in our bibliography. Several recent Ph.D. theses have also focusedon aspects of operator modules with applications to cohomology or harmonicanalysis, for example, those of Neufang [291], Spronk [396], and Wood [434].The reader might also consult [377, 378] for the work of Runde, and for otherreferences.

4

Some ‘extremal theory’

4.1 THE CHOQUET BOUNDARY AND BOUNDARYREPRESENTATIONS

Many problems in functional analysis are best tackled via extreme points. Ex-treme points, particularly in the guise of the Choquet or Shilov boundaries, playa substantial role in the theory of uniform algebras. Since uniform algebras areparticular examples of operator algebras (see 2.2.5) one might expect extremepoints to play a large role for us. Instead, in our book we use a variant of the‘extremal approach’, focused on the ‘noncommutative Shilov boundary’ and the‘injective envelope’. This theory, and its many applications, are developed in thischapter, and in Section 8.4. We already saw in Section 3.7 that function algebrasand function modules may be completely characterized in terms of certain setsof extreme points of the unit ball in the dual space. Later in this chapter we shallsee a noncommutative version of this, which is tightly connected to the ‘noncom-mutative Shilov boundary’, and which gives deep insights into the structure ofoperator algebras and their modules.

For motivational purposes, we first look at the classical Choquet and Shilovboundaries. The material in this first section of the chapter will not be relied onlater, except for inspiration and comparison.

4.1.1 We recall from 2.2.5 that a uniform algebra is a closed unital-subalgebraA of C(Ω), for a compact Hausdorff space Ω. Some authors suppose that Aseparates points of Ω too. There are several good texts on uniform algebras, e.g.see [167, 400]. We use the term function space for a closed subspace of C(Ω). Aunital function space is a closed subspace F of C(Ω) which contains constantfunctions. More generally, we also use this term for a Banach space E with adistinguished element e ∈ E, such that E is isometrically isomorphic to an F asabove, with the isomorphism taking e to 1. A map with the latter property willbe called ‘unital’. For a unital function space (E, e) we write S(E) for the set offunctionals ϕ ∈ E∗ with ϕ(e) = ‖ϕ‖ = 1. Thus S(C(Ω)) is the set of probabilitymeasures on Ω. We write δω for the point mass at ω ∈ Ω, and εω for its restrictionto E, if E ⊂ C(Ω). A Choquet boundary point for E in Ω is defined to be a pointω ∈ Ω satisfying one of the equivalent conditions in the next result:

148 The Choquet boundary and boundary representations

Lemma 4.1.2 Suppose that E is a nonzero closed subspace of C(Ω), for a com-pact Hausdorff space Ω. Suppose that E contains constant functions and separatespoints of Ω. For a point ω ∈ Ω the following are equivalent:

(i) εω ∈ ext(Ball(E∗)),(ii) εω ∈ ext(S(E)),(iii) δω is the only probability measure on Ω extending εω.Moreover, if ψ ∈ ext(S(E)), then ψ = εω for some (necessarily unique) ω ∈ Ω.

Proof The equivalence of (i) and (ii) is an elementary exercise. To prove theequivalence of (ii) and (iii), we will use the well-known fact that

ext(S(C(Ω))) = δω : ω ∈ Ω. (4.1)

Suppose that ω satisfies (iii), and that εω = 12 (ϕ + ψ), with ϕ, ψ ∈ S(E). By

the Hahn–Banach theorem, we may extend ϕ, ψ to probability measures on Ω.Then the average of these measures, by hypothesis, equals δω. By (4.1), bothprobability measures coincide with δω. So ϕ = ψ = εω. This yields (ii).

Let ψ ∈ ext(S(E)), and consider the set Pψ of probability measures on Ωextending ψ. This is a nonempty (by the Hahn–Banach theorem) compact convexsubset of S(C(Ω)), which has an extreme point by the Krein–Milman theorem.An obvious argument shows that such an extreme point is an extreme pointof S(C(Ω)), and is hence by (4.1) a point mass δσ for some necessarily uniqueσ ∈ Ω. This proves the final assertion. If ω satisfies (ii), and ψ = εω, then theabove argument shows that σ = ω, and Pψ is a singleton. This yields (iii).

4.1.3 (The Choquet boundary) For any unital function space (E, e), we con-sider the collection of pairs (Ω, j) consisting of a compact Hausdorff space Ω,and a unital linear isometry j : E → C(Ω), such that j(E) separates points ofΩ. We shall call such a pair a function-extension of E. The above lemma showsthat the set of Choquet boundary points in Ω does not depend essentially onthe function-extension of E, and that this set is homeomorphic to ext(S(E)) astopological spaces. Note that ext(S(E)) is not empty by the Krein–Milman the-orem. Thus we may refer to either ext(S(E)), or to the set of Choquet boundarypoints in Ω, as the Choquet boundary of E. We write Ch(E) for either set.

4.1.4 (The Shilov boundary) As noted in 1.2.21, any Banach space E is linearlyisometric to a closed subspace of C(Ω), for some compact Ω. Indeed, this maybe done so that E separates points. The question arises of finding the smallestΩ which works, that is, what is the minimal or ‘essential’ topological space onwhich E can be supported in this way? For unital function spaces E such aminimal Ω does exist, and it is called the Shilov boundary ∂E of E.

To be a little more careful, we declare two function-extensions (Ω, j) and(Ω′, j′) to be E-equivalent, if there exists a surjective homeomorphism τ : Ω → Ω′

such that j′(x) τ = j(x), for all x ∈ E. This is an equivalence relation on thecollection of function-extensions of E. We define a Shilov boundary for a unital

Some ‘extremal theory’ 149

function space (E, e) to be a pair (∂E, i) having the universal property of thenext theorem.

Theorem 4.1.5 (The boundary theorem for unital function spaces) Let (E, e)be a unital function space. Then there exist a compact Hausdorff space, written∂E, and an isometric unital map i : E → C(∂E) such that i(E) separates pointsof ∂E, with the following universal property: Given any function-extension (Ω, j)of E, there exists a (necessarily unique) topological embedding τ : ∂E → Ω, suchthat j(x) τ = i(x) for all x ∈ E.

4.1.6 (Remarks on the universal property) Before proving the theorem, we willmake five important observations that flow from the universal property in 4.1.5.First, note that any space (∂E, i) having this universal property, has the propertythat there is no proper closed subset Ω of ∂E such that i(·)|Ω is still an isometryon E. This may be seen by letting j = i(·)|Ω and appealing to the universalproperty. One obtains a map τ : ∂E → Ω, which is immediately seen to be theidentity map from the fact that i(E) separates points. Hence Ω = ∂E. Second,we remark that it is a simple exercise to see that a function-extension (Ω, j) of Eis E-equivalent to (∂E, i) if and only if (Ω, j) also has the universal property ofthe theorem. Thus the set of function-extensions having the universal propertyof the theorem, is one equivalence class of the relation we called E-equivalenceabove. Third, we point out that if (Ω, j) is a function-extension of E, and if τis the associated topological embedding τ : ∂E → Ω coming from the universalproperty in the theorem, then letting Ω′ = τ(∂E) and j′(x) = j(x)|Ω′ , we havethat (Ω′, j′) is E-equivalent to (∂E, i). Hence by the second remark, (Ω′, j′) maybe taken to be the Shilov boundary of E. Putting these three remarks togetherwe deduce our fourth remark, namely that the Shilov boundary of E may betaken to be any function-extension (Ω, j) of E with the property that there is noclosed subset Ω′ of Ω such that j(·)|Ω′ is still an isometry on E. Our fifth remarkis that nonetheless there is a canonical choice for the Shilov boundary (∂E, i)(that is, a canonical element of the equivalence class). Namely, we define ∂E tobe the closure of Ch(E) (see 4.1.3), the closure taken in E∗ with respect to thew∗-topology. Clearly ∂E is compact. Since E is unital, the canonical isometryfrom E to C(Ball(E∗)) restricts to an isometry from E to C(S(E)), hence to anisometry i : E → C(∂E) by the Krein–Milman theorem. The proof of Theorem4.1.5 will be complete if we can show that this choice of (∂E, i) has the universalproperty of the theorem.

4.1.7 (Proof of Theorem 4.1.5) Define ∂E as in the last (fifth) remark, andconsider (Ω, j) as in our hypothesis. By Lemma 4.1.2, any element in Ch(E) ‘is’the restriction to j(E) of an evaluation functional δω, for some unique ω ∈ Ω.Thus we have constructed a function τ : Ch(E) → Ω. The map σ taking ω to εω

is a topological embedding of Ω into S(j(E)) ∼= S(E). Indeed since Ω is compact,σ is a homeomorphism onto a closed subset. Also, σ τ is the identity map onCh(E), so that this closed subset contains Ch(E). Moreover, τ is a restriction

150 The Choquet boundary and boundary representations

of the homeomorphism σ−1. Taking closures, we see that a restriction of σ−1 isthe required embedding of ∂E into Ω.

4.1.8 (Representation on the maximal ideal space) For a uniform algebra A,consider the Gelfand representation, which is an isometric unital homomorphismof A into C(MA), where MA is the maximal ideal space (see A.4.2). Clearly Aseparates points of MA. By the third (or fifth) observation in 4.1.6, the Choquetand Shilov boundaries of A may be thought of as subspaces of MA.

4.1.9 (Examples)(1) We denote as usual by T the compact space of complex numbers with

modulus equal to one, and we write z = (z1, . . . , zn) for an element of Tn.

Let E = 1n+1, with its canonical basis (ek)n

k=0. Consider the linear functioni : E → C(Tn) which takes e0 to 1, and otherwise takes ek to the ‘kth coordinatefunction’ z → zk on T

n. This is an isometry such that i(E) separates pointsof T

n. However there is no proper closed subset Ω of Tn so that i(·)|Ω is still

isometric. Indeed if ω = (ωk) ∈ Tn \ Ω, we consider x = (1, ω1, . . . , ωn) ∈ E.

Then ‖x‖ = n + 1. But if |i(x)(z)| = n + 1 for some z ∈ Tn, then we must have

z = ω. Hence i(·)|Ω is not an isometry. By the fourth remark in 4.1.6, this showsthat ∂E = T

n.(2) Another important example is the disc algebra A(D) of 2.2.6. As we noted

there, we may regard A(D) ⊂ C(T) isometrically. However no proper compactsubset of T cannot ‘support A(D) isometrically’, since one may always find afunction in A(D) which peaks off this subset. Thus by the fourth remark in 4.1.6again, we have ∂A(D) = T.

(3) The Shilov boundary of the Hardy space H∞(D) (see 2.7.5 (4)) is themaximal ideal space of L∞(T). This is much more difficult to show (e.g. see [201]).We will see a noncommutative variant of this in 4.3.10.

4.1.10 (C∗-extensions) For a unital-subspace X of a unital C∗-algebra A, thenoncommutative analogue of ‘point separation’ is the assertion that X generatesA as a C∗-algebra. Indeed by the Stone–Weierstrass theorem, to say that a unital-subspace X ⊂ C(Ω) generates C(Ω) as a C∗-algebra, is the same as saying thatX separates points of Ω. Thus we define a C∗-extension of a unital operator spaceX (see 1.3.1) to be a pair (A, j) consisting of a unital C∗-algebra A, and a unitalcomplete isometry j : X → A, such that j(X) generates A as a C∗-algebra.

4.1.11 (Boundary representations) These are Arveson’s noncommutative gen-eralization of Choquet boundary points. If A is a C∗-algebra generated by aunital-subspace X , then a boundary representation of A for X is an irreducible∗-representation π : A → B(H) such that π is the unique completely positivemap A → B(H) extending π|X . This is the noncommutative analogue of Lemma4.1.2 (iii). For convenience below, we shall omit the requirement that boundaryrepresentations be irreducible.

One point about boundary representations, which generalizes what we saw in4.1.3, is that they only depend on the unital operator space structure of X , and


not on the particular C∗-algebra A containing X . In order to see this we recallthat a map v : X → B(K) is a dilation of u : X → B(H) if there is an isometryV : H → K such that u = V ∗v(·)V on X . This dilation is said to be reducingif v(X)V H ⊂ V H and v(X)∗V H ⊂ V H . We say that a completely contractiveunital map u : X → B(H) is in class BX if every completely contractive unitaldilation of u is reducing. We observe that BX is by definition ‘invariant underunital completely isometric isomorphism’. That is, if σ : X1 → X2 is a unitalcompletely isometric isomorphism between unital operator spaces, then u σ isin BX1 if and only if u is in BX2 .

Proposition 4.1.12 Let X be a unital-subspace of a unital C∗-algebra A, whichgenerates A as a C∗-algebra, and let u : X → B(H). Then u is the restriction toX of a boundary representation π : A → B(H), if and only if u ∈ BX .

Proof If u ∈ BX , then by the extension theorems 1.2.10, and 1.3.3, we mayextend u to a completely positive unital map u : A → B(H). By Stinespring’stheorem 1.3.4, we may write u = V ∗θ(·)V , for a ∗-representation θ : A → B(K)and an isometry V : H → K. Thus θ|X dilates u, and hence it is reducing.So V H reduces θ(A), forcing u to be a ∗-representation. Therefore u is uniquelydetermined by u. The above argument shows that u is a boundary representation.

Conversely, suppose that π : A → B(H) is a boundary representation, andthat v : X → B(K) dilates u = π|X . Thus there is an isometry V : H → K suchthat π = V ∗v(·)V on X . Since v is a completely contractive unital map, as beforewe may extend v to a completely positive map Ψ: A → B(K). Then V ∗Ψ(·)Vis completely positive on A, so that by hypothesis we have π = V ∗Ψ(·)V on A.Thus for any a ∈ X , the Kadison–Schwarz inequality 1.3.9 gives

π(aa∗) = V ∗Ψ(aa∗)V ≥ V ∗Ψ(a)Ψ(a∗)V ≥ V ∗Ψ(a)V V ∗Ψ(a∗)V = π(a)π(a)∗.

Since the last expression is again π(aa∗), the second inequality is an equality.This implies, as in the proof of 1.3.11, that V V ∗Ψ(a∗)V = Ψ(a∗)V . That is, V His invariant under Ψ(a∗). An identical argument with a∗a in place of aa∗ showsthat V H is invariant under Ψ(a), so that V H is reducing for Ψ(a) = v(a).

4.1.13 (Remarks on boundary representations)(1) Looking at the proof above yields the following. If A is a C∗-cover in

the sense of 2.1.1 of a unital operator algebra B, then a completely contrac-tive unital homomorphism θ : B → B(H) is the restriction to B of a boundaryrepresentation π : A → B(H), if and only if every completely contractive unitalhomomorphism dilating θ is reducing.

(2) From Proposition 4.1.12 and the remark before it, we see that a mapu : X → B(H) is in BX if and only if for any (or every) C∗-extension (A, j) ofX , the set of completely positive maps from A to B(H) extending u j−1 is asingleton, and that unique extension is a ∗-representation.

(3) One may construct the ‘noncommutative Shilov boundary’ using bound-ary representations. See the Notes on Section 4.1 for references. We will insteaduse the approach developed in the next few sections.

152 The injective envelope

4.2 THE INJECTIVE ENVELOPE

We now quickly describe Hamana and Ruan’s injective envelope of an operatorspace X . This is an injective operator space Z (see 1.2.9 and 1.2.11) containing (acompletely isometric copy of) X , satisfying one of several equivalent additionalproperties which we will spell out momentarily. The next page or so followsclosely the classical development of the (Banach space) injective envelope (e.g.see [237] and references therein).

4.2.1 (Projections and seminorms) Suppose that X is a subspace of an operatorspace W . An X-projection on W is a completely contractive idempotent mapΦ: W → W which restricts to the identity map on X . An X-seminorm on Wis a seminorm of the form p(·) = ‖u(·)‖, for a completely contractive linear mapu : W → W which restricts to the identity map on X .

Define a partial order ≤ on the set of all X-projections, by setting Φ ≤ Ψ ifΦ Ψ = Ψ Φ = Φ. We clearly have Φ ≤ Ψ if and only if Ran(Φ) ⊂ Ran(Ψ) andKer(Ψ) ⊂ Ker(Φ). In the following we use the usual ordering on seminorms.

Lemma 4.2.2 Let X be a subspace of an injective operator space W .(1) Any decreasing net of X-seminorms on W has a lower bound. Hence there

exists a minimal X-seminorm on W , by Zorn’s lemma. Each X-seminormmajorizes a minimal X-seminorm.

(2) If p is a minimal X-seminorm on W , and if p(·) = ‖u(·)‖, for a completelycontractive linear map on W which restricts to the identity map on X, thenu is a minimal X-projection.

Proof We sketch enough of the proof to enable the reader to complete it.Suppose that W ⊂ B(H), and that P is a completely contractive idempotentmap of B(H) onto W (see the proof of 1.2.11). Take a decreasing net (‖ut(·)‖)t asin (1). Since ut is completely contractive, and since CB(W, B(H)) is a dual space(see 1.6.1), a subnet of (ut) converges in the w∗-topology to a u ∈ CB(W, B(H)),say. It is easy to check from assertions in 1.6.1, that ‖u(·)‖ is a lower bound forthe given net. Hence so is the X-seminorm ‖P (u(·))‖ on W . The other assertionsof (1) are evident.

Suppose that p, u are as in (2). Claim: u is an idempotent map. Assumingthis claim, the desired results follow easily; since if Φ ≤ u in the ordering aboveon X-projections, then by minimality we must have ‖u(·)‖ = ‖Φ(·)‖. HenceKer(u) = Ker(Φ), and so Ran(Φ − I) ⊂ Ker(u). Thus u = u Φ = Φ.

To prove the Claim, we consider u as a map into B(H). Let u0 be a w∗-clusterpoint (as in the first paragraph) of the sequence (u(n))n of Cesaro averages of thesequence u, u u, u u u, . . .. We have ‖u(n)(·)‖ ≤ ‖u(·)‖ for any n ≥ 1, hence‖u0(·)‖ ≤ p. If P is as above, then ‖P (u0(·))‖ ≤ ‖u0(·)‖ ≤ p. By minimality,‖u0(·)‖ = p. Then for y ∈ W ,

‖u(y) − u2(y)‖ = p(y − u(y)) = ‖u0(y − u(y))‖ ≤ lim supn

‖u(n)(y − u(y))‖ = 0.

Thus u is idempotent.


4.2.3 (Extensions of operator spaces) An extension of an operator space X isan operator space Y , together with a linear completely isometric map i : X → Y .Often we suppress mention of i, and identify X with a subspace of Y . We saythat Y is a rigid extension of X if IY is the only linear completely contractivemap Y → Y which restricts to the identity map on i(X). We say Y is anessential extension of X if whenever u : Y → Z is a completely contractive mapinto another operator space Z such that u i is a complete isometry, then uis a complete isometry. We say that (Y, i) is an injective envelope of X if Y isinjective, and if there is no injective subspace of Y containing i(X).

Lemma 4.2.4 Let (Y, i) be an extension of an operator space X such that Y isinjective. The following are equivalent:

(i) Y is an injective envelope of X,(ii) Y is a rigid extension of X,(iii) Y is an essential extension of X.

Proof We assume (ii). Suppose that u : Y → Z is a complete contraction suchthat u restricted to i(X) is a complete isometry. Since Y is injective, the inverseof this restricted map may be extended to a complete contraction v : Z → Y .Clearly vu = IY when restricted to i(X), so that by rigidity vu = IY . Hence uis a complete isometry, which proves (iii).

We assume (iii). Suppose that i(X) ⊂ W ⊂ Y , with W injective. Extend IW

to a complete contraction Φ: Y → W . Then Φ is idempotent. By hypothesis Φis also one-to-one, which forces W = Y , and proves (i).

We assume (i). Then there can exist no nontrivial i(X)-projections on Y ,for the range of such a projection is clearly injective. Thus, if p = ‖u(·)‖ is anyminimal i(X)-seminorm on Y , then by 4.2.2 (2) we have that u = IY , so that pis the usual norm on Y . By 4.2.2, there exist minimal i(X)-seminorms, hence theusual norm is a minimal i(X)-seminorm. If v : Y → Y is a complete contractionextending Ii(X), then ‖v(·)‖ is an i(X)-seminorm on Y dominated by the usualnorm. Thus ‖v(·)‖ is a minimal i(X)-seminorm. Hence v = IY by the first linesof this paragraph, which shows (ii).

Lemma 4.2.5 If (Y1, i1) and (Y2, i2) are two injective envelopes of X, then thereexists a surjective complete isometry u : Y1 → Y2 such that u i1 = i2.

Proof Extend i2 i−11 and i1 i−1

2 to complete contractions u : Y1 → Y2 andv : Y2 → Y1 respectively. Since vui1 = i1, by the rigidity property we havevu = IY1 . Similarly, uv is the identity map. Thus u is a completely isometricsurjection by 1.2.7.

Theorem 4.2.6 If an operator space X is contained in an injective operatorspace W , then there is an injective envelope Y of X with X ⊂ Y ⊂ W .

Proof Let Y be the image of a minimal X-projection Φ on W (such a projectionexists by combining (1) and (2) of Lemma 4.2.2). We claim that Y is an injectiveenvelope of X . For if X ⊂ Z ⊂ Y with Z injective, then there is a completely

154 The injective envelope

contractive idempotent map Ψ of Y onto Z. Clearly Ψ Φ ≤ Φ. But Ψ Φ is anX-projection, and so by the minimality of Φ we must have Ψ Φ = Φ. HenceΨ = IY and Z = Y .

4.2.7 (The injective envelope) Combining 1.2.10 and 4.2.6, we see that everyoperator space has an injective envelope. We will often write it as (I(X), j), orI(X) for short. It is essentially unique by 4.2.5.

The following observations, which follow from the last proof and 1.3.13, willbe used frequently. Let B be B(H) (or an injective unital C∗-algebra). If X ⊂ B,then there is an injective envelope R of X with X ⊂ R ⊂ B, and a completelycontractive idempotent map Φ from B onto R. If, further, 1B ∈ X , then Φ iscompletely positive by 1.3.3, and R is a C∗-algebra with the new product Φ

defined bya Φ b = Φ(ab) a, b ∈ R. (4.2)

If further X is a subalgebra of B, then X is also a subalgebra of R. Indeed inthis case Φ(ab) = ab when a, b ∈ X .

If X is a unital operator space, and if X + X is the canonical operatorsystem generated by X (see 1.3.7), then every injective envelope of X + X isan injective envelope of X , and vice versa. Indeed suppose that B is an injectiveenvelope of X + X. By the last paragraph we may suppose that B is a unitalC∗-algebra. Also by the last paragraph there is an injective envelope R of Xwith X ⊂ R ⊂ B, and a unital completely positive projection Φ from B onto R.Clearly Φ is the identity on X + X, and so Φ = IB by rigidity. Hence R = B.The ‘vice versa’ is similar, but uses also 1.3.6. However we will not use this fact.

Corollary 4.2.8(1) If X is a unital operator space (resp. unital operator algebra, approximately

unital operator algebra), then there is an injective envelope (I(X), j) for X,such that I(X) is a unital C∗-algebra and j is a unital map (resp. j is aunital homomorphism, j is a homomorphism).

(2) If A is an approximately unital operator algebra, and if (Y, j) is an injectiveenvelope for A1, then (Y, j|A) is an injective envelope for A.

(3) If A is an approximately unital operator algebra which is injective, then Ais a unital C∗-algebra.

Proof The second paragraph of 4.2.7 proves (1) in the unital case. Assume nowthat A is an approximately unital nondegenerate subalgebra of B(H), and letR ⊂ B(H) and Φ: B(H) → R be as in 4.2.7. Thus R is an injective envelopefor A. If (et)t is a cai for A, then et → IH strongly. If ζ ∈ H with ‖ζ‖ = 1, letϕ(T ) = 〈Φ(T )ζ, ζ〉, for T ∈ B(H). Then ‖ϕ‖ ≤ 1 and ϕ(et) → 1. By (1) of 2.1.18,this implies that ϕ(IH) = 1. Hence Φ(IH) = IH , and A1 = A + CIH ⊂ R. Fromthis it is clear that R is also an injective envelope for A1, and (1) and (3) followat once. For (2), note that the proof in (1) shows that one injective envelope ofA is also an injective envelope of A1. A routine diagram chase using 4.2.5 showsthat any injective envelope of A1 is an injective envelope of A.


The following result is analoguous to a well-known result of Tomiyama [410]:

Corollary 4.2.9 If B is a unital-subalgebra of a unital operator algebra A, andif P : A → B is a completely contractive idempotent map onto B, then P is a‘conditional expectation’. That is, P is a B-bimodule map:

P (b1ab2) = b1P (a)b2, a ∈ A, b1, b2 ∈ B.

Proof Let P : A → B be the idempotent map. Let i : B → A be the inclusion.Let (I(A), J) and (I(B), j) be injective envelopes of A and B respectively, asin 4.2.8 (1). Thus these are unital C∗-algebras, and J, j are unital completelyisometric homomorphisms. By injectivity, we may extend j P J−1 to a com-pletely contractive unital map P : I(A) → I(B) with P J = j P . We may alsoextend J i j−1 to a completely contractive unital map i : I(B) → I(A), withi j = J i. Thus P i j = P J i = j P i = j. Hence P i is the identitymap on j(B), and so by the rigidity property of the injective envelope, P i isthe identity map on I(B). Thus Q = i P is a unital completely contractive (andhence completely positive) idempotent map on I(A). We have

Q(J(a)) = i(P (J(a))) = i(j(P (a)) = J(P (a)), (4.3)

for b ∈ B, a ∈ A, and thus

J(P (ba)) = Q(J(ba)) = Q(J(b)J(a)) = Q(J(b)Q(J(a))), (4.4)

the last step by (2) of 1.3.13. By (4.4), and (4.3) used twice, we deduce that

J(P (ba)) = Q(J(b)Q(J(a))) = Q(J(bP (a))) = J(P (bP (a))) = J(bP (a)).

Hence P (ba) = bP (a), and similarly P (ab) = P (a)b.

4.2.10 (Envelopes of matrix spaces) If Z is an injective operator space, then sois Mm,n(Z). This follows from 1.2.11, since if Z ⊂ B(H), and if P is a completelycontractive idempotent map of B(H) onto Z, then

Pm,n : Mm,n(B(H)) −→ Mm,n(Z)

is a completely contractive idempotent map of the injective operator spaceMm,n(B(H)) (which may be identified with B(Hn, Hm), by (1.2)) onto Mm,n(Z).

More generally, if X is an operator space, then Mm,n(I(X)) is an injectiveenvelope of Mm,n(X). A more general result may be found in 4.6.12, but theassertion may also be argued from what we have done already. (Hint: Use, re-cursively, the much easier special cases n = 2, m = 1; and n = 1, m = 2).

4.2.11 (The Banach space case) As we mentioned earlier, there is a parallelearlier theory of Banach space injective envelopes. This may be derived formallyfrom what we have done above, by replacing B(H) above by ∞I , and complete

156 The C∗-envelope

contractions and complete isometries by contractions and isometries. It is fairlyobvious, by (1.10), that for any Banach space E,

Min(IB(E)) = I(Min(E)) completely isometrically, (4.5)

where IB(E) is the Banach space injective envelope of E. It is well known thatthe injective Banach spaces are the C(K)-spaces for compact Stonean K (e.g.see [237] for this, and for other related theory and references).

4.3 THE C∗-ENVELOPE

In 4.1.10 we defined C∗-extensions of a unital operator space X . We say that twoC∗-extensions (B, i) and (B′, i′) are X-equivalent if there exists a ∗-isomorphismπ : B → B′ such that π i = i′. We define a C∗-envelope of X to be any C∗-extension (B, i) with the universal property of the next theorem.

Theorem 4.3.1 (Arveson–Hamana) If X is a unital operator space, then thereexists a C∗-extension (B, i) of X with the following universal property: Givenany C∗-extension (A, j) of X, there exists a (necessarily unique and surjective)∗-homomorphism π : A → B, such that π j = i.

4.3.2 (Remarks on the universal property) Before we begin the proof of theArveson–Hamana theorem, we will make a series of important but simple remarksconcerning the universal property of 4.3.1. First, if (B, i) is a C∗-extension ofX with this universal property, then there exists no nontrivial closed two sidedideal I of B such that q i is a complete isometry on X , where q : B → B/Iis the natural quotient map. This assertion follows by applying the universalproperty with j = q i. One obtains a ∗-homomorphism π : B/I → B withπ q i = i. Since i(X) is generating we see that π q = IB , which implies that qis one-to-one. Thus I = (0). The second remark is that the set of C∗-extensions(B, i) satisfying the universal property of the theorem, is one entire equivalenceclass of the relation of X-equivalence defined above 4.3.1. This is easily seen bya routine diagram chase. The third remark is that if (A, j) is any C∗-extensionof X , if π : A → B is the ∗-epimorphism provided by the universal property, andif I = Ker(π), then (A/I, q j) is clearly X-equivalent to (B, i) (from which itfollows that q j is completely isometric). Here q is the quotient map from A toA/I. Thus by the second remark, (A/I, qj) may be taken to be a C∗-envelope ofX . Putting these remarks together, we get our fourth and final remark, namelythat the C∗-envelope of X may be taken to be any C∗-extension (A, j) of X forwhich there exists no nontrivial closed two-sided ideal I of A such that q j iscompletely isometric on X , where q is the quotient map from A to A/I.

4.3.3 (Proof of Theorem 4.3.1) By 4.2.8, we may choose an injective envelope(I(X), i) of X , with I(X) a unital C∗-algebra and i a unital map. We defineC∗

e (X) to be the C∗-subalgebra of I(X) generated by i(X); then (C∗e (X), i) is a

C∗-extension of X .


Suppose now that (A, j) is any C∗-extension of X , and suppose that A isa unital ∗-subalgebra of B(H). Then j(X) ⊂ B(H), and by 4.2.7, there is acompletely positive idempotent map on B(H) whose range is an injective en-velope R of j(X), and R is a C∗-algebra with respect to a new product. Withrespect to the usual product on B(H), the C∗-subalgebra of B(H) generated byR contains A, the C∗-subalgebra of B(H) generated by j(X). We let B be theC∗-subalgebra (in the new product) of R generated by j(X). Thus by 1.3.13 (3),π = Φ|A is a ∗-homomorphism from A to R, with respect to the new product onR. Since π extends the identity map on j(X), it clearly also maps into B.

The final point is that the natural unital completely isometric surjectionR → I(X) guaranteed by 4.2.5, is a ∗-homomorphism by 1.3.10. Hence it is clearthat (B, j), as a C∗-extension of X , is X-equivalent to (C∗

e (X), i). Putting thesefacts together, we see that (C∗

e (X), i) has the announced universal property.

4.3.4 (C∗-envelopes and the Shilov boundary) We use the notation (C∗e (X), i)

for any C∗-envelope of a unital operator space X .Suppose that X is a uniform algebra, or more generally a unital function

space. From the universal property of the last theorem, C∗e (X) is a homomorphic

image of a commutative unital C∗-algebra, and hence C∗e (X) is commutative.

Thus C∗e (X) = C(Ω) for some compact space Ω. The universal property of the

Arveson–Hamana theorem, together with basics of the duality between compactspaces and commutative unital C∗-algebras (e.g. see A.5.4), translates to showthat Ω satisfies the universal property of Theorem 4.1.5. Hence in this case,C∗

e (X) coincides with the continuous functions on the usual Shilov boundary∂X of X . This suggests that in the general case of a unital operator space X ,the C∗-algebra C∗

e (X) should be regarded as a noncommutative Shilov boundary.We define the C∗-envelope of a nonunital operator algebra A to be a pair

(B, i), where B is the C∗-subalgebra generated by the copy i(A) of A inside aC∗-envelope (C∗

e (A1), i) of the unitization A1 of A (see 2.1.11).

Proposition 4.3.5 Let A be an operator algebra, and let (C∗e (A), i) be a C∗-

envelope of A. Then i is a homomorphism, and C∗e (A) has the following universal

property: Given any C∗-cover (B, j) of A, there exists a (necessarily unique andsurjective) ∗-homomorphism π : B → C∗

e (A) such that π j = i.

Proof The result reduces to 4.3.1 if A is unital. Assume that A is nonunital andlet (B, j) be a C∗-cover of A. Then B is nonunital as well, by 2.1.8, and j extendsby 2.1.15 to a completely isometric unital homomorphism j0 : A1 → B1, whoserange generates B1 as a C∗-algebra. Thus by the Arveson–Hamana theorem 4.3.1,there is a surjective ∗-homomorphism ρ : B1 → C∗

e (A1) such that ρj0 = i, wherei : A1 → C∗

e (A1) is the canonical embedding. Let π be ρ restricted to B, then πis a ∗-homomorphism with

π(j(a)) = ρ(j0(a)) = j(a) ∈ C∗e (A),

for all a ∈ A. Thus π maps B into C∗e (A), and the above shows that π j = i.

The last identity forces i to be a homomorphism.


4.3.6 (Properties of the C∗-envelope) If A is an approximately unital operatoralgebra, or unital operator space, then its C∗-envelope (C∗

e (A), i) is easily seento be both a rigid and an essential extension of A, in the sense of 4.2.3. One needonly to recall that C∗

e (A) is a subspace of the injective envelope (using 4.3.3, andalso 4.2.8 (2) in the operator algebra case), and that the injective envelope hasthese properties (see Lemma 4.2.4). For example, to see that C∗

e (A) is essential,we take a complete contraction u : C∗

e (A) → B(H), which restricts to a completeisometry on the copy of A. Extend u to a complete contraction I(A) → B(H)by 1.2.10, and then use the ‘essential’ property of I(A).

Although we will not use this, it follows from the last paragraph in 4.2.7that for a unital operator space X , C∗

e (X) and C∗e (X + X) coincide. Also, if

u : X → Y is a surjective unital linear complete isometry onto another unitaloperator space, then one may ‘extend’ u to a ∗-isomorphism between any C∗-envelopes (C∗

e (X1), j1) and (C∗e (X2), j2). Indeed a routine ‘diagram chase’ shows

that (C∗e (X2), j2 u) is a C∗-envelope for X1.

4.3.7 (Examples) In the rest of this section, we look at several examples of theC∗-envelope or noncommutative Shilov boundary.

(1) The first example to consider is T n, the upper triangular n×n matrices,sitting inside Mn. Since Mn is simple (i.e. has no nontrivial two-sided ideals), itfollows from the fourth remark in 4.3.2 that Mn is the C∗-envelope of T n.

(2) More generally, we consider the C∗-envelope of a unital subspace X ofMn. The C∗-subalgebra of Mn generated by X is finite-dimensional, and hence is∗-isomorphic to a finite direct sum B of full ‘matrix blocks’ Mnk

. Some of theseblocks are redundant. That is, if p is the central projection in B correspondingto the identity matrix of this block, then the map x → x(1B − p) is completelyisometric on X . If one eliminates such blocks, then the remaining direct sum ofblocks is the C∗-envelope of X .

The next example we consider is the ‘noncommutative version’ of Example4.1.9 (1). In the latter, we showed that the classical Shilov boundary of 1

n is the(n−1)-torus T

(n−1). Here we consider 1n with its natural operator space structure

(that is, its Max structure). Let C∗(Fn−1) be the full C∗-algebra of the free groupon n−1 generators u1, . . . , un−1, and consider the map i : 1

n → C∗(Fn−1), whichtakes e1 to u0 = 1 and ek to uk−1, for k = 2, . . . , n. We will show that i is the‘Shilov representation’ of Max(1

n).

Proposition 4.3.8

(1) Max(1n) is a unital operator space (with unit e1). Indeed the above map

i : Max(1n) → C∗(Fn−1) is a unital complete isometry, whose range gener-

ates C∗(Fn−1).(2) C∗

e (Max(1n)) = C∗(Fn−1).

Proof Note that uk : k ≤ n is a set of contractions, so that the map iabove is clearly a contraction. Therefore i is a complete contraction by (1.12).Conversely, suppose that Max(1

n) ⊂ B(H) completely isometrically, with ek


corresponding to contractions Tk. For each k, form the canonical ‘unitary di-lation’ Uk ∈ M2(B(H)), whose 1-1-corner is Tk. Let Vj−1 = UjU

−11 , these

are also unitary. By the universal property of C∗(Fn−1), there exists a unique∗-homomorphism π : C∗(Fn−1) → M2(B(H)) such that π(uk) = Vk for allk = 0, 1, . . . , n − 1. If Φ is the projection from M2(B(H)) onto its 1-1-corner,then Φ(π(·)U1) is a complete contraction extending i−1. Thus i−1 is a completecontraction, and i is a complete isometry. Clearly Ran(i) is generating.

To prove (2), we let (A, j) be a C∗-envelope of Max(1n). By 4.3.1, there exists

a surjective unital ∗-homomorphism π : C∗(Fn−1) → A such that π i = j. LetVk = π(uk) for k = 0, 1, . . . , n − 1. Thus the Vk are unitaries which generateA. Suppose that C∗(Fn−1) is represented as a unital ∗-subalgebra of B(H).By 1.2.10 we may extend the map i j−1 to a completely contractive mapu : A → B(H). Since u is unital, it is completely positive by 1.3.3. Note thatVk = π(i(ek+1)) = j(ek+1), so that u(Vk) = i(ek+1) = uk. Therefore

u(V ∗k Vk) = u(1) = IH = u∗

kuk = u(Vk)∗u(Vk),

and similarly u(VkV ∗k ) = u(Vk)u(Vk)∗. Thus by 1.3.11 we have that

u(Vka) = u(Vk)u(a) and u(V ∗k a) = u(Vk)∗u(a) = u(V ∗

k )u(a), a ∈ A.

Thus u is a ∗-homomorphism. Hence u maps into C∗(Fn−1), and is clearly aninverse for π. Thus π is one-to-one.

4.3.9 (Noncommutative Dirichlet and logmodular algebras) Suppose that A isa closed unital-subalgebra of a unital C∗-algebra B. We say that A is a Dirichletalgebra if A+A is norm dense in B. We say that A is left convexly approximatingin modulus if every positive b ∈ B is a uniform limit of terms of the form∑n

k=1 a∗kak for ak ∈ A. Here the n are varying too. The word ‘left’ here refers

to the preference of products a∗a as opposed to aa∗; thus the reader may guessthe meaning of right convexly approximating in modulus. We say that A hasfactorization (resp. is logmodular) if every element b ∈ B such that b ≥ ε1 forsome ε > 0, is of the form (resp. is a uniform limit of terms of the form) a∗a wherea ∈ A−1. A logmodular algebra is both left and right convexly approximating inmodulus. To see this, first approximate a positive b ∈ B with b + 1

n .To illustrate these notions, we recall that the disc algebra A(D) is a Dirich-

let subalgebra of C(T), and that any Dirichlet uniform algebra is logmodular(see [167]). The algebra of n × n upper triangular matrices in Mn is Dirichlet,and is known to have factorization (this is the Choleski factorization). Thus it islogmodular. The Hardy space H∞(D), or any of its usual uniform algebra gen-eralizations, or more generally still Arveson’s noncommutative H∞ spaces, areknown to have factorization, and are therefore logmodular. We will say a littlemore about these spaces in the Notes section for 4.3. For the reader’s conveniencewe will simply recall the definition: a noncommutative H∞ is a w∗-closed unital-subalgebra A of a von Neumann algebra B, such that B has a faithful normaltracial state τ , A+A is w∗-dense in B, and such that the unique faithful normal


conditional expectation Φ of B onto ∆(A) for which τ Φ = τ , is multiplicativeon A. A simple example is the upper triangular n × n matrices inside B = Mn.

Our point is simply that the examples above satisfy the hypotheses of thenext two results. Both results generalize useful facts well known for logmodularuniform algebras. The second result is a generalization of the important ‘unique-ness of representing measure’ for point evaluations on logmodular algebras.

Proposition 4.3.10 Suppose that A is a unital-subalgebra of a unital C∗-algebraB, which is either Dirichlet, or is left or right convexly approximating in modulus.Then B = C∗

e (A).

Proof In the case that A is Dirichlet, the canonical ∗-epimorphism from B toC∗

e (A) is by 1.3.6 an isometry, and is therefore one-to-one.Suppose that A is left convexly approximating in modulus (the ‘right’ case

is similar). We apply the fourth remark in 4.3.2. Assume that I ⊂ B is a closedtwo-sided ideal such that the canonical map Q : A → B/I factoring throughthe canonical maps A → B

qI→ B/I is a complete isometry. It suffices to checkthat I = (0). Fix b ∈ I with b ≥ 0. Then b is a limit of terms of the form∑n

k=1 a∗kak, for ak ∈ A. Since qI is a ∗-homomorphism, qI(b) is the limit of

terms∑n

k=1 qI(ak)∗qI(ak) =∑n

k=1 Q(ak)∗Q(ak). Also, ‖b‖ is a limit of terms‖∑n

k=1 a∗kak‖. However the last quantity is, by 1.2.5, the square of the norm of

the column [ak] in Cn(A). Since Q is a complete isometry, this norm coincideswith the square of the norm of the column [Q(ak)] in Cn(B/I). We have

‖b‖ = lim∥∥∥ n∑

k=1

a∗kak

∥∥∥ = lim∥∥∥ n∑

k=1

Q(ak)∗Q(ak)∥∥∥ = ‖qI(b)‖ = 0.

Thus b = 0, which implies that I = (0).

Theorem 4.3.11 Suppose that A is either a Dirichlet or logmodular unital-subalgebra of a unital C∗-algebra B.(1) Any unital completely contractive (resp. completely isometric) homomor-

phism π : A → B(H) has a unique extension to a completely positive andcompletely contractive (resp. and completely isometric) map from B intoB(H).

(2) Every ∗-representation of B is a boundary representation for A (see 4.1.11).

Proof Assume that π : A → B(H) is a unital completely contractive homomor-phism. The claim regarding the existence of a completely contractive, and hencecompletely positive, extension of π is clear, by 1.2.10 and 1.3.3. The completelyisometric statement is obtained by applying 4.3.10, and the fact that C∗

e (A) isan essential extension of A (see 4.3.6). The claim regarding uniqueness is clearin the Dirichlet algebra case, by 1.3.6.

Thus we now assume that A is logmodular and shall prove uniqueness. Wesuppose that Φ and Ψ are two completely positive extensions of π to all of B.Let ζ be a unit vector in H , and suppose that a, b ∈ A with ba = 1. Then


1 = 〈ζ, ζ〉 = 〈π(ba)ζ, ζ〉 = 〈Φ(b)Ψ(a)ζ, ζ〉 = 〈Ψ(a)ζ, Φ(b)∗ζ〉.

By the Cauchy–Schwarz inequality, we have 1 ≤ 〈Ψ(a)∗Ψ(a)ζ, ζ〉〈Φ(b)Φ(b)∗ζ, ζ〉.By the Kadison–Schwarz inequality (see 1.3.9), we have

1 ≤ 〈Ψ(a∗a)ζ, ζ〉〈Φ(bb∗)ζ, ζ〉.

Since A is logmodular, this yields

1 ≤ 〈Ψ(x)ζ, ζ〉〈Φ(x−1)ζ, ζ〉

for all strictly positive x ∈ B. Writing x = eh for h ∈ Bsa, we may then replaceh with th for real t, to obtain

1 ≤ 〈Ψ(eth)ζ, ζ〉〈Φ(e−th)ζ, ζ〉.

Let f(t) = g(t)k(t), where g(t) = 〈Ψ(eth)ζ, ζ〉 and k(t) = 〈Φ(e−th)ζ, ζ〉. By theabove, f has a local minimum at t = 0. Thus if f ′(0) exists, then f ′(0) = 0. Bythe undergraduate ‘product rule’, f ′(0) equals

limt→0

k(t)⟨Ψ(1t(eth−1)

)ζ, ζ

⟩+ g(t)

⟨Φ(1t(e−th−1)

)ζ, ζ

⟩= 〈Ψ(h)ζ, ζ〉−〈Φ(h)ζ, ζ〉.

Thus 〈Ψ(h)ζ, ζ〉 = 〈Φ(h)ζ, ζ〉. Hence Ψ(h) = Φ(h) for all selfadjoint h ∈ B. ThusΨ = Φ. This proves (1), and (2) follows from (1) and the definitions.

4.4 THE INJECTIVE ENVELOPE, THE TRIPLE ENVELOPE, ANDTROS

This section is a little more technical (and terse) than some of the others inthis chapter. The reader should feel free to skim through these results: they willbe used to give quick proofs of a few results, and then after this the injectiveenvelope will be rarely mentioned again.

4.4.1 (TROs and their morphisms) We recall that a ternary ring of operatorsor TRO is a closed linear subspace Z of B(K, H) (or of a C∗-algebra) satisfyingZZZ ⊂ Z. For x, y, z ∈ Z, we sometimes write xy∗z as [x, y, z], this is the tripleproduct on Z. The basic example of a TRO is pA(1− p), for a C∗-algebra A anda projection p in A (or in M(A)). A subtriple of a TRO Z is a closed subspaceY of Z satisfying Y Y Y ⊂ Y . A triple morphism between TROs is a linear mapwhich respects the triple product: thus T ([x, y, z]) = [Tx, T y, T z]. These are thenatural morphisms between TROs, and we note that they naturally arise whenrestricting a ∗-homomorphism from a C∗-algebra A to a subtriple of A. Howeverthere is another important class of maps from any TRO Z to itself. Recall fromExample 3.1.2 (6) that a TRO Z is a bimodule over the C∗-algebras ZZ andZZ. Thus we may consider, for example, the bounded right module maps on Z.We will not use this in this chapter, but by a result of Lin (see 8.1.16 (3)), these

162 The injective envelope, the triple envelope, and TROs

module maps are in a canonical bijective correspondence with the left multipliers(in the usual sense of 2.6) of the C∗-algebra ZZ. Indeed in Section 8.3 we willstudy TROs and their morphisms in much more detail.

Some authors substitute the word ‘ternary’ for ‘triple’ in the definitions above,because of the potential confusion with the JB∗-triple literature, where ‘triple’has a different connotation. We allow ourselves the use of the word ‘triple’ herebecause we shall not discuss JB∗-triples in this text, and hence there should belittle danger of confusion.

4.4.2 (An important construction) The following notation will appear fre-quently in the next several pages. We fix an operator space X ⊂ B(H). Theconstruction and results which follow are not altered essentially if we take X tobe a subspace of B(K, H), or of an injective C∗-algebra say, but for simplicitywe take B(H) here. Consider the Paulsen system S(X) ⊂ M2(B(H)) in theusual way (see 1.3.14). By 4.2.7, there is a completely positive idempotent mapΦ on M2(B(H)) whose range is an injective envelope I(S(X)) of S(X). Also,I(S(X)) is a unital C∗-algebra in a new product Φ (see (4.2)). Write p and qfor the canonical projections IH ⊕ 0 and 0⊕ IH . Since Φ(p) = p and Φ(q) = q, itfollows from 2.6.16 that Φ is ‘corner-preserving’, and that its ‘1-2-corner’ P is anidempotent map on B(H). By definition of the product Φ, it is clear that p and qare complementary projections in the C∗-algebra I(S(X)). With respect to theseprojections, I(S(X)) may be viewed as consisting of 2× 2 matrices, in the usualway (see 2.6.14). Let Ikl(X), or simply Ikl, denote its ‘k-l-corner’, for k, l = 1, 2.Thus I11 is the unital C∗-algebra pI(S(X))p, I22 is (1 − p)I(S(X))(1 − p), andI12 = pI(S(X))(1 − p) = Ran(P ). Note that by 1.3.12, Φ(pa) = pΦ(a) = pa forany a ∈ I(S(X)). Thus pa belongs to I(S(X)), as does ap by the same reason-ing. Also, multiplication by p or 1 − p in I(S(X)) is the same in the product ofM2(B(H)) or in the new product Φ.

We write J for the canonical map from X into I12(X); we shall see shortlythat (I12(X), J) is an injective envelope of X . We have the following diagram:

X → S(X) =[

C XX C

]→ I(S(X)) =

[I11(X) I(X) = I12(X)

I(X) = I21(X) I22(X)

].

We temporarily write Z for I12. According to Example 3.1.2 (5), we see thatthe ‘corner’ Z of I(S(X)) is an operator I11-I22-bimodule. Similarly, by 4.4.1, Zis a TRO with triple product [x, y, z] defined from the product of the C∗-algebraI(S(X)). With this product we have ZZZ ⊂ Z, ZZ ⊂ I11, and ZZ ⊂ I22.Note that in terms of the product in B(H), we have

[x, y, z] = P (xy∗z), x, y, z ∈ Z. (4.6)

This follows from the definition of the new product on I(S(X)) (as given by(4.2)), and the definition of P as the 1-2-corner of Φ.

Theorem 4.4.3 (Hamana–Ruan) If X is an operator space, let A be the in-jective C∗-algebra I(S(X)) considered above, and let p and 1 − p be the two


complementary diagonal projections in A mentioned above. Then pA(1−p) is aninjective envelope of X.

Proof We suppose that X ⊂ B(H), and we use the notation established above.By 1.2.11, it is easy to see that Z = pA(1 − p) is injective. Let v : Z → Z be acompletely contractive map extending the identity map on J(X). By 4.2.4, weneed to show that v = IZ . By Paulsen’s lemma 1.3.15, v gives rise to a canonicallyassociated map on S(Z), the latter viewed as a subset of A in the obvious way.Since A is injective, we may extend further to a complete contraction Ψ fromA to itself. Note that the restriction of Ψ to S(X) is the identity map. By therigidity property of A (see 4.2.4), both Ψ and v are the identity map.

Corollary 4.4.4 (Hamana–Ruan) An operator space X is injective if and onlyif X ∼= pA(1 − p) completely isometrically, for a projection p in an injectiveC∗-algebra A.

Proof If X is injective, then the 1-2-corner of the injective C∗-algebra I(S(X))above, is simply X itself. Indeed if (Y, i) is any injective envelope of an injectivespace X , then i is necessarily surjective. The converse is obvious.

4.4.5 (Remarks)(1) We saw in 4.4.2 that for any operator space X , the space Z = I12(X)

may be viewed as a TRO. If X happened originally to be a TRO inside B(K, H),and if one traces through the construction in 4.4.2, using B(H ⊕ K) in place ofM2(B(H)), one finds that the ‘triple product’ [x, y, z] in X coincides with its‘triple product’ in Z. This follows from (4.6).

(2) Although the C∗-algebras I11, I22, I(S(X)) in 4.4.2 seem to depend ona particular embedding X ⊂ B(H), in fact up to appropriate isomorphism theydo not. Indeed if u : X → B(K) is a complete isometry, and if Y = u(X),then it is easy to see that u ‘extends’ to a ∗-isomorphism θ between I(S(X)) andI(S(Y )) (for example, see the proof of 4.4.6 below). This ∗-isomorphism restrictsto a triple isomorphism between I12(X) and I12(Y ), and also to ∗-isomorphismsbetween I11(X) and I11(Y ), and between I22(X) and I22(Y ).

Corollary 4.4.6 (Hamana–Kirchberg–Ruan) A surjective complete isometrybetween TROs is a triple morphism.

Proof Such a surjective complete isometry u : X → Y gives rise, by 1.3.15, to acanonical complete order isomorphism between the operator systems S(X) andS(Y ). This isomorphism extends, by 4.2.5, to a completely isometric unital sur-jection θ between I(S(X)) and I(S(Y )). We assume that these latter C∗-algebrashave been chosen as in 4.4.2 and Remark 4.4.5 (1), so that the triple productsof X and Y coincide with those on I12(X) and I12(Y ) respectively. By 1.3.10, θis a ∗-isomorphism. Since θ(1 ⊕ 0) = 1 ⊕ 0, θ is ‘corner-preserving’ as in 2.6.16.Since the 1-2-corner θ12 of θ is the restriction of θ to a subtriple, it is a triplemorphism. However, θ12 agrees with u on the copy of X .


4.4.7 (TROs and the triple envelope) From 4.4.6, it follows that an operatorspace can have at most one ‘triple product’ with respect to which it is completelyisometrically ‘triple isomorphic’ to a TRO. From this fact and Theorem 4.4.3, itfollows that any injective envelope of an operator space X has a unique tripleproduct with respect to which ‘it is a TRO’. Moreover, given any two injectiveenvelopes, the map given by 4.2.5 is necessarily a triple morphism. In the sequelwe can therefore regard I(X) as a TRO without any ambiguity.

We now turn to the triple envelope T (X) of an operator space X . This is thegeneralization to nonunital operator spaces of the C∗-envelope or ‘noncommuta-tive Shilov boundary’ (see Section 4.3). In Section 8.3 we will revisit this topicin more detail, and we shall see there that the triple envelope has most of theproperties of the injective envelope. Also it has an additional universal propertythat is very useful, namely, T (X) is the smallest TRO or C∗-module containingX (see 8.3.9 for a precise statement). Since it is much smaller, it is often muchmore tractable than I(X). Most occurrences of the injective envelope in the nextcouple of sections could be replaced by the triple envelope, and in some placesthis would be preferable. We will not do this here however, mainly for pedagog-ical reasons, namely to avoid using basic facts about C∗-modules that will befully developed in Chapter 8. For now, we simply give one construction of T (X),and make some deductions.

We view I(X) as a TRO as above, and we define T (X) to be the smallestsubtriple of I(X) containing X . We claim that

T (X) = Spanx1x∗2x3x

∗4 · · ·x2n+1 : n ≥ 0, x1, x2, . . . , x2n+1 ∈ X. (4.7)

Note that the ‘odd products’ appearing in (4.7), are formed by repetitions ofthe ‘triple product’ of I(X). To prove (4.7), note that any subtriple of I(X)containing X must contain such ‘odd products’, and hence also the closure oftheir span. Conversely, if u, v, w are three such ‘odd products’, then uv∗w isanother ‘odd product’. Thus the right side of (4.7) is indeed a subtriple of I(X)containing X . By the discussion at the end of the second last paragraph, we seethat as a triple system, T (X) does not really depend on the particular injectiveenvelope I(X) chosen above.

If X is a C∗-algebra or TRO, then by Remark 4.4.5 (1) and (4.7), it followsthat X is its own triple envelope. On the other hand, if X is a unital operatorspace, then we saw in 4.2.8 that I(X) may be taken to be a unital C∗-algebra,containing X as a unital subspace. By adding ‘1s’ to the ‘odd products’ in (4.7)it is clear that T (X) is simply the C∗-subalgebra of I(X) generated by i(X). Bythe first few lines of 4.3.3 we conclude that C∗

e (X) is a triple envelope of X .

4.4.8 (Shilov inner product) Regard I(X) as the 1-2-corner of the C∗-algebraI(S(X)) as in 4.4.2, T (X) ⊂ I(X), and define B = T (X)T (X). By our conven-tion in 1.1.2, this is a closed set, indeed this is a C∗-subalgebra of I22(X), henceof I(S(X)). Then the triple envelope T (X) is a right operator B-module, andmoreover we may define a B-valued ‘inner product’ 〈y|z〉 = y∗z, for y, z ∈ T (X).


The ensuing map X ×X → T (X)T (X) ⊂ I22(X) is sometimes referred to as aShilov inner product on X .

Theorem 4.4.9 (Youngson) Let P be a completely contractive idempotent mapon a TRO Z. For x, y, z ∈ Z we have

P (P (x)P (y)∗P (z)) = P (xP (y)∗P (z)) = P (P (x)y∗P (z)) = P (P (x)P (y)∗z).

Furthermore Ran(P ) is completely isometrically isomorphic to a TRO.

Proof Suppose that Y is the range of P , and regard P as valued in Y . Leti : Y → Z be the inclusion. Using 1.3.15, we have associated unital completecontractions P and i between the Paulsen systems S(Z) and S(Y ), such thatP i = I on S(Y ). By definition of injectivity, we may extend these to completecontractions P ′ : I(S(Z)) → I(S(Y )) and i′ : I(S(Y )) → I(S(Z)). Since P ′ i′

restricted to S(Y ) is the identity map on S(Y ), by the rigidity property of theinjective envelope we have that P ′ i′ is the identity map on I(S(Y )). We willassume that I(S(Z)) has been written as a 2×2 matrix C∗-algebra as in (1), 4.4.2,and 4.4.3, so that the 1-2-corner of this C∗-algebra is an injective envelope I(Z)of Z, and so that the ‘triple product’ on Z agrees with the one on I(Z). DefineQ = i′ P ′, this is a completely positive unital idempotent map on I(S(Z)).Also, all of our maps fix the diagonal projections p = 1 ⊕ 0 and q = 0 ⊕ 1, andare therefore ‘corner-preserving’ as in 2.6.16. Let c be the canonical map takingZ into the 1-2-corner of I(S(Z)). We have

Q(c(z)) = i′(P ′(c(z))) = i′(P (c(z))) = c(i(P (z))), z ∈ Z. (4.8)

By 1.3.13, the range of Q is a C∗-algebra with product given by

Q(Q(x)Q(y)) = Q(Q(x)y) = Q(xQ(y)) (4.9)

for x, y ∈ I(S(Z)). The products here are the product of the C∗-algebra I(S(Z)).As in the discussion in 4.4.2, the projections p, q allow us to write Ran(Q) as a2 × 2 matrix C∗-algebra, Q being corner-preserving. By the reasoning in 4.4.2,the 1-2-corner W of this C∗-algebra may be regarded as a TRO, with ‘tripleproduct’ given by the formula [x, y, z] = u(xy∗z), for x, y, z ∈ W , where u is the1-2-corner of Q. By (4.8), the restriction of u to the copy of Z in the 1-2-corner,is i P . Thus it follows that this triple product, restricted to the copy of Y , issimply P (xy∗z), for x, y, z ∈ Y . Thus Y is completely isometrically isomorphicto a subtriple of W .

Using (4.9), it is easy to see that Q(Q(x)Q(y)Q(z)) equals

Q(Q(Q(x)Q(y))Q(z)) = Q(Q(xQ(y))Q(z)) = Q(xQ(y)Q(z)),

for x, y, z ∈ I(S(Z)). Similarly,

Q(Q(x)Q(y)Q(z)) = Q(Q(x)yQ(z)) = Q(Q(x)Q(y)z).

The desired formulae follow from these last relations upon choosing the x, y∗, zfrom the copy of Z inside I(S(Z)), and using equation (4.8).


Corollary 4.4.10 Suppose that P is a completely contractive idempotent mapfrom a TRO Z onto a subspace Y for which Y Y Y ⊂ Y . Then P is a ‘conditionalexpectation’ in the sense that

P (xy∗z) = xy∗P (z), P (xz∗y) = xP (z)∗y, P (zy∗x) = P (z)y∗x

for all x, y ∈ Y and z ∈ Z.

4.4.11 (An essential ideal) We define two C∗-algebras C = C(X) = I(X)I(X)

and D = D(X) = I(X)I(X), in the notation in 4.4.2. The products here arein the C∗-algebra I(S(X)), as in 4.4.8. These are C∗-subalgebras of I11(X) andI22(X) respectively. Facts in the last paragraph of 4.4.2 show that the subset

L =[C(X) I(X)I(X) D(X)

]of the C∗-algebra I(S(X)), is an ideal. In fact L, is an essential ideal (see 2.6.10).To see this, suppose that K is an ideal in I(S(X)) with K ∩ L = (0). Letπ : I(S(X)) → I(S(X))/K be the canonical ∗-epimorphism. Since π|L is one-to-one, it is completely isometric by 1.2.4. By 2.6.15, it is clear that π maps each ofthe four corners into a matching corner of I(S(X))/K. Let πij be the i-j-cornermap of π. Since π is a unital ∗-homomorphism, we have π21(z) = π12(z∗)∗, andπ11 and π22 are unital maps between the diagonal corners. Let Φ (resp. Φij) bethe restriction of π (resp. πij) to S(X) (resp. to the appropriate corner of S(X)).Then Φii is the identity map of C, and Φ12 = Φ

21 is completely isometric (sinceπ|L is). Thus by 1.3.15, Φ is a complete isometry. Since I(S(X)) is an essentialextension of S(X), π is a complete isometry as well, and hence K = (0). HenceL is an essential ideal by 2.6.10.

Proposition 4.4.12 Let X be an operator space, and suppose that a ∈ I11(X),or that a is a right module map on the triple envelope T (X) of X (with respectto the module action discussed in 4.4.8). If ax = 0 for all x ∈ X, then a = 0.

Proof If a is a right module map on T (X), and if a(X) = 0, then we clearlyhave a(x1x

∗2x3x

∗4 · · ·x2n+1) = a(x1)x∗

2x3x∗4 · · ·x2n+1 = 0, for x1, x2, . . . , x2n+1 in

X . Thus a = 0, by (4.7).For the other assertion, we may assume that ‖a‖ ≤ 1. By replacing a by a∗a,

we may also suppose that 0 ≤ a ≤ 1, where this last ‘1’ is the identity of I11.Define u(z) = (1 − a)z, for z ∈ I(X). Since u(x) = x for x ∈ X , we see that u isthe identity map, by rigidity. Thus aI(X) = 0, and so (a ⊕ 0)L = 0, where L isas in 4.4.11. By the last line of 4.4.11, and the first definition in 2.6.10, it followsthat a ⊕ 0 = 0, so a = 0.

Proposition 4.4.13 Suppose that X is an operator space with an injective en-velope which is a C∗-algebra B. Then M2(B) is an injective envelope for thePaulsen system S(X). Moreover, we may take I11(X) = I22(X) = B.


Proof By 4.2.8 (3), B is unital, with identity 1 say. The C∗-algebra M2(B) isinjective by 4.2.10, and has as subsystems

S(X) ⊂ S(B) ⊂ M2(B),

where we have identified the diagonal idempotents in S(X) and S(B) with 1.To see the first assertion, it suffices to show that the identity map is the onlyS(X)-projection Ψ: M2(B) → M2(B). Such a Ψ fixes the diagonal idempotentsof S(X), hence is ‘corner-preserving’: indeed as in 2.6.16, we may write

Ψ =[

v1 uu v2

].

Now u|X = IX , so by the rigidity property of the injective envelope, u = IB .Thus Ψ is a S(B)-projection. Now one sees that Ψ fixes 1⊗M2, which is a unitalC∗-algebra. Hence, by 1.3.12, Ψ is a M2-bimodule map. Thus v1 = v2 = u = IB ,and so Ψ is the identity map.

The last assertion is clear from the variant of the construction in 4.4.2 withB(H) replaced by B (if necessary use also Remark 4.4.5 (2)).

4.5 THE MULTIPLIER ALGEBRA OF AN OPERATOR SPACE

We concentrate on ‘left multipliers’ in this section; the ‘right-handed’ variant isleft to the reader, and the associated ‘left adjointable multipliers’ will be morefully developed in Section 8.4. We will also begin to see the applications of thesefruitful concepts. The left multiplier algebra Ml(X) of an operator space X ,(resp. the left adjointable multiplier algebra Al(X)) turns out to be an abstractoperator algebra (resp. C∗-algebra) whose elements are maps on X . To illustratethe versatility of this construction, we point out:(1) If A is an approximately unital operator algebra, then Ml(A) is the usual

left multiplier algebra LM(A) considered in Section 2.6. If A is a C∗-algebra,then Al(A) = M(A).

(2) If E is a Banach space, then Ml(Min(E)) coincides with the classical func-tion multiplier algebra M(E) considered in Section 3.7, whereas Al(Min(E))is the classical ‘centralizer algebra’ considered there.

(3) If Z is a right Hilbert C∗-module, then we shall show in 8.4.2 that Ml(Z)and Al(Z) are respectively the algebras of bounded right module maps, andadjointable maps, on Z.

Thus although Ml(X) and Al(X) are defined purely in terms of the matrixnorms and vector space structure on X , they often encode ‘operator algebrastructure’. Operator spaces X with trivial (one-dimensional) multiplier algebrasare exactly the spaces lacking ‘operator algebraic structure’ in a sense which onecan make precise.

4.5.1 (The left multiplier algebra of an operator space) For an operator spaceX , we define a left multiplier of X to be a linear map u : X → X such that

168 The multiplier algebra of an operator space

there exists a Hilbert space H , an S ∈ B(H), and a linear complete isometryσ : X → B(H) with σ(ux) = Sσ(x) for all x ∈ X . We define the multipliernorm of u, to be the infimum of ‖S‖ over all possible H, S, σ as above. We defineMl(X) to be the set of left multipliers of X . Notice that we may replace theB(H) in the definition of Ml(X) by an arbitrary C∗-algebra, without any realchange in meaning. Or, we may replace B(H) by B(K, H), where K is anotherHilbert space. We leave it to the reader to check that this allowance does notenlarge the set Ml(X) or change the multiplier norm.

The ‘multiplier norm’ clearly satisfies all the properties of a norm, exceptfor the triangle inequality. In fact from the next theorem, the triangle inequalityproperty is easily seen (using the fact that the set of maps u characterized inthat theorem is clearly convex). Indeed from this theorem it is easily seen thatMl(X) is a normed algebra, and a unital subalgebra of CB(X), but with apossibly larger norm. These last facts will be reproved later in the proof of 4.5.5too. Notice also that if we fix such σ : X → B(K, H), and define Mσ

l (X) to bethe set of S ∈ B(K) such that Sσ(X) ⊂ σ(X), then Mσ

l (X) is an operatoralgebra. Although we shall not use this explicitly, later we shall see that there isa ‘universal embedding’ σ0 such that Ml(X) = Mσ0

l (X). Indeed this also followsfrom Theorem 4.5.5 and its proof.

For any linear map u : X → X , we define τu : C2(X) → C2(X) to be the mapu ⊕ IX ; that is:

τu

([xy

])=[

u(x)y

], x, y ∈ X.

‘Left multipliers’ are best viewed as a sequence of equivalent definitions, asin the following theorem. To explain the notation in this result: in (iii)–(v), weare viewing X ⊂ T (X) ⊂ I(X) ⊂ I(S(X)) as in 4.4.2 and 4.4.8. Here T (X) isthe triple envelope (see 4.4.7). Thus I11(X) is the C∗-algebra defined in 4.4.2,and the inner product in (v) is the one mentioned in 4.4.8. The matrices in (v)are indexed on rows by i, and on columns by j. Thus the inequality in (v) takesplace in the C∗-algebra Mn(B), where B = T (X)T (X) (see 4.4.8).

Theorem 4.5.2 Let u : X → X be a linear map. The following are equivalent:

(i) u is a left multiplier of X with multiplier norm ≤ 1.(ii) τu is completely contractive.(iii) There exists a (necessarily unique, by 4.4.12) a ∈ I11(X) of norm ≤ 1, such

that u(x) = ax for all x ∈ X.(iv) u is the restriction to X of a (necessarily unique, by 4.4.12) contractive

right module map a on T (X) with a(X) ⊂ X.(v) [〈u(xi)|u(xj)〉] ≤ [〈xi|xj〉], for all n ∈ N and x1, . . . , xn ∈ X.

The infimum in the definition of the ‘multiplier norm’ (see 4.5.1) is achieved.

Proof Assume (i) and let σ, S, H be chosen as in the definition of Ml(X) above.Then for x, y ∈ X , we have

Some ‘extremal theory’ 169∣∣∣∣∣∣∣∣τu

([xy

])∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣[σ(ux)

σ(y)

]∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣[S 0

0 I

] [σ(x)σ(y)

]∣∣∣∣∣∣∣∣ ≤ max‖S‖, 1∣∣∣∣∣∣∣∣[x

y

]∣∣∣∣∣∣∣∣ .Thus ‖τu‖ ≤ max‖S‖, 1. A similar argument shows that ‖τu‖cb ≤ max‖S‖, 1.Then (ii) follows by the definition of the ‘multiplier norm’.

We now prove that (ii) implies (iii). We give Paulsen’s short proof of thisresult found by Blecher, Effros, and Zarikian [57]. Another short proof will besketched in 8.4.5. Using notation in 4.4.2, let

S =

C1 0 X0 C1 X

X X C1

⊂

I11 I11 I(X)I11 I11 I(X)

I(X) I(X) I22

.

To explain this notation, if we let M be the last space, then M should be viewedas a corner, and ∗-subalgebra, of M2(I(S(X))) in the obvious way. Thus M isan injective C∗-algebra. We assume that ‖τu‖cb ≤ 1 and define a map on S by

Φ′ :

λ 0 x1

0 µ x2

y∗1 y∗

2 ν

−→

λ 0 u(x1)0 µ x2

u(y1)∗ y∗2 ν

.

Let Dn be the diagonal ∗-subalgebra of Mn. Since τu is a completely contrac-tive left D2-module map on C2(X), Φ′ is completely positive, by the discussionin 3.6.1. By Arveson’s extension theorem 1.3.5, we may extend this map to acompletely positive map Φ: M → M . Since Φ fixes the diagonal D3, fact 2.6.17tells us that Φ is ‘corner-preserving’. Write Φij for the ‘i-j-corner’ of Φ, herei, j = 1, 2, 3. View the lower 2 × 2 corner of Φ as a map Ψ on I(S(X)). Therestriction of Ψ to S(X) is the identity map, thus by rigidity Ψ is the identitymap on I(S(X)). Hence Φ fixes the C∗-subalgebraC 0 0

0 I11 I(X)0 I(X) I22

of M . By 1.3.12, Φ is a bimodule map over this subalgebra. Thus Φ(wz) = Φ(w)z,where w is the matrix with a 1 in the 1-2-corner and zeroes elsewhere, and z isthe matrix with an element from X in the 2-3-corner and zeroes elsewhere. Weimmediately obtain (iii) by inspection of the relation Φ(wz) = Φ(w)z, and thefact that Φ extends Φ′. The desired element a in I11(X), is Φ12(1).

That (iii) implies (i) is clear from the definitions in 4.5.1, if we regard X as asubspace of the C∗-algebra I(S(X)), as discussed in 4.4.2, and taking S in 4.5.1to be a ⊕ 0 ∈ I(S(X)). This also shows the final assertion of the theorem. Wedo not need this, but (iv) implies (i) easily via the theory of C∗-modules.

In the rest of the proof we set Y = T (X), viewed as a subtriple of I(X),with X identified as a subspace of Y . We let B be the C∗-algebra Y Y , andwe denote the canonical B-valued inner product in (v) as y∗z, for y, z ∈ Y . We


recall that products like these, and the ones met below, make sense and may beinterpreted, for example, as products in the C∗-algebra I(S(X)).

To show that (iii) implies (iv), let a be as in (iii). Using (4.7) we clearly have

aXXXX · · ·X = u(X)XXX · · ·X ⊂ Y,

and aY ⊂ Y . Also, axy∗z = (ax)(y∗z) for x, y, z ∈ Y . Thus the map z → az isa contractive right B-module map on Y , whose restriction to X is u.

If (iv) holds, then a well-known result of Paschke implies that

a(z)∗a(z) ≤ z∗z, z ∈ Y. (4.10)

We sketch the quick argument for (4.10) here. Set bn = (z∗z + 1n )−

12 , the inverse

taken in the unitization of B (which in this case may be thought of as the span ofB and the identity of I22(X)). Then (zbn)∗zbn = bnz∗zbn ≤ 1 (by basic spectraltheory for the selfadjoint z∗z). Thus ‖zbn‖ ≤ 1, so that ‖a(zbn)‖ ≤ 1. We deducethat a(zbn)∗a(zbn) ≤ 1. However a is a B1-module map clearly, and it followsthat a(z)∗a(z) ≤ z∗z + 1

n . Letting n → ∞ yields (4.10).For b1, . . . , bn ∈ B and x1, . . . , xn ∈ X , set z =

∑i xibi. By (4.10),∑

i,j

b∗i u(xi)∗u(xj)bj = a(z)∗a(z) ≤ z∗z =∑i,j

b∗i x∗i xjbj.

A well-known positivity criterion (see [407, IV.3.2] or 8.1.12) yields (v).Let [xij ], [yij ] ∈ Mn(X), and let k ≤ n. If (v) holds, then we have

[u(xki)∗ u(xkj) + y∗ki ykj ] ≤ [x∗

ki xkj + y∗ki ykj ] .

Summing over k, and taking the norm, we obtain∥∥∥[∑k

u(xki)∗ u(xkj) + y∗ki ykj

]∥∥∥ ≤∥∥∥[∑

k

x∗ki xkj + y∗

ki ykj

]∥∥∥ =∣∣∣∣∣∣∣∣[xij

yij

]∣∣∣∣∣∣∣∣ .This is equivalent to saying that ‖τu‖cb ≤ 1. Hence (v) implies (ii).

4.5.3 (Matrix norms on Ml(X)) For u = [uij ] ∈ Mn(Ml(X)), write Lu forthe operator on Cn(X) given by the usual formula for the product of a matrixand a column. This is a left multiplier on Cn(X) (for if aij is associated to uij

as in Theorem 4.5.2 (iii), then a variant of the proof that (iii) implies (i) in 4.5.2shows that Lu ∈ Ml(Cn(X))). We define ‘multiplier matrix norms’ on Ml(X),by assigning to u = [uij ] above, the multiplier norm of Lu (that is, its ‘norm’ inMl(Cn(X))). In the proof of 4.5.5, we show that these quantities are well defined,and are norms defining an operator algebra structure on Ml(X). Moreover

Mn(Ml(X)) ∼= Ml(Cn(X)) as operator algebras, (4.11)

via the map u → Lu above. Indeed, note that if (4.11) holds isometrically for ev-ery n ∈ N, then it holds completely isometrically. This may be seen by composingthe isometric isomorphisms

Mm(Mn(Ml(X))) ∼= Mmn(Ml(X)) ∼= Ml(Cmn(X)) ∼= Mm(Ml(Cn(X))).


4.5.4 (A matricial generalization of Theorem 4.5.2) Theorem 4.5.2 has a variantcharacterizing ‘matrices of multipliers’. For example, if u : Cn(X) → Cn(X) is alinear map, then the following are equivalent:(i) u is a left multiplier of Cn(X) with multiplier norm ≤ 1,(ii) u ⊕ IX : Cn+1(X) → Cn+1(X) is completely contractive,(iii) there exists an a ∈ Ball(Mn(I11(X))) such that u(x) = ax for all x ∈ Cn(X).To prove this, one may mimic the proof of 4.5.2. We give further hints for thedifficult implication that (ii) implies (iii) here; one follows the proof above closely,replacing the space M in that proof by the following space: Mn(I11) Cn(I11) Cn(I(X))

Rn(I11) I11 I(X)Rn(I(X)) I(X) I22

,

which may be regarded as a unital ∗-subalgebra of M2n(I(S(X))). The last fewsteps are modified by considering Φ applied to the sum over j of products of amatrix with an ej in the 1-2-corner and zeroes elsewhere, and the matrix withan element xj from X in the 2-3-corner and zeroes elsewhere. Here (ej) is thecanonical basis of Cn. The desired element a is the matrix with jth columnΦ12(ej). Hence

‖a‖ ≤ ‖Φ12‖cb ‖[e1 · · · en]‖ ≤ 1.

Theorem 4.5.5 If X is an operator space, then the ‘multiplier norms’ definedin 4.5.1 and 4.5.3 are norms. With these matrix norms Ml(X) is an operatoralgebra, and X is a left operator Ml(X)-module. Moreover, the inclusion mapfrom Ml(X) to CB(X) is a one-to-one completely contractive homomorphism.

Proof Again we use notation from 4.4.2, as well as the completely isometricembeddings X ⊂ I(X) ⊂ I(S(X)). We saw there that I(X) is a left operatorI11(X)-module. We define

IMl(X) = a ∈ I11(X) : aX ⊂ X.

This is clearly an operator algebra, a subalgebra of the C∗-algebra I11(X). More-over, X is a left operator IMl(X)-module (as can be seen by using 3.1.2 (2) and3.1.12, if necessary).

The equivalence of (i) and (iii) in 4.5.2, shows that Ml(X) is a subalgebraof CB(X), and yields an isometric isomorphism θ : IMl(X) → Ml(X). Namelyθ(a)(x) = ax for any x ∈ X . Indeed θ is a complete isometry, using 4.5.4.Therefore, the ‘multiplier matrix norms’ are norms, and indeed

Ml(X) ∼= IMl(X) as operator algebras. (4.12)

Since every operator module is a matrix normed module (see 3.1.5 (1)), thecanonical embedding from Ml(X) to CB(X) is completely contractive, and it isone-to-one by 4.4.12.


4.5.6 (Right multipliers) There are definitions and results analoguous to theabove, but with ‘left’ replaced by ‘right’. Thus we may regard Mr(X) ⊂ CB(X).The latter inclusion is as sets (not necessarily isometrically), and we need to putthe reverse of the usual composition multiplication on this subset of CB(X). Thereader can see the reason for this by considering, for example, the ‘right-handversion’ of (4.12), namely that Mr(X) ∼= b ∈ I22(X) : Xb ⊂ X.

From these descriptions it is clear that as operators on X , any v ∈ Mr(X)commutes with every u ∈ Ml(X).

4.5.7 (Adjointable multipliers) We write Al(X) for the C∗-algebra ∆(Ml(X))(see 2.1.2 for this notation). Similarly, Ar(X) = ∆(Mr(X)). The operators inAl(X) are called left adjointable multipliers, for reasons that will be clear inSection 8.4, where we will give a more thorough development of Al(X), and itslinks to C∗-modules. For now we give just one quick result about Al(X):

Proposition 4.5.8 Let X be an operator space.(1) A linear map u : X → X is in Al(X) if and only if there exist a Hilbert

space H, an S ∈ B(H), and a linear complete isometry σ : X → B(H) withσ(ux) = Sσ(x) for all x ∈ X, and such that also S∗σ(X) ⊂ σ(X).

(2) Al(X) ∼= a ∈ I11(X) : aX ⊂ X and a∗X ⊂ X as C∗-algebras.(3) If u, S and σ are as in (1), then the involution u∗ in Al(X) is the map

x → σ−1(S∗σ(x)) on X.(4) The canonical inclusion map from Al(X) into CB(X) (or into B(X)) is

an isometric homomorphism.

Proof Assertion (2) is clear from (4.12), since the algebras in (2) are the diag-onal C∗-algebras of the algebras in (4.12). Thus if u ∈ Al(X), then we have acorresponding a in the set on the right in (2) such that u(x) = ax for all x ∈ X .If we regard I(S(X)) as a C∗-subalgebra of some B(H), we see that the requiredcondition in (1) holds.

Conversely, suppose that S, H, σ are as in (1). If Mσl (X) is as in 4.5.1, then

the restriction to ∆(Mσl (X)) of the canonical contractive homomorphism from

Mσl (X) to Ml(X), is a ∗-homomorphism into Al(X), by the last paragraph of

2.1.2. Since S ∈ ∆(Mσl (X)), u ∈ Al(X). This completes the proof of (1). Note

that S∗ ∈ ∆(Mσl (X)) is taken by this ∗-homomorphism to u∗ ∈ Al(X). This

gives (3). Finally, (4) follows from the above and from A.5.9.

4.5.9 (Restrictions of multipliers) It follows immediately from the definitions4.5.1, that if u ∈ Ml(X), and if Z is a closed subspace of X such that u(Z) ⊂ Z,then u|Z ∈ Ml(Z). Moreover, the multiplier norm of u|Z is smaller than themultiplier norm of u. A similar argument using 4.5.8 shows that if u ∈ Al(X),and if u(Z) ⊂ Z and u∗(Z) ⊂ Z, then u|Z ∈ Al(Z).

4.5.10 (The Banach space case) If E is a Banach space, then

M(E) = Ml(Min(E)) and Z(E) = Al(Min(E)) (4.13)

(see 3.7.1 for this notation). We will prove this momentarily.


If X is any operator space with an injective envelope (B, j) such that B is aunital C∗-algebra, then putting (4.12) together with 4.4.13 yields the fact thatMl(X) ∼= a ∈ B : aj(X) ⊂ j(X) as operator algebras. Using this fact, togetherwith the fact that I(Min(E)) is a commutative C∗-algebra C(Ω) (see 4.2.11), weconclude that Ml(Min(E)) ∼= a ∈ C(Ω) : aj(E) ⊂ j(E). This shows thatMl(Min(E)) is a minimal operator space, and that the inclusion of Ml(Min(E))in M(E) is a contraction (by the implication (iv) ⇒ (i) in 3.7.2). Conversely, by3.7.2 again and definition 4.5.1, we also have a contractive inclusion of M(E)in Ml(Min(E)). Thus Ml(Min(E)) = M(E). Also, Al(Min(E)) = Z(E), sincethese are the ‘diagonals’ of the previous two equal algebras.

Proposition 4.5.11 Let A be an approximately unital operator algebra. ThenMl(A) = CBA(A) = LM(A), and Mr(A) = RM(A), as operator algebras.

Proof Identify LM(A) = CBA(A) (see Section 2.6). Let λ : LM(A) → CB(A)be the canonical completely isometric embedding. From the characterization ofLM(A) in 2.6.2 (2), and the definition in 4.5.1, we see that λ(u) ∈ Ball(Ml(A)),for all u ∈ Ball(LM(A)). Thus λ, viewed as a map into Ml(A), is contractive. Asimilar argument with matrices, using Corollary 2.6.6, shows that λ is completelycontractive. Since the inclusion Ml(A) ⊂ CB(A) is a complete contraction, by4.5.5, we see that λ : LM(A) → Ml(A) is a complete isometry. Similarly, we havea completely isometric homomorphism ρ : RM(A) → Mr(A). If u ∈ Ml(A), andb ∈ A, then using 4.5.6 we see that u(ab) = u(ρ(b)a) = ρ(b)(u(a)) = u(a)b, sothat u ∈ CBA(A). Thus λ is surjective.

Proposition 4.5.12 If ν : X → Y is a linear surjective complete isometry be-tween operator spaces, then the map taking u to νuν−1 is a completely isometricisomorphism from Ml(X) to Ml(Y ).

Proof The isometry follows from the definition in 4.5.1: if u = σ−1(Sσ(·)),for σ and S as in 4.5.1, then ν(u(ν−1(y))) = (σ ν−1)−1(S(σ ν−1)(y)). Thus‖νuν−1‖Ml(Y ) ≤ ‖u‖Ml(X), and the other direction follows by symmetry. Thecomplete isometry is similar, and is left as an exercise using (4.11).

The last result is an almost tautological ‘Banach–Stone type’ result for oper-ator spaces. In fact many theorems of this type may be deduced from it. As anexample, we give the following ‘Banach–Stone theorem’ for operator algebras:

Theorem 4.5.13 Let v : A → B be a completely isometric linear surjectionbetween approximately unital operator algebras. Then there exists a completelyisometric homomorphism θ from A onto B, and a unitary U with U and U−1 inM(B), such that v = θ(·)U .

Proof If A and B are unital, then LM(A) = A and LM(B) = B. Henceby 4.5.11 and 4.5.12, there is a completely isometric isomorphism θ : A → Bsuch that vλ(a)v−1 = λ(θ(a)), for any a ∈ A. Here λ denotes the left regularrepresentation either on A or B. This gives v(a) = θ(a)v(1A). Hence if we setU = v(1A), then there exists an a0 ∈ Ball(A) with 1B = v(a0) = θ(a0)U , so that


U∗ = θ(a0) ∈ B, by an early fact in A.1.1. By 2.1.3, v(1A) is an isometry, andit lies in ∆(B). By symmetry v(1A) must also be a coisometry, and therefore aunitary, in ∆(B). One way to see this is to consider vop : Aop → Bop as in 1.2.25,and use the fact that ∆(B)op = ∆(Bop).

In the general case, clearly v∗∗ : A∗∗ → B∗∗ is a completely isometric linearsurjection. By the first part, there is a unital completely isometric isomorphismθ : A∗∗ → B∗∗, and a unitary U ∈ ∆(B∗∗) such that v(a) = θ(a)U for a ∈ A.Inside B∗∗, we have that BU−1 = θ(A) is an operator algebra with cai. We haveBU−1BU−1 = BU−1, by A.6.2. Thus BU−1B = B. Also, BBU−1 = BU−1,by A.6.2 again. Thus by 2.5.10, BU−1 = B. Thus BU = B and θ(A) = B.Hence U, U−1 ∈ RM(B). A similar argument applied to U−1B, shows thatU, U−1 ∈ LM(B). So U, U−1 ∈ M(B).

The asymmetry in the statement of 4.5.13 is easily removed by setting θ′ equalto U∗θ(·)U . Then θ′ is also an isomorphism of A onto B, and θ(·)U = Uθ′(·).4.5.14 (One-sided M -projections) If X is an operator space, then a linearidempotent P : X → X is said to be a left M-projection if the map

σP (x) =[

P (x)x − P (x)

]is a complete isometry from X → C2(X). A similar definition (involving R2(X))pertains to right M-projections.

These projections are connected to the noncommutative variant on M -idealsstudied in Section 4.8. Here we will prove just one result concerning them:

Theorem 4.5.15 If P is an idempotent linear map on an operator space X,then the following are equivalent:(i) P is a left M -projection.(ii) The map τP introduced above 4.5.2 is completely contractive.(iii) P is a (selfadjoint) projection in the C∗-algebra A(X).(iv) There exist a completely isometric embedding σ : X → B(H), and a projec-

tion e ∈ B(H), such that σ(Px) = eσ(x) for all x ∈ X.(v) There exists a completely isometric embedding σ : X → B(H) such that

σ(x)∗σ(y) = 0, x ∈ P (X), y ∈ (I − P )(X).

Proof To show that (i) implies (iv), suppose that X ⊂ B(H), and view C2(X)as a subset of M2(X) ⊂ M2(B(H)) in the usual way. Then take σ = σP (themap in 4.5.14), regarded as a map X → M2(B(H)), and set

e =[

1 00 0

].

That (iv) implies (v) is clear, since e(1 − e) = 0.


Assume (v). If x ∈ X , then by the C∗-identity, ‖x‖2 equals

‖σ(x)‖2 = ‖σ(Px) + σ(x − Px)‖2 = ‖σ(Px)∗σ(Px) + σ(x − Px)∗σ(x − Px)‖.But the latter is the norm of σP (x) in C2(X). Thus σP is an isometry. A similarargument shows that it is a complete isometry, which shows (i).

To prove the equivalence of (ii) and (iii), note that by 4.5.2, (ii) holds if andonly if P ∈ Ball(Ml(X)). Since P = P 2, and since Ml(X) is an operator algebra,this occurs when P = P ∗ (see 2.1.3), in which case P ∈ ∆(Ml(X)) = A(X).

That (iv) implies (iii) follows by 4.5.8 (1) and (3). Conversely, assume (iii).By 4.5.2, there exists an a ∈ Ball(I11) such that Px = ax for all x ∈ X . SinceP is an idempotent map, we have (a2 − a)X = 0, which implies that a2 = a by4.4.12. As in the last paragraph, a is a (selfadjoint) projection. Now one maydeduce (iv) by regarding I(S(X)) as a C∗-subalgebra of some B(H).

We refer the reader to [73], for example, for explicit calculation of the multi-plier algebras for some other classes of operator spaces.

4.6 MULTIPLIERS AND THE ‘CHARACTERIZATION THEOREMS’

It is very simple to deduce the BRS and CES theorems from 4.5.2. In fact wegive a more general result which has such theorems as immediate consequences.

4.6.1 (Oplications) If X and Y are operator spaces, then a (left) oplication ofY on X is a completely contractive bilinear map m : Y × X → X (see 1.5.4),such that there is an element e ∈ Ball(Y ) (or more generally, a net of elements(et)t in Ball(Y )) such that m(e, x) = x (resp. m(et, x) → x) for all x ∈ X .The word ‘oplication’ is intended to be a contraction of the phrase ‘operatormultiplication’. The element e is called a left identity for the oplication.

Theorem 4.6.2 (Oplication theorem) Let m : Y ×X → X be an oplication.(1) There exists a (necessarily unique) linear complete contraction θ from Y

into Ml(X) such that θ(y)(x) = m(y, x) for all y ∈ Y, x ∈ X. If e is a leftidentity for the oplication, then θ(e) = IX .

(2) If Y is an algebra (resp. C∗-algebra), and if m is a module action, then θis a homomorphism into Ml(X) (resp. ∗-homomorphism into Al(X)).

(3) If Y is an operator system, whose unit is a left identity for m, then θ mapsinto Al(X), and is completely positive.

Proof We assume the presence of a left identity e, and leave the case involvingthe net as an exercise (in which case one needs to use limits in the formulaebelow). Let θ : Y → B(X) be the map associated with m. Clearly θ(e) = IX . Ifτ is as in 4.5.1, if m2 is as in 1.5.4, and if y ∈ Ball(Y ) and x, x′ ∈ X , then∣∣∣∣∣∣∣∣τθ(y)

([xx′

])∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣[m(y, x)

x′

]∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣m2

([y 00 e

],

[x 0x′ 0

])∣∣∣∣∣∣∣∣ ≤ ∣∣∣∣∣∣∣∣[ xx′

]∣∣∣∣∣∣∣∣ .Thus τθ(y) is contractive, and an analoguous argument with matrices shows thatit is completely contractive. By 4.5.2, we deduce that θ is a contractive map into

176 Multipliers and the ‘characterization theorems’

Ml(X). Given [yij ] ∈ Ball(Mn(Y )), we consider [θ(yij)] as a map v on Cn(X),via the usual formula for the product of a matrix and a column. Arguing asabove, but now using larger matrices, we see that v ∈ Ball(Ml(Cn(X))). Thusθ is a complete contraction into Ml(X). This proves (1).

Since θ(y)(x) = m(y, x) for all y ∈ Y, x ∈ X , the first assertion in (2) is quiteclear. The second one follows from the last paragraph of 2.1.2, and the fact thatAl(X) = ∆(Ml(X)). In a similar way, (3) follows from 1.3.8 and 1.3.3.

4.6.3 The BRS theorem 2.3.2 follows immediately from the last result. For ifA is an algebra satisfying the BRS hypotheses, then 4.6.2 provides a completelycontractive homomorphism θ from A into the operator algebra Ml(A). That θis a complete isometry is obvious: for example, ‖θ(a)‖ ≥ ‖a1‖ = ‖a‖ for a ∈ A.

Indeed one sees easily from this approach, that the ‘associativity’ conditionfor the algebra is not needed in the BRS characterization (this was proved inthe original paper [69] using the injective envelope). One way to see this is toconsider the product on A both as a left and as a right oplication; and then the‘automatic associativity’ follows from 4.6.2 (1) and the last assertion in 4.5.6.

Corollary 4.6.4 Let A be a unital Banach algebra, and let A1 and A2 be oper-ator space structures on A (keeping the same norm). Suppose that the canonicalmap A1 ⊗h A2 → A2, induced by the product on A, is completely contractive.Then there is a third operator space structure A3 on A between A1 and A2, suchthat A3 is completely isometrically isomorphic to a unital operator algebra.

Proof By the oplication theorem 4.6.2, there is a unital completely contractivehomomorphism θ : A1 → Ml(A2) with θ(a)(b) = ab for all a, b ∈ A. On the otherhand, we have a canonical complete contraction Ml(A2) → CB(A2) by 4.5.5,and a canonical complete contraction CB(A2) → A2 given by ‘evaluation at 1A’.Thus we have a sequence of completely contractive maps

A1θ−→ Ml(A2) −→ CB(A2) −→ A2

which compose to the identity map IA. Thus θ is an isometric homomorphism.Since Ml(A2) is an operator algebra, we may define an operator algebra A3 tobe A with the matrix norms pulled back via θ from Ml(A2). The other assertionsare now evident from the above.

4.6.5 (The commutative case) The last result is general enough to explainmany of the results found in the 1970s, distinguishing general operator algebrasamongst the Banach algebras. As an example, we show how it gives Theorem3.7.9. Thus consider a unital Banach algebra A such that the product on A iscontractive as a map from A ⊗g2 A to A. We remarked in 3.7.7 that this isequivalent to it being completely contractive as a map from Max(A)⊗h Min(A)to Min(A). Thus we deduce from 4.6.4 that A is isometrically isomorphic toan operator algebra. Indeed, the proof of 4.6.4 shows that A is isomorphic toa subalgebra of Ml(Min(A)). However from (4.13), Ml(Min(A)) is a uniformalgebra, which implies that A is a uniform algebra.


4.6.6 (Applications to operator modules) The reader may want to glance atSection 3.1 to recall the definition of an operator module. We will assume thatall operator bimodules are nondegenerate in this section.

Any operator space X is an operator Ml(X)-Mr(X)-bimodule, and is thusalso an operator Al(X)-Ar(X)-bimodule. This follows, for example, from thetechnique in the proof in 4.5.5 showing that X is an operator Ml(X)-module.We call these bimodule structures the extremal operator module actions on X .Conversely, if A is a unital Banach algebra which is also an operator space, thenby the oplication theorem, the left operator module actions of A on X are in one-to-one correspondence with the completely contractive unital homomorphismsfrom A into Ml(X). If A is a C∗-algebra then we may replace Ml(X) by Al(X)here. This generalizes the phenomena in Section 3.7.

We may now give a generalization of the Christensen–Effros–Sinclair theorem3.3.1; an abstract characterization of nondegenerate operator bimodules:

Theorem 4.6.7 Suppose that X is an operator space, that A and B are operatorspaces which are also approximately unital Banach algebras, and that X is anondegenerate left A-module and a nondegenerate right B-module. Then X isan operator A-B-bimodule if and only if

‖ax‖n ≤ ‖a‖n‖x‖n, and ‖xb‖n ≤ ‖x‖n‖b‖n,

for every n ∈ N, a ∈ Mn(A), b ∈ Mn(B) and x ∈ Mn(X). In particular, thecentered equations imply that a(xb) = (ax)b for all a ∈ A, x ∈ X, b ∈ B.

Proof The centered equation here is simply saying that X is both a left and aright h-module in the sense of 3.1.3; and then the one direction follows from aremark in 3.1.3. Conversely, if the centered equation holds, then the two moduleactions are oplications on X . By the oplication theorem, the two actions are‘prolongations’ (in the sense of 3.1.12) of the extremal Ml(X)-Mr(X)-bimoduleactions (see 4.6.6). Since X is an operator Ml(X)-Mr(X)-bimodule, and sinceprolongations of operator modules remain operator modules (see 3.1.12), we de-duce that X is an operator A-B-bimodule. Since these extremal actions commute,so do the original module actions. This proves the last assertion.

4.6.8 (Examples) The following examples illustrate the main point in 4.6.6.(1) Let Hc be a column Hilbert space (see 1.2.23). We leave it as an exercise

that Ml(Hc) = B(H) and Mr(Hc) = C (see also 8.4.2 for a more generalresult). Thus if A is a unital Banach algebra which is also an operator space,the left operator module actions of A on Hc are exactly in correspondence withthe completely contractive unital homomorphisms θ : A → B(H). Indeed we sawthis already in Proposition 3.1.7. On the other hand, the right operator moduleactions are all ‘scalar’: in correspondence with the characters of A (see A.4.2).

(2) Consider the possible left module actions on X = 1n. We equip 1

n

with its natural (maximal) operator space structure, so that X is the operatorspace dual of ∞n (see (1.30)). There are plenty of Banach module actions of a

178 Multipliers and the ‘characterization theorems’

Banach algebra A on X , corresponding to contractive unital homomorphismsfrom A to B(1

n). Similarly, there are plenty of ‘matrix normed module actions’(see 3.1.4) on X . For example, since X = (∞n )∗ we have from Example 3.1.5(2) that Max(1

n) is canonically a matrix normed ∞n -module. Considering thecorresponding homomorphism from ∞n into B(X), A.5.9 shows that there is anisometric copy of the Banach algebra ∞n inside B(X) or CB(X). One mightfirst guess that this copy equals Al(X). In fact this is false, and indeed we shallshow that Ml(Max(1

n)) = C. Equivalently, by the main point in 4.6.6, thereare no ‘nonscalar’ (see the last line of (1) above) operator module actions onX = Max(1

n). We recall that in 4.3.8 we showed that X is a unital operatorspace, and that C∗

e (X) = C∗(Fn−1). By the last paragraph of 4.4.7, the latter isalso a triple envelope of X . Hence, by 4.5.2 (iv), any u ∈ Ml(X) is necessarilythe restriction to X of a right module map on C∗(Fn−1). Since the latter is aunital C∗-algebra, any right module map on it is given by left multiplication bya fixed a ∈ C∗(Fn−1). Recall from 4.3.8 that X is identified with the span of1 and the generators u1, . . . , un−1 of C∗(Fn−1). Since u(X) ⊂ X , we have thata = a1 ∈ X , and auj belongs to X for any j. Hence we must have a ∈ C1, andtherefore u ∈ CIX . Hence Ml(Max(1

n)) = C as promised.

4.6.9 (CES-representations of operator modules) If X is an operator bimoduleover algebras A and B as in 4.6.7), then one may extract from the proof of 4.6.7an explicit CES-representation for X (see 3.3.3). Indeed let J be the canonicalcompletely isometric embedding from X into I(S(X)) (see 4.4.2). By the oplica-tion theorem 4.6.2 and (4.12), there exist completely contractive homomorphismsθ : A → I11(X) and π : B → I22(X), with

J(ax) = θ(a)J(x) and J(xb) = J(x)π(b), a ∈ A, b ∈ B, x ∈ X.

Thus if we represent I(S(X)) faithfully and nondegenerately on a Hilbert spaceL, and if we regard J, θ and π as valued in B(L), then (J, θ, π) is an explicitCES-representation of the operator A-B-bimodule X .

We next improve on the extension theorem from Section 3.6.

Theorem 4.6.10 Let X be a nondegenerate operator A-B-bimodule over unitalC∗-algebras A and B. Then X is an injective operator space if and only if Xhas the following extension property:

For every nondegenerate operator A-B-bimodule W , closed A-B-submodule Yof W , and completely contractive A-B-bimodule map u : Y → X, there existsa completely contractive A-B-bimodule map u : W → X extending u.

Proof Suppose that X satisfies the above extension property. By the CEStheorem 3.3.1, X may be realized as an A-B-submodule of some B(K, H), whereH is a Hilbert A-module and K is a Hilbert B-module. By the hypothesis appliedto the inclusion map X ⊂ B(K, H), there is a completely contractive idempotentmap from B(K, H) onto X . Since B(K, H) is injective, so is X .


If X is injective, then by the proof of 4.4.4, I12(X) = X . Represent theC∗-algebra I(S(X)) faithfully and nondegenerately on a Hilbert space. The twodiagonal projections 1⊕0 and 0⊕1 determine a splitting of the Hilbert space asH ⊕ K, say (e.g. see 2.6.14). Hence I11(X) and I22(X) are unital ∗-subalgebrasof B(H) and B(K) respectively, so that we have

I(S(X)) =[

I11(X) XX I22(X)

]⊂

[B(H) B(K, H)

B(H, K) B(K)

]= B(H ⊕ K).

By injectivity, there is a completely positive idempotent map Φ from B(H ⊕K)onto I(S(X)). By 2.6.16, Φ decomposes as a 2× 2 matrix of maps. Let P be the‘1-2-corner’ of Φ, then P : B(K, H) → X is a completely contractive idempotentmap onto X . By 1.3.12, Φ is a ‘conditional expectation’ onto I(S(X)). It followsthat P is a left I11-module map. However if X is an operator A-B-bimodule,then as in 4.6.9 there is a unital completely contractive homomorphism θ fromA to I11 ⊂ B(H) implementing the left module action of A on X . Hence H is aHilbert A-module, with A-action given by θ. Since θ maps into I11, we see thatP is a left A-module map onto X . Similarly, P is a right B-module map ontoX . We saw in 3.6.4 that B(K, H) is injective as an operator A-B-bimodule. Itis now easy to see that so is X .

4.6.11 (Bimodule injective envelopes) In the following discussion, A and Bare unital C∗-algebras. Suppose that X is an operator A-B-bimodule. We saw in4.4.2 that the injective envelope I(X) = I12(X) considered above, is an operatormodule over I11. Hence, for example via the maps θ and π in 4.6.9, I(X) isan operator A-B-bimodule. By 4.6.10, it follows that I(X) is injective in thecategory of operator A-B-bimodules and completely bounded bimodule maps(see 3.6.4).

In fact, I(X) is also an ‘injective envelope’ of X in the latter category. Tobe more precise: one may adapt all the definitions in 4.2.3 to the A-B-bimodulesituation, simply by changing the word ‘map’ to ‘A-B-bimodule map’, ‘operatorspace’ to ‘operator A-B-bimodule’, and ‘subspace’ to ‘A-B-submodule’. Thuswe have notions of ‘A-B-rigid extension’, and so on. What we said in the lastparagraph clearly implies that the operator space injective envelope I(X) is alsoan A-B-injective extension of X . Since I(X) is rigid, it is an A-B-rigid extensionof X . A trivial modification of the proof of 4.2.5 shows that there is an essentiallyunique A-B-rigid injective A-B-extension of X . Conversely, putting the last twosentences together, an obvious diagram chase shows that any A-B-rigid injectiveextension of X , is also rigid as an operator space extension, and is hence aninjective envelope of X in the old sense (of 4.2.3).

Such observations are often useful, as applications such as the following show:

Corollary 4.6.12 If I(X) is an injective envelope of X, then for any cardinalsI and J , MI,J(I(X)) is an injective envelope of MI,J(X), and of KI,J(X).

Proof Note that MI,J(X), KI,J(X), and MI,J(I(X)) are operator bimodulesover ∞I and ∞J , by 3.1.14 (3). Also MI,J(I(X)) is an injective operator space, by

180 Multipliers and duality

the proof in the first paragraph of 4.2.10. By 4.6.10, it is also injective in the cate-gory of operator ∞I -∞J -bimodules. Suppose that u : MI,J (I(X)) → MI,J(I(X))is a completely contractive ∞I -∞J -bimodule map extending the identity map onthe image of KI,J(X). It is fairly obvious that this bimodule condition impliesthat u is of the form u([zpq]) = [upq(zpq)], for maps upq : I(X) → I(X). Clearlyupq is a completely contractive map extending the identity map on i(X). Byrigidity, upq is the identity map, for all p, q. Hence u is the identity map. ThusMI,J(I(X)) is an injective ∞I -∞J -rigid extension of KI,J(X), and MI,J(X). Bythe observation above 4.6.12, we obtain the desired result.

4.7 MULTIPLIERS AND DUALITY

Operator space multipliers yield deep facts about w∗-topologies. For example:

Theorem 4.7.1 (Magajna) If X is a dual operator space, then any operator inMl(X) is w∗-continuous.

Proof This proof is again rather technical. As in the proof of 2.7.13, there isa canonical w∗-continuous projection q : X∗∗ → X , which induces a completelyisometric w∗-homeomorphism v : X∗∗/Ker(q) → X (using A.2.5 (3) if necessary).If u ∈ Ml(X), then u∗∗ ∈ B(X∗∗). We claim that:

q(u∗∗(η)) = u q(η), η ∈ X∗∗. (4.14)

Assuming (4.14), it follows that u∗∗ induces a map u in B(X∗∗/Ker(q)), namelyu(η) = (u∗∗(η))˙, where η is the equivalence class of η ∈ X∗∗ in the quotient.Since u∗∗ is w∗-continuous, so is u, by basic Banach space duality principles(e.g. A.2.4). Using (4.14) it is easy to see that u = v−1uv. Since u, v, and v−1

are w∗-continuous, so is u, and thus the theorem is proved.In order to prove (4.14), we will use the notation in 4.4.2, and we will be

silently using the principles in A.2.3. We let A be the C∗-algebra I(S(X)), andlet p be the projection in A corresponding to the identity of I11. We regardA ⊂ A∗∗, then p may be viewed as a projection in A∗∗, and A∗∗ has four cornerswith respect to p, as in 2.6.14. Let c11 and c12 be, respectively, the canonicalembeddings of I11 = pAp and I(X) = pA(1 − p) in A. We will write C11 = c∗∗11and C12 = c∗∗12, these embed I∗∗

11 and I(X)∗∗ into A∗∗. We view X as a subspaceof pA(1 − p) ⊂ pA∗∗(1 − p) ⊂ A∗∗, and thus we may identify X∗∗ with thew∗-closure of X in the corner pA∗∗(1 − p). Via this embedding and 1.3.15, weview S(X∗∗) ⊂ A∗∗. Also by 1.3.15, q induces a canonical complete contractionQ : S(X∗∗) → S(X) ⊂ A, with Q|S(X) = I|S(X). Since A is injective, we mayextend Q to a complete contraction R : A∗∗ → A. Since R|S(X) = I|S(X), itfollows by rigidity (see 4.2.3) that R|A = IA. By 1.3.12, R is an A-bimodulemap.

Let a ∈ I11 correspond to u, as in 4.5.2 (iii). Then C11(a) = c∗∗11(a) = c11(a).Similarly C12(z) = c12(z) if z ∈ I(X). Since R is an A-bimodule map, we have


R(C11(a)C12(η)) = c11(a)R(C12(η)) = c11(a) c12(q(η)) = c12(uq(η)), (4.15)

for η ∈ X∗∗ ⊂ I(X)∗∗. On the other hand, we claim that

C11(a)C12(η) = C12(u∗∗(η)), η ∈ X∗∗. (4.16)

Indeed since the product in a W ∗-algebra is separately w∗-continuous, both sidesof (4.16) may be viewed as w∗-continuous functions taking η ∈ X∗∗ into the W ∗-algebra A∗∗. Since A is a subalgebra of A∗∗, if z ∈ X ⊂ I(X) then

C11(a)C12(z) = c11(a) c12(z) = c12(az) = c12(uz) = C12(u∗∗(z)).

Thus (4.16) follows by a routine w∗-density argument. By (4.15) and (4.16),c12(uq(η)) = R(C12(u∗∗(η))) = c12(q(u∗∗(η))). This proves (4.14).

Corollary 4.7.2 If B is an operator algebra which is also a dual operator space,then the product on B is separately w∗-continuous.

Proof If a ∈ B, then the map b → ab on A is clearly a left multiplier, andtherefore is w∗-continuous by 4.7.1. Similarly the product is w∗-continuous inthe first variable.

Next we turn to a general functional analytic result:

Lemma 4.7.3

(1) Let T : E → F be a one-to-one linear map between Banach spaces, andsuppose that F is a dual Banach space. The following are equivalent:(i) E is a dual Banach space and T is w∗-continuous,(ii) T (Ball(E)) is w∗-compact.

(2) Let X and Y be operator spaces, with Y a dual operator space, and letu : X → Y be a one-to-one linear map such that un(Ball(Mn(X))) is w∗-compact for every positive integer n. Then the predual of X given in theproof of (1), is an operator space predual of X, and u is w∗-continuous.

Proof (1) That (i) implies (ii) is clear. Given (ii), we see by the Principle ofUniform Boundedness that T (Ball(E)), and therefore also T , is bounded. Wemay assume that T is a contraction. Suppose that Z is the predual of F , viewedas a subset of F ∗, and let W = T ∗(Z), a linear subspace of E∗. The canonicalmap j : E → W ∗ is one-to-one and contractive, and

〈j(x), T ∗(z)〉 = 〈T ∗(z), x〉 = 〈T (x), z〉.

We will show that j is an isometric isomorphism. Given ψ ∈ Ball(W ∗), define amap ψ(z) = ψ(T ∗(z)), for z ∈ Z. Then ψ ∈ Ball(Z∗) = Ball(F ). If we can showthat ψ = T (x), for some x ∈ Ball(E), then we are done, for in this case it is clearthat ψ = j(x). So suppose, by way of contradiction, that ψ /∈ T (Ball(E)). Byassumption, T (Ball(E)) is w∗-closed, and so by the Hahn–Banach theorem there

182 Multipliers and duality

exists z ∈ Z such that ψ(z) > 1 and |〈T (x), z〉| ≤ 1 for all x ∈ Ball(E). Thelatter condition implies that ‖T ∗(z)‖ ≤ 1, whereas the former condition impliesthe contradictory assertion that ψ(T ∗(z)) > 1.

That T is w∗-continuous with respect to the predual W is now clear.(2) We will use 1.6.4 to check that the predual from (1), is an operator space

predual. Let xλ = [xλij ] be a net in Ball(Mn(X)), and x = [xij ] ∈ Mn(X), with

< xλij , w >→< xij , w > for all i, j and w ∈ W . Equivalently, u(xλ

ij) → u(xij)in the w∗-topology of Y . By 1.6.3 (1), the matrices [u(xλ

ij)] converge in the w∗-topology to [u(xij)] in Mn(Y ). By hypothesis, [u(xij)] ∈ un(Ball(Mn(X))), sothat x ∈ Ball(Mn(X)), and we are done.

Theorem 4.7.4 Let X be a dual operator space.(1) Ml(X) is a dual operator algebra, and Al(X) is a W ∗-algebra.(2) A bounded net (at)t in Ml(X) or Al(X) converges in the w∗-topology to an

element a in the same space, if and only if at(x) → a(x) in the w∗-topologyin X for all x ∈ X.

Proof For the first assertion in (1), it suffices by Theorem 2.7.9 to show thatMl(X) is a dual operator space. To do this we shall use 4.7.3 applied to thecanonical embedding u : Ml(X) → CB(X), and basic facts about CB(X) (see1.6.1). We will show that if (at)t is a net in Ball(Mn(Ml(X))) converging inthe w∗-topology in Mn(CB(X)) to a, then a ∈ Ball(Mn(Ml(X))). Let µ be theisomorphism in (4.11). We will test if µ(a) ∈ Ball(Ml(Cn(X))) using 4.5.2 (ii).Choosing x, y in Mm(Cn(X)), we need to check that∣∣∣∣∣∣∣∣[µ(a)m(x)

y

]∣∣∣∣∣∣∣∣ ≤∣∣∣∣∣∣∣∣[x

y

]∣∣∣∣∣∣∣∣ . (4.17)

We do have ∣∣∣∣∣∣∣∣[µ(at)m(x)y

]∣∣∣∣∣∣∣∣ ≤∣∣∣∣∣∣∣∣[x

y

]∣∣∣∣∣∣∣∣ . (4.18)

Let z (resp. zt) be the matrix on the left side of (4.17) (resp. (4.18)). Thesematrices are in M2m,m(Cn(X)) = M2mn,m(X), and zt → z in the w∗-topology inM2m,m(X), by 1.6.3 (1). Hence (4.17) follows from (4.18). Thus a ∈ Ball(Ml(X)).

We have now proved that Ml(X) is a dual operator space. Thus Ml(X) is adual operator algebra, by Theorem 2.7.9. That is, there is a w∗-homeomorphiccompletely isometric isomorphism of Ml(X) onto a w∗-closed unital-subalgebraB of some B(H). Clearly ∆(B) = B ∩ B∗ is a von Neumann algebra by 2.1.2;hence Al(X) ∼= ∆(B) is a W ∗-algebra.

Assertion (2) follows from (1), and the definition of the w∗-topologies con-cerned (see the proof of 4.7.3).

Lemma 4.7.5 Any dual operator space X is a normal dual bimodule, and henceis also a dual operator bimodule (see 3.8.3), over Ml(X) and Mr(X) (and alsoover Al(X) and Ar(X)).


Proof The fact that X is an operator bimodule follows from 4.6.6. The otherfacts follow immediately from 4.7.4 (2) and 4.7.1.

Theorem 4.7.6 Let X and M be dual operator spaces, and let m : M ×X → Xbe an oplication. Suppose that m is w∗-continuous in the first variable. Then wemay conclude:(1) m is separately w∗-continuous.(2) There exist Hilbert spaces H, K, a unital w∗-continuous completely contrac-

tive map π : M → B(K), and a w∗-continuous complete isometry v from Xinto B(H, K), with v(m(b, x)) = π(b)v(x) for all x ∈ X, b ∈ M .

(3) If M is also a W ∗-algebra (resp. algebra) and if m is a module action, thenπ in (2) may be taken to be a ∗-homomorphism (resp. homomorphism).

Proof First apply Theorem 3.8.3 to the Ml(X) action on X (after appealingto 4.7.5). This provides the Hilbert spaces H, K, and the map v. It also providesa w∗-continuous completely contractive homomorphism ρ : Ml(X) → B(K),with ρ(u)v(x) = v(ux) for all x ∈ X, u ∈ Ml(X). Next apply the oplicationtheorem 4.6.2, to get a unital completely contractive θ : M → Ml(X) such thatθ(b)x = m(b, x) for all x ∈ X, b ∈ M . It is elementary to check that θ is w∗-continuous using, first, 4.7.4 (2), second, the fact that m is w∗-continuous in thefirst variable, and third, the Krein–Smulian theorem A.2.5 (2). Let π = ρ θ,and now (2) is clear.

If b ∈ M , and xt → x in the w∗-topology in X , then π(b)v(xt) → π(b)v(x) inthe w∗-topology in B(H, K). This follows since v is a w∗-w∗-homeomorphism,and operator multiplication is separately w∗-continuous. Thus m(b, xt) → m(b, x),which gives (1). Adding the ideas in 4.6.2 (2), and 1.2.4, gives (3).

The last result generalizes Theorem 3.8.3, and allows us to characterize dualoperator modules over algebras which need not be operator algebras:

Corollary 4.7.7 Suppose that X is a left operator module over a unital Banachalgebra and operator space A, that A and X are dual operator spaces, and that themodule action is w∗-continuous in the first variable. Then X is a dual operatormodule in the sense of 3.8.1.

4.8 NONCOMMUTATIVE M -IDEALS

The theory of M -ideals of Banach spaces emerged in the early 1970s with thepaper [4] of Alfsen and Effros. It is an important tool in functional analysis andapproximation theory (see [195] for an extensive survey of results). A one-sidedvariant of this classical theory was recently developed in [57,440,71,73,442], and isintimately connected with the operator space multipliers met in the last sections.Indeed, we already met in 4.5.14 the ‘left M -projections’: these are exactly theorthogonal projections in the C∗-algebra Al(X). If X is a dual operator space,then we saw that Al(X) is a von Neumann algebra. Applying basic von Neumannalgebra theory then leads to a cogent ‘noncommutative’ generalization of the

184 Noncommutative M -ideals

classical theory of M -ideals. The aim of this short section is to give some of theflavor of this subject, and to hopefully act as a conduit to the more satisfyingaccount in the cited papers. En route, we will also compute the various M -idealsof operator algebras.

4.8.1 (M -ideals: classical and one-sided) We begin with some definitions. In3.7.4 we defined M -projections. A subspace Y of a Banach space E is called anM -summand of E if Y is the range of an M -projection. A subspace Y of E iscalled an M -ideal in E if Y ⊥⊥ is an M -summand in E∗∗. Of course Y ⊥⊥ is justthe w∗-closure of Y in E∗∗, and may be identified with Y ∗∗ (see A.2.3).

If X is an operator space, then a map P on X is a complete M -projectionif the amplification Pn is an M -projection on Mn(X) for every n ∈ N. Thisimmediately leads, by straightforward analogy with the last paragraph, to thenotion of complete M -summand and complete M -ideal.

In 4.5.14 we defined left M -projections. By analogy to the above, a rightM -summand of X is the range of a left M -projection on X . We say that Y is aright M -ideal if Y ⊥⊥ is a right M -summand in X∗∗. Similar definitions pertainto the ‘other-handed’ case, i.e., right M -projections, left M -ideals, and so on.

We shall see that all these definitions are interconnected in a tidy way.

4.8.2 (Transferring the classical theory) Few of the proofs of results in theextensive classical M -ideal theory transfer verbatim to the one-sided case; mostrequire new, noncommutative, arguments. We will see an example of the latterin Theorem 4.8.6. As an illustration of an ‘uncomplicated transfer’, we mentionthe fact that for any one-sided M -summand J of an operator space X , thereexists a unique contractive linear idempotent of X onto J , namely the one-sided M -projection P defining J . To prove this, note that it suffices to showthat Ker(P ) ⊂ Ker(Q) for any contractive idempotent Q onto Ran(P ), for thenQ(x) = Q(x− (x−P (x))) = P (x) for every x ∈ X . If y ∈ Ker(P ), let z = Q(y).For t > 0, we have

‖(t + 1)z‖ = ‖Q(tz + y)‖ ≤ ‖tz + y‖ =∣∣∣∣∣∣∣∣[ tz

y

]∣∣∣∣∣∣∣∣ ≤ √t2‖z‖2 + ‖y‖2 ,

the second equality because the map σP in 4.5.14 is an isometry. Squaring, weobtain (2t + 1)‖z‖2 ≤ ‖y‖2. Since t > 0 was arbitrary, z = 0.

4.8.3 (Examples) We showed in 3.7.4 that the M -projections on a Banachspace E are exactly the projections in the centralizer algebra Z(E) discussedin Section 3.7. We showed in (4.13) that Z(E) = Al(Min(E)). Thus, by 4.5.15,the right M -projections on Min(E) are exactly the classical M -projections onE. Therefore, left M -summands (resp. left M -ideals) of Min(E) are exactly thesame as the classical M -summands (resp. M -ideals) of E. Similar assertions holdfor the ‘other-handed’ variants.

Hence classical M -ideals and summands are particular examples within theone-sided M -ideal theory. We see next that complete M -ideals and summands(see 4.8.1) are also particular examples within the one-sided theory.


Proposition 4.8.4 Suppose that X is an operator space.(1) A projection P : X → X is a complete M -projection if and only if it is both

a left M -projection and a right M -projection.(2) A subspace Y of X is a complete M -summand if and only if it is both a left

M -summand and a right M -summand.(3) A subspace Y of X is a complete M -ideal if and only if it is both a left

M -ideal and a right M -ideal.

Proof (1) If P is both a left and right M -projection, then by first applyingthe map σP from 4.5.14, and then the ‘row variant’ of σP , we obtain

‖x‖ =∥∥∥∥[ P (x)

x − P (x)

]∥∥∥∥ =∥∥∥∥[ P 2(x) (I − P )P (x)

P (I − P )(x) (I − P )2(x)

]∥∥∥∥ =∥∥∥∥[P (x) 0

0 x − P (x)

]∥∥∥∥ ,

which is simply max ‖P (x)‖ , ‖x − P (x)‖, for x ∈ X . Similar identities hold formatrices, and so P is a complete M -projection. Conversely, if P is a completeM -projection, then the mapping

u : X −→ X ⊕∞ X ⊂ M2(X) : x −→ (Px) ⊕ (x − Px)

is completely isometric. It follows that if u2,1 is as in 1.2.1, then∥∥∥∥[ P (x)x − P (x)

]∥∥∥∥ =∥∥∥∥u2,1

([P (x)

x − P (x)

])∥∥∥∥ = max ‖Px‖ , ‖x − Px‖ = ‖x‖ .

Again it is easy to generalize this to matrices, so that P is a left M -projection.A similar argument shows that P is a right M -projection.

One direction of (2) follows from (1), the other follows from (1) together withthe ‘uniqueness’ assertion proven in 4.8.2. Now (3) is also clear.

Theorem 4.8.5 Let A be an approximately unital operator algebra.(1) The M -ideals in A are the complete M -ideals. These are exactly the two-

sided ideals in A which possess a two-sided contractive approximate identity.(2) The M -summands in A are the complete M -summands. These are exactly

the principal ideals Ae for a central projection e ∈ M(A).(3) The right M -ideals in A are exactly the right ideals in A which possess a

left contractive approximate identity.(4) The right M -summands in A are exactly the principal right ideals eA for a

projection e ∈ LM(A).

Proof (4) By 4.5.15 together with 4.5.11, the left M -projections on A areexactly the projections e in LM(A) (or equivalently in ∆(LM(A))).

(3) If J is a right M -ideal of A, then J∗∗ = J⊥⊥ = Jw∗

is a right M -summand. Hence by (4), J⊥⊥ = eA∗∗ for a projection e ∈ A∗∗. Considered assubsets of A∗∗, we have JA ⊂ J∗∗. But also JA ⊂ A. Thus JA ⊂ J∗∗ ∩ A,and the latter space (by A.2.3 (4)) equals J . So J is a right ideal of A. Since

186 Noncommutative M -ideals

A∗∗ is unital, we must have e ∈ J∗∗. Therefore e is a left identity for J∗∗. ByProposition 2.5.8, J has a left contractive approximate identity.

Conversely, if J is a closed right ideal of A with a contractive left approximateidentity, then by Proposition 2.5.8 and 2.5.2, we have that J∗∗ is a subalgebra ofA∗∗, and J∗∗ has a left identity e of norm 1. Note that e is a projection in A∗∗.Moreover, J∗∗A∗∗ ⊂ J∗∗ by a routine separate w∗-continuity argument similarto those encountered in Section 2.5. We obtain

J∗∗ = eJ∗∗ ⊂ eA∗∗ ⊂ J∗∗.

Thus J∗∗ = eA∗∗. Since e is a projection in A∗∗, we see by (4) that J∗∗ is a rightM -summand of A∗∗. Hence J is a right M -ideal of A.

(2) If e is a central projection in M(A), then Ae is a complete M -summandby (4), the ‘left-handed’ version of (4), and 4.8.4 (2). Thus it is an M -summand.Conversely, suppose that P is an M -projection on A. We may assume that Ais unital (the general case follows easily from this case by considering the M -projection P ∗∗ on A∗∗, and using a definition of M(A) from Section 2.6). We setz = P (1). As in the third centered equation in the proof of 3.7.3, we have

‖P ∗(ϕ)‖ + ‖(I − P )∗(ϕ)‖ ≤ 1,

for any ϕ ∈ A∗ of norm 1. If ϕ is a state, we deduce that

1 = ϕ(1) = P ∗(ϕ)(1) + (I − P )∗(ϕ)(1) ≤ ‖P ∗(ϕ)‖ + ‖(I − P )∗(ϕ)‖ = 1.

If 0 ≤ a ≤ c and 0 ≤ b ≤ d, and a + b = c + d, then a = c and b = d. Thus

ϕ(z) = P ∗(ϕ)(1) = ‖P ∗(ϕ)‖ ≥ 0. (4.19)

Thus z is Hermitian in A (see A.4.2). Claim 1: z2 = z, or equivalently by A.4.2,

ϕ(z2 − z) = 0. (4.20)

for all states ϕ on A. To prove this we will need Claim 2: If ϕ is a state on A withϕ(z) = 1, then (4.20) holds. Claim 2 follows by spectral theory: regard ϕ|C∗(1,z)

as a probability measure µ on [−1, 1]. Since ϕ(z) = 1, µ is the point mass δ1.To see Claim 1, we may assume that ϕ(z) and ϕ(1 − z) are nonzero (for if

ϕ(z) = 0, for example, then ϕ(1 − z) = 1, so that by Claim 2 for 1− z, we haveϕ((1 − z)2) = ϕ(1 − z), which is equivalent to (4.20)). From (4.19) we see firstthat ψ = P ∗(ϕ)/‖P ∗(ϕ)‖ is a state of A, and second that

ψ(z) =ϕ(P (z))‖P ∗(ϕ)‖ =

ϕ(P (1))‖P ∗(ϕ)‖ = 1.

By Claim 2, we have ψ(z2) = ψ(z). Equivalently, ϕ(P (z2)) = ϕ(P (z)) = ϕ(z).Also, ϕ((I − P )((1 − z)2)) = ϕ(1 − z) (replacing P by I − P and z = P (1) by(I − P )(1)). A line of algebra shows that ϕ(z2 − z) = 0, establishing Claim 1.


By 2.1.3, z is a (selfadjoint) projection in A. If ϕ is any state with P ∗(ϕ) = 0,and if ψ is as defined above, extend ψ to a state ψ on some C∗-cover of A. Bythe Cauchy–Schwarz inequality we obtain, for any a ∈ A, that

|ψ(a(1 − z))|2 ≤ ψ(aa∗) ψ(1 − z) = 0.

Thus ϕ(P (a(1 − z))) = 0. Since this holds for an arbitrary state, A.4.2 impliesthat P (a(1 − z)) = 0. Thus A(1 − z) ⊂ (I − P )(A). Symmetrical argumentsshow that (1 − z)A ⊂ (I − P )(A), Az ⊂ P (A), and zA ⊂ P (A). If a ∈ A, thena = az + a(1 − z), so that P (a) = az. Similarly, P (a) = za. Now (2) is clear.

(1) This is similar to the proof of (3), but uses (2) instead of (4).

The following is a sample result from the theory of one-sided M -ideals:

Theorem 4.8.6 Let X be an operator space (resp. dual operator space) andsuppose that Jλ : λ ∈ Λ is a family of right M -ideals (resp. right M -summands)of X. Then the closure of Span∪λ∈ΛJλ in the norm (resp. w∗-) topology is aright M -ideal (resp. right M -summand) of X.

Proof (Sketch) By basic functional analysis, (∪λ∈ΛJλ)⊥⊥ is the w∗-closure ofthe span of ∪λ∈ΛJ⊥⊥

λ . Thus it suffices to prove the ‘respectively’ assertion, andwe may assume henceforth that X is a dual operator space.

For the remainder of the proof, we rely heavily on the fact that Al(X) is aW ∗-algebra (see 4.7.4), and on facts about projections in von Neumann algebras.We claim first that if P, Q are left M -projections on X , then the w∗-closure ofP (X) + Q(X) is the right M -summand (P ∨ Q)(X), where P ∨ Q is the usual‘join’ of projections in the von Neumann algebra Al(X). Since (P ∨ Q)P = P ,and similarly for Q, it follows that P (X)+Q(X) ⊂ (P ∨Q)(X). Now (P ∨Q)(X)is w∗-closed, since P ∨ Q is a w∗-continuous projection (see 4.7.1), and so

P (X) + Q(X)w∗

⊂ (P ∨ Q)(X).

To see the reverse inclusion, we will use the formula

p ∨ q = w∗− limn→∞(p + q)1/n,

valid for any two projections p, q in a von Neumann algebra. This formula followseasily from e.g. [214, Lemma 5.1.5]. It follows from this and from Theorem 4.7.4(2), that if x ∈ (P ∨ Q)(X), then

x = (P ∨ Q)(x) = w∗− limn→∞(P + Q)1/n(x). (4.21)

By the spectral theorem, for each fixed n ∈ N, there exist polynomials fk withoutconstant terms, such that fk(P+Q) → (P+Q)1/n. Hence (P+Q)1/n(x) lies in the

norm closure of P (X)+Q(X). Thus from (4.21) we see that x ∈ P (X) + Q(X)w∗

.We have now proved the above claim.


Finally, suppose that Jλ : λ ∈ Λ is a family of right M -summands asabove. Suppose that Pλ is the associated family of left M -projections, andset P = ∨λ∈Λ Pλ. We claim that the right M -summand P (X) equals the w∗-closure W of Span∪λ∈ΛJλ. Indeed, the argument at the beginning of the lastparagraph shows immediately that P (X) contains W . For any finite subset ∆ ofΛ, define P∆ = ∨λ∈∆ Pλ. Clearly P∆ is an increasing net in the W ∗-algebraAl(X), and P∆ → P in the w∗-topology. Thus, using Theorem 4.7.4 (2) again,it follows that P∆(x) → P (x) in the w∗-topology of X , for any x ∈ X . By theprevious paragraph, P∆(x) lies in W . Thus P (x) ∈ W . Hence W = P (X).

Combining facts from the last two results, we have:

Corollary 4.8.7 Let A be an approximately operator algebra. The closed span inA of any family of right ideals of A, each possessing a left contractive approximateidentity for itself, is a right ideal also possessing a left contractive approximateidentity.

It is interesting that the last result fails badly for Banach algebras: indeed itis easy to find finite-dimensional counterexamples.

In 8.5.19 (resp. 8.5.16) we will show that the right M -ideals (resp. right M -summands) in a right Hilbert C∗-module are exactly the closed right submodules(resp. orthogonally complemented right submodules). Results in the noncommu-tative M -ideal theory may then be seen to be generalizations of some aspects ofthe theory of C∗-modules. See [56] for more on this perspective.


4.1: Again the story begins with Arveson, and his introduction of noncommu-tative generalizations of the Choquet and Shilov boundaries of unital operatorspaces in the foundational papers [21,22]. Arveson built up a formidable array ofmachinery, including the theory of boundary representations, multivariable dila-tion theory, and much more. Remark 4.1.13 (2) is from [21]. Proposition 4.1.12is a refinement due mostly to Muhly and Solel, of an advance made in [21]. Itis adapted from [283] and [128]. Putting 4.1.12 together with ideas of Woronow-icz [435] permits a construction of the noncommutative Shilov boundary differentfrom the one presented in Section 4.3. Namely, any completely isometric unitalrepresentation π of a unital operator algebra B has a dilation ρ which is a com-pletely isometric boundary representation. It is easy to see that the C∗-algebragenerated by ρ(B) is a C∗-envelope of B. This is presented in [128] from theperspective of Agler’s model theoretic approach [1], and a simplification of theargument appears in [24]. Kirchberg has a related approach we believe (but weare not sure if this is in print).

There are many equivalent definitions of Choquet boundary points in theliterature. Indeed the Choquet boundary, the classical Shilov boundary, and theboundary theorem are a crucial part of the theory of function spaces and uniformalgebras. For example, see the texts [167,400]. The Shilov boundary of a nonunital


function space always exists too, in the form of a line bundle—see [53, Section4] and the notes for 8.3.

4.2: Injective envelopes of unital operator spaces were first considered byHamana in [184] in the late 1970s; many of these proofs are valid for generaloperator spaces. He also studied injective envelopes of operator systems in sev-eral of his papers (see reference list), and gave various excellent applications.Ruan studied the injective envelope of a general operator space in [371]. Hamanaclaims some of the same results at a similar date; he refers to some conferencetalks which he gave, and in [189,193] he refers to [188] which we have not seen.Some of this was published in [190, 191], and finally in [193]. We have followedthe presentation from [189] more closely than that of [371] (although they aresimilar). Injective envelopes have been used to study nonselfadjoint operator al-gebras for a long time (e.g. see [69, 40]). The fact that an injective (envelope ofa) C∗-algebra (or approximately unital operator algebra) is a unital C∗-algebra,we could only find in [65]. The result 4.2.9 appears in [61]. In the C∗-algebracase this result also follows from 1.3.12. If we drop the hypothesis in 4.2.9 thatB = P (A) is a subalgebra of A (we still require P (1A) = 1A), then an identicalproof shows that P (a1P (a2)) = P (P (a1)a2) = P (P (a1)P (a2)) for all a1, a2 ∈ A.Such results fail for general Banach algebras. For work on ‘separable injectivity’see [367], for example. In [365], Robertson classifies injective Hilbertian spaces.

4.3: The result 4.3.1 was proved by Arveson in several important casesin [21, 22]. The general case of 4.3.1, the name ‘C∗-envelope’, and 4.3.5 and4.3.6 in the unital case, are from [184]. The general case of 4.3.5 was observedin [72]. The observation in Example 4.3.7 (2) has been put to effect by Zarikian,who has developed software to compute C∗-envelopes, multipliers, and so on,for subspaces of Mn (see [441]). Proposition 4.3.8 appears in [310,445,446]; theproof of (2) given here is due to Pisier. The C∗-envelope is generally consider-ably smaller than the injective envelope, and is therefore often more tractable.Some interesting C∗-envelopes have been computed in [21, 22, 276, 284], for ex-ample, and in the literature on nonselfadjoint direct limit algebras. A fruitfuland important concept is that of a Shilov module, or Shilov representation, ofan operator algebra A (e.g. see [127, 281]). This is a Hilbert A-module which isan A-submodule of a Hilbert C∗

e (A)-module. Every boundary representation isa Shilov representation, but not vice versa.

Algebras with factorization have been studied by many authors, e.g. see [20,108,340,349]. The term convexly approximating in modulus we have seen used forfunction algebras in [127]. For more on Arveson’s noncommutative H∞ algebras(also known as finite maximal subdiagonal algebras) see, for example, [20, 275,270, 339]. Items 4.3.10 and 4.3.11 are from [61]. In the commutative case, theimportant results about generalized H∞ algebras are known to hinge on thecommutative version of 4.3.11. E.g. see [258] for a discussion of this point.

4.4: For 4.4.3 and 4.4.4, see the Hamana and Ruan references in the Notes to4.2. Result 4.4.6 appears in Ruan’s thesis [369], but was noticed independently byHamana (see Notes for 4.2), and Kirchberg in (we believe) the 1980s. The remark


in the last paragraph of 4.4.7 is from [445]. Youngson’s theorem is from [439].The proof here is different, but it is not better than the original, which useda useful fact due to Effros and Størmer [151]. There are related results at theBanach space level (e.g. see [151, 164, 379], and references therein). Corollary4.4.10 is also true with the word ‘completely’ removed [141]. The triple envelopeis due to Hamana (see 8.3, and references in the Notes to 8.3 for more historicaldetails). The triple envelope, or noncommutative Shilov boundary, is generallymuch smaller than I(X); for example if X is a subspace of a C∗-algebra A, thenT (X) may be realized as a subtriple of A∗∗ (see [58]).

The other unattributed results in this section are from [68], except for theobservations 4.4.13, and part of 4.4.12 (from [53]). The use of essential idealsin 4.4.11 and 4.4.12, was suggested by [163] (see also the proof of [194, Lemma3.2]). It is shown in [68] that I11(X) = M(C(X)) = LM(C(X)) = RM(C(X)),and similarly for I22(X) and D(X) (in the notation of 4.4.2 and 4.4.11).

Neal and Russo have recently given remarkable intrinsic characterizations, upto complete isometry, of one-sided ideals of C∗-algebras, TROs, and C∗-algebrasas operator spaces with certain affine geometric properties. See [289,290].

4.5: The multipliers discussed in this section have a complicated history, inpart because three of the papers concerned did not appear in the journals theywere first submitted to. Kirchberg (e.g. see [229–231]), and K. H. Werner [425],had considered multipliers of special classes of operator spaces or operator sys-tems. Around 1998, W. Werner considered one-sided multipliers of nonunital op-erator systems (c.f. [427]). Using the injective envelope, he proved the analogueof some parts of 4.5.2, 4.5.8, and 4.5.9 for such multipliers. Hamana in [187],and Frank and Paulsen in [163], considered respectively the multiplier algebraM(A), and LM(A) and ‘local multiplier algebras’, in terms of the injective enve-lope. These authors considered the case when A is a C∗-algebra. In early 1999,independently to Werner, and inspired by the characterizations in Section 3.7of Chapter 3, Blecher introduced multipliers of general operator spaces [53], inorder to improve and unify the analoguous noncommutative characterizationtheorems. (We learned of Werner’s work on operator system multipliers in themiddle of 2000, but believed it to be only distantly related to the general op-erator space setting. In fact there is a direct connection. The point is that anoperator space X may be embedded in the selfadjoint subspace of the Paulsensystem with zero main diagonal entries. The latter nonunital operator systemfalls within the framework of his ‘nonunital system’ theory [426]. By makingappropriate choices, and applying Werner’s original theorem to certain 4 × 4matrices, one may recover Ml(X) and some of the implications in 4.5.2. Thesearguments may be found in the more recent paper [428].)

The paper [53] introduced the conceptual framework that Section 4.5 and4.6.1–4.6.8 fall within. This paper was written with an emphasis on the tripleenvelope (since this may be viewed as the ‘noncommutative Shilov boundary’),as opposed to the much larger injective envelope. For those familiar with C∗-modules this is a natural approach (see also [56] for a recent survey from the


perspective of C∗-modules of some topics in the second half of the present chap-ter). After the main conceptual advances in [53] were made, work also began onthe tidy alternative approach based directly on the injective envelope presentedin [68]. About a year later, the basic list of characterizations of operator spacemultipliers was completed in [57]. The route which we have followed in thesesections is a combination of the approaches of the last three papers mentioned.The equivalence of (i), (iv), and (v) in Theorem 4.5.2 is from [53], the equiva-lence with (iii) is from [68], whereas the equivalence with (ii) was proved in [57](although the later short proof presented here comes from [314]). In fact (ii)was inspired by Werner’s theorem discussed above, and is a noncommutativeanalogue of the pretty norm formula in 3.7.3 (iii). Because of the simplicity of4.5.2 (ii), it is easy to see that the class of left multipliers is closed with respectto most of the ‘usual constructions’, for example, quotients, tensor products,duality, interpolation, amplification (e.g. see [57, 73]).

Formula (4.11) is from [53]. As C∗-algebras, Mn(Al(X)) ∼= Al(Cn(X)), since

Mn(∆(Ml(X))) = ∆(Mn(Ml(X))) ∼= ∆(Ml(Cn(X))) = Al(Cn(X)),

the ‘∼=’ here following from 2.1.2 and (4.11). If X is a dual operator space thenthe reader may find in [73] the relation MI(Ml(X)) ∼= Ml(MI,J(X)) as dual op-erator algebras, for any cardinals I, J . A similar relation holds with Ml replacedby Al. These are valid for any operator space if I, J are finite.

The generalization in 4.5.4 of Paulsen’s trick is from [440]. The statement inTheorem 4.5.5 is from [53], but the proof here, centered on the relation (4.12),is from [68]. Similarly, for 4.5.7–4.5.13: these appear in [53], but many of theproofs here are from [68]. Also, 4.5.11 is closely related to [163]. Indeed someproofs in [68] were inspired by some of Frank and Paulsen’s techniques in [163],and conversely [68] shows that some of those authors results on multipliers of C∗-algebras in terms of the injective envelope, fit harmoniously within the injectiveenvelope approach to multipliers on general operator spaces. One may recover,for example, the fact that LM(A) ⊂ I(A) by combining 4.5.11 with (4.12). Thedefinition of a left M -projection P in 4.5.14 may be restated informally as thestatement that X is a ‘column sum’ of P (X) and (I − P )(X). Items 4.5.14 and4.5.15 are from [57, 440]. See [53, 56, 57, 73], for example, for further theory ofone-sided multipliers and adjointable maps of operator spaces, and computationof these multiplier algebras for certain classes of operator spaces.

The first ‘noncommutative Banach–Stone theorem’, for surjective isometriesbetween C∗-algebras, is due to Kadison [212]. A good survey of isometries withmore recent proofs may be found in [158]. Arveson in [21, 22] proved varioustheorems of this type for unital completely isometric maps between certain op-erator algebras. Once Hamana had established the existence of the C∗-envelopein general, it was fairly obvious that Arveson’s techniques showed that unitalsurjective completely isometric maps between general unital operator algebrasare multiplicative (see [40, 144]). Isometries between general nonselfadjoint op-erator algebras were studied in [196, 278, 14], for example. We do not have the


space to list all the papers on the subject of isometries between particular classesof operator algebras. There are also ‘Banach–Stone theorems’ for algebras withone-sided identity, such as ideals in C∗-algebras [55]; and further generalizationsmay be found in [219]. The case of nonsurjective complete isometries was dis-cussed in [58, 59, 61]. See also [10, 96, 86], for example. Kirchberg has some deepcharacterizations of certain classes of maps in his papers cited here. Junge, Ruan,and Sherman have recently characterized complete isometries between noncom-mutative Lp spaces (see [210] and references therein).

The Banach–Stone theorem 4.5.13 may also be proved by extending u to asurjective complete isometry on a containing C∗-algebra (e.g. the injective orC∗- envelope), and then using 4.4.6 or a Banach–Stone theorem for C∗-algebras.See, for example, 8.3.13. This is related to the topic of ‘recovering’ the producton a unital operator algebra from its operator space structure, which is discussedfully in Section 6 of [56]. Indeed, Theorem 4.5.13 shows that an approximatelyunital operator algebra can have only one operator algebra product (compatiblewith the operator space structure and the cai).

If X is a unital operator space then Ml(X) ⊂ X completely isometrically, viathe map T → T (1). This follows easily from the fact that, as operator algebras,Ml(X) ∼= a ∈ C∗

e (X) : aX ⊂ X, which in turn follows from the first fact inthe second paragraph of 4.5.10.

4.6: The results in the first two thirds of this section are due to Blecher [53](although admittedly some of the proofs in early versions of that paper were toocomplicated). As we said in the Notes to 4.5, operator space multipliers wereintroduced in order to establish a ‘better route’ to the BRS and CES theorems;thus it was not surprising that 4.5.2 (ii) yields these theorems (as was observedby Blecher–Effros–Zarikian, and Paulsen, independently). Indeed 4.5.2 (ii) yieldsa characterization of operator algebras with a one-sided contractive approximateidentity (see [54]). Some of these ideas were taken further with the introductionin [218, 219] of quasimultipliers, a variant of the notion of one-sided multiplier.Perhaps the main result in [218] is a bijective correspondence between operatoralgebra products on an operator space X , and contractive quasimultipliers. Someother applications of the techniques in this section to operator modules may befound in [53], for example.

Results 4.6.9–4.6.11 are from [68], and 4.6.12 is from [57] (but see 3.11 in[186]). See [163] for the case of 4.6.10 when X is a C∗-algebra. In connectionwith 4.6.10, many papers studying injectivity in some other categories of operatormodules are referenced in the Notes to 3.6; and we also note that the analogueof 4.2.4 holds for operator bimodules, with most parts of the proof unchanged(see [68,217]). However, we would like to mention here the surprising and usefulresults of Frank and Paulsen [163]. They study a variant of injectivity where oneextends completely bounded module maps, but do not insist that the extensionhas the same norm. One of their results implies that if A is a C∗-algebra, andif u is any completely bounded A-module map on I(A) whose restriction to Ais IA, then u is the identity map (cf. 4.2.3). This has some lovely consequences,


such as the fact that if there exists a completely bounded A-module projectionfrom B(H) onto a ∗-subalgebra A, then A is injective in the usual sense.

Finally, we prove two characterizations of uniform algebras. First, we give theoriginal proof from [40] of Corollary 3.7.10. If A is a unital operator algebra andalso a minimal operator space, then it is completely isometrically isomorphic by4.2.8 to a subalgebra of the C∗-algebra I(Min(A)). However, we saw in 4.2.11 thatI(Min(A)) is linearly completely isometric, and hence ∗-isomorphic by 4.5.13,for example, to a C(K)-space. This gives the result. One may also characterizeuniform algebras as the unital operator algebras for which the multiplicationviewed as a map on A⊗A is completely contractive with respect to the minimaltensor product (see 1.5.1). Indeed together with Zeibig, we noted that since theminimal tensor product is ‘injective’ (see 1.5.1) we may extend the product toa completely contractive map M : I(A) ⊗min I(A) → I(A). However I(A) is aC∗-algebra as we saw in 4.2.8. By the rigidity property of the injective envelope, itfollows that M(1, x) = M(x, 1) = x for all x ∈ I(A). Since the Haagerup tensorproduct dominates the minimal tensor product (see 1.5.13), we have that Msatisfies the conditions of the BRS theorem (except associativity). As is pointedout in 4.6.3, the BRS theorem does not require the associativity hypothesis, sothat M is an operator algebra product on I(A). By the Banach–Stone theorem foroperator algebras, M is the usual product on I(A). Next we point out that since⊗min is ‘commutative’ (see 1.5.1), applying the last part of the above argumentto the reversed product on I(A), we see that the reversed product is the usualproduct. Thus I(A) is a commutative C∗-algebra.

4.7: Theorem 4.7.1 was proved in the Al(X) case, and 4.7.2 in the unitalcase, by Blecher–Effros–Zarikian and Blecher [57,54]. The general cases are newresults from a joint paper in preparation with Magajna. We thank Magajna forpermission to include these here. All the results in this section are from thesepapers, to which we refer for further applications, and other facts about w∗-topologies on operator spaces and algebras, such as the facts which follow: Notethat Al(X) need not even be an AW ∗-algebra if X is an operator space which isa dual Banach space. However in the latter case, it is true that any T ∈ Al(X) isw∗-continuous, and if A is a unital operator algebra with a Banach space predualthen ∆(A) is a W ∗-algebra. It is worth pointing out that although multiplierswork very well with respect to operator space duality, the closely related tripleand injective envelopes do not. Indeed if A is a dual operator algebra, then C∗

e (A)and I(A) need not be W ∗-algebras. For example, take A = U(X) (see 2.2.10),where X is the span of 1, x, x2 in C([0, 1]), or the span of the two generatorsof the free group F2 inside the reduced group C∗-algebra. In the former case, theC∗-envelope is M2(C([0, 1])), and so the injective envelope is M2(I(C([0, 1]))).However, it is well known folklore that I(C([0, 1])) is the Dixmier algebra, acommutative AW ∗-algebra which is not a W ∗-algebra (see 3.9.7 in [320]). Similararguments pertain in the free group case (see [185, Corollary 3.8]).

4.8: See [195] and [31] for surveys of the classical theory of M -ideals, andreferences to the extensive literature. Effros and Ruan introduced and studied


complete M -ideals of operator spaces in [148]. Assertions (1) and (2) of 4.8.5 inthe case of C∗-algebras are from [4,3,393], the general case of (1) and (2) is from[144], which relied on results from [393]. The argument given here for these itemsis adapted from [437], which is a simplification of the original argument from[393]. Arias and Rosenthal showed in [17] that complete M -ideals in operatoralgebras (and more general operator spaces) have in fact a quite strong kindof approximate identity. M -ideals in nonselfadjoint operator algebras have alsobeen considered, for example, in [10, 115, 144, 343]. One nice application of M -ideal theory is the following observation: if I is a closed two-sided ideal in anoperator algebra, and if both A and I have contractive approximate identities,then for each a ∈ A the distance from a to I is achieved. This follows from 4.8.5(1), and the ‘proximinality’ property that every M -ideal possesses [195].

The other results in this section are from [57, 73, 440]. See also [71, 442].One-sided M -ideals were intended as a generalization of classical M -ideals to‘noncommutative functional analysis’, where one might expect to have left andright ‘ideals’. What we call ‘one-sided M -ideals’ here were called ‘complete one-sided M -ideals’ in [57]. One-sided M -ideals and summands are stable underquotients, tensors, interpolation, duality, and so on. The reason for this is usuallythat multipliers work well under these constructions, as observed earlier. Thus,for example, using interpolation one can show that in contrast to the classicalcase, Lp spaces and their noncommutative variants can possess nontrivial one-sided M -ideals. Or as another example, one can show that one-sided M -idealsin a dual operator space X satisfy a Kaplansky density theorem. It is not hardto see that a left M -ideal in a left operator module is necessarily a submodule.See [73] for these, and very many other such results. Some other applications of‘one-sided M -structure’ may be found in [54, 55], for example.

For references on noncommutative convexity and ‘matricial extreme points’,see for example [2,152,153,156,157,223,266,422,429], and references therein. Non-commutative Choquet theory will play an exciting role in the future. A sample ofother interesting related topics which we had no space to include in this chapter:the question of the uniqueness of operator algebra structures on a given Banachalgebra, and the companion question of when contractive maps are automaticallycompletely contractive (e.g. see [21, 109,282,337], and references therein); ques-tions involving algebraic or linear isomorphism, or peturbations, of nonselfadjointalgebras (e.g. see [342] or Chapter 18 of [108]); ‘approximately finite operatoralgebras’ and related questions about K-theory (e.g. see [341, 353]); Popescu’snoncommutative disc algebra and H∞ (not connected to Arveson’s subdiago-nal algebras discussed earlier) and dilation theory (see [347,348], and referencestherein); the Muhly–Solel work on tensor algebras, correspondences and dilations(see [284–287], and references therein); and the operator space aspects of ‘inter-polation’ as studied by Agler, Cole, McCarthy, McCullouch, Paulsen, Wermer,and others (see [276] and references therein). Some results of the flavor in thischapter, but for algebras with a one-sided cai, or no kind of identity, may befound in [55, 60, 218,219].

5

Completely isomorphic theory ofoperator algebras

5.1 HOMOMORPHISMS OF OPERATOR ALGEBRAS

In the first four chapters, we studied operator algebras up to completely isometricisomorphism. There, the most important mappings between operator algebras(the morphisms in the language of categories) were the completely contractivehomomorphisms. In this chapter we turn to the study of operator algebras ‘upto isomorphism’ or ‘up to complete isomorphism’. Here we must deal with ho-momorphisms which are merely bounded or completely bounded. Theorem 5.1.2below, which is the main result of Section 5.1, clarifies the relationship betweencompletely bounded and completely contractive representations.

5.1.1 (Similarities) Let A be an operator algebra, let H be a Hilbert spaceand let π : A → B(H) be a bounded homomorphism. If S : H → H is anyinvertible bounded operator, then the mapping πS : A → B(H) defined by lettingπS(a) = S−1π(a)S for any a ∈ A, is a bounded homomorphism. Moreover, weclearly have ‖πS‖ ≤ ‖S−1‖‖S‖‖π‖. If further π is completely bounded, then πS

is completely bounded as well, with ‖πS‖cb ≤ ‖S−1‖‖S‖‖π‖cb. We say that tworepresentations π, ρ : A → B(H) are similar to each other if there exists S asabove such that ρ = πS (in which case π = ρS−1).

Theorem 5.1.2 (Paulsen) Let A be an operator algebra and let π : A → B(H)be a completely bounded representation. Then there exists a bounded invertibleoperator S : H → H such that πS : A → B(H) is completely contractive. Iffurther A is unital and π is unital, then this may be achieved with S satisfying‖S−1‖‖S‖ = ‖π‖cb.

Proof We first assume that A is unital and π : A → B(H) is unital. By therepresentation theorem for completely bounded maps (see 1.2.8) we may find aHilbert space K, a unital completely contractive representation θ : A → B(K)and linear maps V : H → K and W : K → H such that ‖V ‖ ‖W‖ ≤ ‖π‖cb andπ(a) = Wθ(a)V for any a ∈ A. We introduce two (closed) subspaces F ⊂ E ⊂ Kdefined by E = [θ(A)V (H)] and F = E ∩ Ker(W ). We claim that these twospaces are θ(A)-invariant. Indeed this is clear for E. To prove it for F , fix x in F ,and consider a sequence (xi)i≥1 in the linear span of θ(a)V ζ : a ∈ A, ζ ∈ H

196 Homomorphisms of operator algebras

converging to x ∈ F . For each i ≥ 1, we may find finite families (aik)k in A and

(ζik)k in H such that xi =

∑k θ(ai

k)V ζik. By assumption Wx = 0, and so

limi

∑k

π(aik)ζi

k = limi

∑k

Wθ(aik)V ζi

k = 0.

If a ∈ A then Wθ(a)x is the limit of Wθ(a)xi. For each i ≥ 1,

Wθ(a)xi =∑

k

Wθ(a)θ(aik)V ζi

k =∑

k

Wθ(aaik)V ζi

k

=∑

k

π(aaik)ζi

k = π(a)∑

k

π(aik)ζi

k

because π is a homomorphism. In the limit we have that Wθ(a)x = 0, that is,θ(A)x belongs to F .

Since E and F are θ(A)-invariant, by 3.2.2 the space L = E F is semi-invariant in the sense of 3.2.1. Let p be the projection from E onto L. By 3.2.2θ induces a completely contractive homomorphism θ : A → B(L) satisfying

θ(a)p = pθ(a), a ∈ A. (5.1)

Since θ is unital, the range of V is contained in E. Let T = pV : H → L. SinceWV = Wθ(1)V = π(1) = IH and W (IE − p) = 0, we have for ζ ∈ H that

ζ = WV ζ = W(pV ζ + (IE − p)V ζ

)= WTζ.

Thus W|LT = IH . Since W|L is one-to-one, we deduce that T is an isomorphism,with T−1 = W|L. Clearly ‖T ‖ ≤ ‖V ‖ and ‖T−1‖ ≤ ‖W‖, and hence ‖T ‖‖T−1‖is less than or equal to ‖π‖cb. We claim that

T−1θ(a)T = π(a), a ∈ A. (5.2)

Indeed, for a ∈ A and ζ ∈ H , we have

T−1θ(a)Tζ = T−1θ(a)pV ζ = T−1pθ(a)V ζ

by (5.1). Hence using the fact that T−1p is equal to W|E , we have

T−1θ(a)Tζ = Wθ(a)V ζ = π(a)ζ.

Since H and L are isomorphic (via T ), they are actually isometrically isomor-phic, and hence there exists a unitary U : H → L. We let S = U−1T : H → H ;this is an isomorphism and ‖S‖‖S−1‖ = ‖T ‖‖T−1‖ ≤ ‖π‖cb. By (5.2), we haveπ(a) = S−1U∗θ(a)US for any a ∈ A. Thus πS−1 : a → U∗θ(a)U is a completecontraction. This concludes the unital case.

Assume now that A is unital, but π(1) = IH . Then q = π(1) is an idempotent.Since every idempotent operator is similar to an orthogonal projection (this is

Completely isomorphic theory of operator algebras 197

an easy exercise), we may assume without loss of generality that q is selfadjoint.Let K = Ran(q). We have π = qπ(·)q, and in particular, K is π(A)-invariant.Moreover the induced ‘restriction’ of π to K is unital, and so by the first partof the proof, there is an isomorphism S : K → K such that the mapping πS iscompletely contractive from A into B(K). Then if S ′ = S ⊕ I : H → H , thenπS′ : A → B(H) is a complete contraction. Finally if A is nonunital, we canreduce to the unital case by means of the unitization A1 (see 2.1.11). Indeed ifπ : A → B(H) is any completely bounded homomorphism, and if π : A1 → B(H)is the extension of π obtained by letting π(1) = IH , then π is a completelybounded homomorphism as well. Hence π is similar to a complete contractionand this clearly implies that π itself is similar to a complete contraction.

5.1.3 (Similarity problems) Paulsen’s theorem 5.1.2 is the key to many im-portant similarity problems. We do not intend to discuss such problems in thebook, and refer the reader to [335], which is devoted to this topic, to [314], andto the Notes to Section 5.1. We merely mention that most similarity problemsreduce to the following question: for a fixed operator algebra A, is every boundedrepresentation π : A → B(H) similar to a contractive one? It clearly follows fromTheorem 5.1.2 that this property holds provided that every bounded homomor-phism on A is automatically completely bounded.

We note for later use that nuclear C∗-algebras satisfy the above property.Indeed, if A is a nuclear C∗-algebra and π : A → B(H) is a bounded homomor-phism, then π is completely bounded and ‖π‖cb ≤ ‖π‖2. This result was provedby Bunce [78] and Christensen [91] independently (see also [335, Chapter 7] fora proof). This applies in particular when A is a commutative C∗-algebra.

5.1.4 (Isomorphisms of operator algebras) Let A be a Banach space which isalso an algebra. We say that A is isomorphic to an operator algebra if there is anoperator algebra B ⊂ B(H), and a bounded invertible homomorphism ρ from Aonto B. In that case, ρ−1 : B → A also is a bounded homomorphism, and A is aC-Banach algebra (see A.4.1) with C ≤ ‖ρ‖2‖ρ−1‖. If ρ is an isometry we say thatA is isometrically isomorphic to an operator algebra. These classes have variousstability properties. Assume that A is isomorphic (resp. isometrically isomorphic)to an operator algebra. Clearly the same holds true for any subalgebra of A. Onthe other hand, if I is any index set, then the algebra ∞I (A) with coordinatewiseproduct is isomorphic (resp. isometrically isomorphic) to an operator algebra.Lastly, if J ⊂ A is any closed two-sided ideal, then the quotient algebra A/J isisomorphic (resp. isometrically isomorphic) to an operator algebra. Indeed thisfollows from the isometric version of Proposition 2.3.4.

5.1.5 (A counterexample) It is far from true that every Banach algebra isisomorphic to an operator algebra. One way to see this is to recall that opera-tor algebras are Arens regular (see 2.5.2 and 2.5.4), and to observe that Arensregularity is preserved under isomorphism. Thus a Banach algebra which is notArens regular cannot be isomorphic to an operator algebra. We refer the reader

198 Homomorphisms of operator algebras

to the Notes to Section 2.5 for further discussion of Arens regularity. In particu-lar A = 1

Z, equipped with convolution product, is not Arens regular, and hence

it is not isomorphic to an operator algebra. We will next provide a direct proofof this last fact without using Arens regularity.

Assume to the contrary that there exists, for some Hilbert space H , a boundedhomomorphism ρ : A → B(H) whose range B = ρ(A) is closed. Since A is unital,replacing H by [BH ] if necessary, we may assume that ρ(1) = IH . Let (en)n

denote the canonical basis of A = 1Z, and let T = ρ(e1) ∈ B(H). The element

e1 is invertible and en1 = en for any n ∈ Z. Since ρ is unital, this implies that

T is invertible and that T n = ρ(en) for any n ∈ Z. Thus we have ‖T n‖ ≤ ‖ρ‖for any n ∈ Z. By Nagy’s similarity theorem [403], this implies that T is similarto a unitary operator in B(H), that is, there is an invertible bounded operatorS : H → H such that S−1TS is a unitary. Applying von Neumann’s inequality forunitaries (the simplest case), we deduce that T is polynomially bounded. Thatis, there is a constant K ≥ 1 such that ‖f(T )‖ ≤ K sup|f(z)| : z ∈ C, |z| ≤ 1,for any polynomial f . Since f(e1) = ρ−1

(f(T )

)and en

1 = en for any n, we derivean estimate ∑

k≥0

|αk| ≤ C sup ∣∣∣∑

k≥0

αkzk∣∣∣ : z ∈ T

,

for any finite family (αk)k≥0 of complex numbers. This estimate implies that theFourier series of any f ∈ C(T) is absolutely convergent, a contradiction.

5.1.6 (Complete isomorphisms of operator algebras) Let A be an operator spacewhich is also an algebra. We say that A is completely isomorphic to an operatoralgebra if there is an operator algebra B ⊂ B(H), and a completely boundedinvertible homomorphism ρ : A → B such that ρ−1 is completely bounded. Thecase when ρ is a complete isometry corresponds to A being an abstract operatoralgebra in the sense of Section 2.1. Stability properties mentioned in 5.1.4 holdas well in the completely bounded setting. Namely if A is completely isomorphicto an operator algebra then the same holds for subalgebras of A, for any directsum ∞I (A) (I being an index set), and for quotients of A. The latter result usesthe full statement of Proposition 2.3.4.

An operator space structure (in the sense of 1.2.2) on an algebra A for whichA is completely isomorphic to an operator algebra, will be called an operatoralgebra structure on A. Recall that if A already has a norm, we demand that theoperator space structure is compatible with this one.

Proposition 5.1.7 Let A be a commutative C∗-algebra. Then an operator spacestructure on A is an operator algebra structure (for the usual product) if and onlyif it is completely isomorphic to the minimal one.

Proof Min(A) is the canonical structure of A (see 1.2.3) and our statementsays that up to complete isomorphism, this is actually the only one. Assumethat A is equipped with a operator space structure for which it is completelyisomorphic to an operator algebra B. We let ρ : A → B ⊂ B(H) be the corre-sponding completely bounded invertible homomorphism. By the last paragraph


of 5.1.3, ρ is completely bounded on Min(A). Hence ρ : Min(A) → B is a completeisomorphism by (1.10), which concludes the proof.

5.1.8 (A necessary condition) Let m : A ⊗ A → A be the linear mapping in-duced by the multiplication on an algebra A. If A is an operator algebra, thenm extends to a completely contractive map on A ⊗h A (see 2.3.1). Hence if A iscompletely isomorphic to an operator algebra, the mapping m extends to a com-pletely bounded map m : A ⊗h A → A. The key result for the study of operatoralgebras up to complete isomorphism will be Theorem 5.2.1, which asserts thatthe converse is true. In the next statement, we give a simple application of theabove necessary condition.

Proposition 5.1.9 Let X be an operator space and let CB(X) have its canon-ical matrix norms (1.6). Then CB(X) is completely isomorphic to an operatoralgebra if and only if X is completely isomorphic to a column Hilbert space.

Proof The ‘if’ part follows from (1.14). Conversely, if CB(X) is completelyisomorphic to an operator algebra, then we have a completely bounded multi-plication m : CB(X)⊗h CB(X) → CB(X) (see 5.1.8). By (1.35) we may regardX ⊗min X∗, and hence X ⊗ X∗, as a subspace of CB(X). Then we have

m(x′ ⊗ ϕ, x ⊗ ϕ′) = 〈ϕ, x〉x′ ⊗ ϕ′, x, x′ ∈ X, ϕ, ϕ′ ∈ X∗.

Fix x′ ∈ X and ϕ′ ∈ X∗ with ‖x′‖ = ‖ϕ′‖ = 1. For any x1, . . . , xn in X andϕ1, . . . , ϕn ∈ X∗, we have∣∣∣ n∑

k=1

〈ϕk, xk〉∣∣∣ =

∥∥∥ n∑k=1

〈ϕk, xk〉x′ ⊗ ϕ′∥∥∥

CB(X)=∥∥∥ n∑

k=1

m(x′ ⊗ ϕk, xk ⊗ ϕ′)∥∥∥

CB(X),

and by (1.40) the latter quantity is dominated by

‖m‖∥∥[x′ ⊗ ϕ1 · · · x′ ⊗ ϕn]

∥∥Rn(CB(X))

∥∥[x1 ⊗ ϕ′ · · · xn ⊗ ϕ′]t∥∥

Cn(CB(X)).

Since the embedding ϕ → x′⊗ϕ is a complete isometry from X∗ to the ‘subspace’X⊗minX∗ of CB(X), the norm of [x′⊗ϕ1 · · · x′⊗ϕn] in Rn(CB(X)) is equal tothe norm of [ϕ1 . . . ϕn] in Rn(X∗). Likewise the norm of [x1⊗ϕ′ · · · xn ⊗ϕ′]t inCn(CB(X)) is equal to the norm of [x1 · · · xn]t in Cn(X). Thus we have proved∣∣∣ n∑

k=1

〈ϕk, xk〉∣∣∣ ≤ ‖m‖

∥∥[ϕ1 · · · ϕn]∥∥

Rn(X∗)

∥∥[x1 · · · xn]t∥∥

Cn(X).

Thus the duality pairing extends to a (completely) bounded functional

X∗ ⊗h X −→ C.

By the CSPS theorem for bilinear forms (see 1.5.7 and 1.5.8), there exist a Hilbertspace H and completely bounded maps v : X → Hc and w : Hc → X∗∗ such thatwv : X → X∗∗ is the canonical embedding iX . Since iX is a complete isometry,‖x‖ = ‖wnvn(x)‖ ≤ ‖w‖cb‖vn(x)‖ for any n ≥ 1 and any x ∈ Mn(X). Thus vinduces a complete isomorphism between X and the range of v.

200 Completely bounded characterizations

5.1.10 (Interpolation) Let (A0, A1) be a compatible couple of Banach algebrasand for any θ ∈ (0, 1), let Aθ = [A0, A1]θ denote the Banach algebra obtainedfrom A0 and A1 by the complex interpolation method (2.3.6). Using functorialproperties of interpolation, one can easily deduce from Proposition 2.3.7 that:

(1) If A0, A1 are isomorphic to operator algebras, then Aθ is isomorphic toan operator algebra.

(2) If A0 and A1 are also operator spaces and if A0, A1 are completely isomor-phic to operator algebras, then Aθ equipped with its canonical operator spacestructure (see 1.2.30) is completely isomorphic to an operator algebra.

5.2 COMPLETELY BOUNDED CHARACTERIZATIONS

The following theorem, which is the main result of this section, is a completelyisomorphic counterpart of the BRS theorem (see 2.3.2).

Theorem 5.2.1 Let A be an operator space which is also an algebra, and letm : A⊗A → A denote the multiplication on A. Then A is completely isomorphicto an operator algebra if and only if m extends to a completely bounded mapm : A ⊗h A → A.

The proof of this will be completed in 5.2.9 below. We divide it into severalintermediate steps which provide interesting related results and generalizations.

5.2.2 (Polynomials in noncommuting variables) For any integer n ≥ 1, welet Pn be the algebra of all polynomials in n2 noncommuting variables withoutconstant terms. These variables will be denoted by Xij (1 ≤ i, j ≤ n). Let A be analgebra and assume that A is equipped with two operator space structures, whichwe denote by A1 and A2. For any F in Pn and any [aij ] in the algebra Mn(A),we let F

([aij ]

)∈ A be defined by substituting the aij ’s in for the variables. Next,

if N ≥ 1 is an integer, if F = [Fkl]1≤k,l≤N is any element of MN (Pn) (each entryFkl being in Pn), and if a = [aij ]1≤i,j≤n is in Mn(A), we define

F (a) =[Fkl

([aij ]

)]∈ MN (A).

Then given any positive number δ > 0 and F ∈ MN (Pn) as above, we set

‖F‖A1,A2,δ = sup‖F (a)‖MN (A2) : a ∈ Mn(A), ‖a‖Mn(A1) ≤ δ

.

This definition is not changed if the supremum is taken over all a ∈ Mn(A) with‖a‖Mn(A1) < δ. If A1 = A2 then ‖F‖A1,A2,δ will be denoted by ‖F‖A,δ, and wewill use the symbol A for A1 and A2. Moreover we let ‖F‖A = ‖F‖A,1 .

Lemma 5.2.3 Let B be an operator algebra, let I be an index set, and let C be asubalgebra of ∞I (B). Suppose that J ⊂ C is a closed two-sided ideal, and considerthe operator algebra A = C/J . Then for any n, N ≥ 1 and any F ∈ MN(Pn),we have ‖F‖A ≤ ‖F‖B .


Proof Let c ∈ Ball(Mn(C)). Since Mn

(∞I (B)

)= ∞I

(Mn(B)

)isometrically

(see 1.2.17), we may write c = (bλ)λ∈I with bλ ∈ Mn(B) and ‖bλ‖ ≤ 1. Since themultiplication on ∞I (B) is obtained by taking the multiplication of B coordi-natewise, we see that F (c) =

(F (bλ)

)λ

for any F ∈ Pn. Thus if F ∈ MN(Pn) wealso have F (c) =

(F (bλ)

)λ

and since MN

(∞I (B)

)= ∞I

(MN(B)

)isometrically,

we obtain that ‖F (c)‖ = supλ‖F (bλ)‖. Hence ‖F (c)‖ ≤ ‖F‖B. Taking thesupremum over c ∈ Ball(Mn(C)), we deduce that ‖F‖C ≤ ‖F‖B .

Fix a ∈ Mn(C/I), with ‖a‖ < 1. Let q : C → C/I = A be the canonicalquotient map. By definition of the quotient (see 1.2.14), we may find c ∈ Mn(C)such that ‖c‖ < 1 and qn(c) = a. Since q is a homomorphism, qn(F (c)) = F (a)for any F ∈ MN (Pn). Thus ‖F (a)‖ ≤ ‖F (c)‖, from which we deduce that‖F‖A ≤ ‖F‖C. The result therefore follows by the first part of this proof.

5.2.4 (Homogeneous polynomials) For any integer r ≥ 1, we consider theset Λr =

(1, . . . , n × 1, . . . , n

)r. For any α = ((i1, j1), . . . , (ir, jr)) ∈ Λr,we set αp = (ip, jp). We say that a polynomial F ∈ Pn is homogeneous ofdegree r if it lies in the linear span of Xα1 · · ·Xαr : α ∈ Λr. We say that amatrix valued polynomial F = [Fkl] ∈ MN(Pn) is homogeneous of degree r ifeach entry Fkl is homogeneous of degree r. Clearly any F ∈ MN (Pn) admitsa unique decomposition as a finite sum F =

∑r≥1 Fr , where Fr ∈ MN (Pn) is

homogeneous of degree r. If B is an operator algebra then for any b ∈ Mn(B)and any complex number z we have Fr(zb) = zrF (b). Thus

Fr(b) =12π

∫ 2π

0

e−irt F (eitb) dt .

This implies that ‖Fr‖B ≤ ‖F‖B for any r ≥ 1.

Proposition 5.2.5 Let B be an operator algebra and let A be an algebra equippedwith two operator space structures denoted by A1 and A2.(1) Let M, δ > 0. The following assertions are equivalent:

(i) For any integers n, N ≥ 1 and any F ∈ MN (Pn), we have

‖F‖A1,A2,δ ≤ M‖F‖B. (5.3)

(ii) There exist an index set I, a subalgebra C ⊂ ∞I (B), a closed two-sidedideal J ⊂ C, and an algebra isomorphism ρ : A → C/J such that

‖ρ : A1 −→ C/J‖cb ≤ δ−1 and ‖ρ−1 : C/J −→ A2‖cb ≤ M.

(2) Let M, δ > 0, and assume that for any n, N ≥ 1, (5.3) is satisfied for anymatrix valued homogeneous polynomial F ∈ MN (Pn). Then there exists anisomorphism ρ : A → C/J as in (ii) with

‖ρ : A1 −→ C/J‖cb ≤ 2δ−1 and ‖ρ−1 : C/J −→ A2‖cb ≤ M.


Proof We will first deduce (2) from (1). Assuming (1), let F =∑

r≥1 Fr bean arbitrary element of MN (Pn), with Fr homogeneous of degree r (see thediscussion in 5.2.4). Since Fr(a) = 2−rFr(2a) for any a ∈ Mn(A) and any r ≥ 1,we have ‖Fr‖A1,A2,δ/2 ≤ 2−r‖Fr‖A1,A2,δ for any r ≥ 1. Thus

‖F‖A1,A2,δ/2 ≤∑r≥1

‖Fr‖A1,A2,δ/2 ≤∑r≥1

2−r‖Fr‖A1,A2,δ .

Thus if (5.3) holds for any homogeneous polynomial, we conclude by 5.2.4 that

‖F‖A1,A2,δ/2 ≤∑r≥1

2−rM‖Fr‖B ≤ M‖F‖B .

This shows (5.3) with δ replaced by δ/2. Hence (2) follows from (1).Turning to (1), we now prove the easy fact that (ii) implies (i). Assume

(ii), and consider F ∈ MN (Pn) and a ∈ Mn(A), with ‖a‖Mn(A1) ≤ δ. Then‖ρn(a)‖Mn(C/J) ≤ 1, and so Lemma 5.2.3 ensures that ‖F (ρn(a))‖ ≤ ‖F‖B .

Since ρ is a homomorphism, we have F(ρn(a)

)= ρN

(F (a)

). Hence

‖F (a)‖MN (A2) ≤ ‖ρ−1 : C/J −→ A2‖cb ‖F (ρn(a))‖ ≤ M‖F‖B .

Taking the supremum over a, we obtain (5.3).We now assume (i) and shall prove (ii), which is the main implication. Let

Σ1 = a ∈ K(A1) : ‖a‖ ≤ δ, let Σ2 be the closed unit ball of K(B), and letI = Σ2

Σ1 be the set of all functions from Σ1 into Σ2. If a (resp. b) is an elementof Σ1 (resp. Σ2), we denote its entries by aij (resp. bij), i, j ≥ 1. It will beconvenient to regard ∞I (B) as the space of bounded functions from I into B.For any a ∈ Σ1 and any integers i, j ≥ 1, we define such a function fa

ij : I → Bby letting fa

ij(λ) = λ(a)ij . We let V be the subalgebra of ∞I (B) generated byall the fa

ij , with a ∈ Σ1 and i, j ≥ 1. Let C = V . Our goal is to show that wecan define a completely bounded homomorphism q : C → A2 such that

q(faij) = aij , a ∈ Σ1, i, j ≥ 1. (5.4)

To this end, let N ≥ 1 be an integer and let v = [vkl] ∈ MN (V ). Accordingto the definition of V , there exist pairwise distinct a(1), . . . , a(r) in Σ1, and aninteger m ≥ 1, such that each vkl lies in the algebra generated by the finite set

fa(p)ij : 1 ≤ p ≤ r, 1 ≤ i, j ≤ m

.

Let n = mr. Here it is convenient to denote the generators of Pn as Xpqij , for

1 ≤ i, j ≤ n and 1 ≤ p, q ≤ m. Thus we regard Mn(Pn) as consisting of r2

‘block matrices’ each of size m × m. With this notation we see that there existpolynomials Fkl ∈ Pn only depending on the variables

[X11ij ] ⊕ · · · ⊕ [Xrr

ij ]

[X11ij ]

. . .[Xrr

ij ]

,


such that vkl = Fkl

([fa(1)

ij ] ⊕ · · · ⊕ [fa(r)ij ]

). For 1 ≤ k, l ≤ N and λ ∈ I,

vkl(λ) = Fkl

([λ(a(1))ij

]⊕ · · · ⊕

[λ(a(r))ij

]). (5.5)

The reader may verify this algebraic relation by substituting in an uncomplicatedsample polynomial v. We let F = [Fkl] ∈ MN(Pn), and we observe that

‖F‖B = sup∥∥F (

[β(1)] ⊕ · · · ⊕ [β(r)])∥∥

MN (B)

,

where the supremum runs over all β(1), . . . , β(r) ∈ Mn(B) such that∥∥[β(1)] ⊕ · · · ⊕ [β(r)]∥∥

Mn(B)= sup

p‖β(p)‖Mm(B) ≤ 1.

We claim that‖v‖ = ‖F‖B. (5.6)

Indeed the isometric identification MN

(∞I (B)

)= ∞I

(MN(B)

)yields

‖v‖ = sup∥∥v(λ)

∥∥MN (B)

: λ ∈ I

= sup∥∥F

([λ(a(1))ij ] ⊕ · · · ⊕ [λ(a(r))ij ]

)∥∥MN (B)

: λ ∈ I

by (5.5). However the last quantity also equals

sup∥∥F (

[b(1)ij ] ⊕ · · · ⊕ [b(r)ij ])∥∥

MN (B): b(1), · · · , b(r) ∈ Ball(K(B))

.

The claim then follows by the preceding calculation.For any 1 ≤ p ≤ r, let α(p) ∈ Mm(A1) be the truncation of a(p) defined by

α(p)ij = a(p)ij for 1 ≤ i, j ≤ m. Then∥∥α(1) ⊕ · · · ⊕ α(r)∥∥

Mn(A1)≤ sup

p‖a(p)‖ ≤ δ,

hence it follows from our assumption (5.3) and from (5.6) that∥∥∥F ([a(1)ij ] ⊕ · · · ⊕ [a(r)ij ]

)∥∥∥MN (A2)

≤ M‖v‖.

This shows that (5.4) defines a bounded homomorphism q from V into A2, andthat passing to the closure, we have∥∥q : C −→ A2

∥∥cb

≤ M.

Let n ≥ 1 be an integer and let a ∈ Mn(A1), with ‖a‖ = δ. We regard a as anelement of Σ1 in the obvious way, and we let v = [fa

ij ] ∈ Mn(V ). Then ‖v‖ ≤ 1and qn(v) = a. This shows that q is onto and that if we let J = Ker(q), thecorresponding inverse mapping ρ from A into C/J is completely bounded on A1,with ‖ρ : A1 → C/J‖cb ≤ δ−1. This shows (ii).


5.2.6 Let X be an operator space, let H be a Hilbert space, and let v be acompletely bounded map from X into B(H). For any r ≥ 1, we let

v • · · · • v : X ⊗ · · · ⊗ X −→ B(H)

be the linear mapping on the r-fold tensor product X⊗· · ·⊗X taking x1⊗· · ·⊗xr

to v(x1) · · · v(xr) for any x1, . . . , xr ∈ X . By 1.5.8, this mapping extends to acompletely bounded map on X ⊗h · · · ⊗h X , with ‖v • · · · • v‖cb ≤ ‖v‖r

cb.

Lemma 5.2.7 Let X be an operator space, let r, N ≥ 1 be two integers, and letz ∈ MN(X ⊗ · · · ⊗ X), the tensor product having r factors. Then

‖z‖MN(X⊗h···⊗hX) = sup∥∥(IMN ⊗ (v • · · · • v))(z)

∥∥MN (B(H))

,

where the supremum runs over all separable Hilbert spaces H and all completelycontractive maps v : X → B(H).

Proof That the left-hand side is greater than the right was observed in 5.2.6.Conversely, let z ∈ MN (X ⊗ · · · ⊗ X), and let j : X ⊗h · · · ⊗h X → B(H0) bea completely isometric embedding of the r-fold Haagerup tensor product of X ,for some Hilbert space H0. Let Hr = H0. By the CSPS theorem (see 1.5.7 and1.5.8), there exist Hilbert spaces H1, . . . , Hr−1 and completely contractive mapsvk : X → B(Hk, Hk−1), 1 ≤ k ≤ r, such that j(x1 ⊗ · · · ⊗ xr) = v1(x1) · · · vr(xr)for any x1, . . . , xr ∈ X . Let H = Hr ⊕ Hr−1 ⊕ · · · ⊕ H0 be the Hilbertian directsum of the Hk’s, and let v : X → B(H) be defined by letting

v(x)(ζr, . . . , ζ0

)=(0, vr(x)(ζr), vr−1(x)(ζr−1), . . . , v1(x)(ζ1)

)for any x ∈ X and ζk ∈ Hk. It is plain that v is completely contractive. Moreoverfor any x1, . . . , xr ∈ X and any ζ ∈ Hr = H0, we have

v(x1) · · · v(xr)(ζ, 0, · · · , 0

)=(0, . . . , 0, v1(x1) · · · vr(xr)(ζ)

)=(0, . . . , 0, j(x1 ⊗ · · · ⊗ xr)(ζ)

)Since j is a complete isometry, we deduce that v • · · · • v is a complete isometryon the r-fold Haagerup tensor product of X , and hence

‖z‖h =∥∥(IMN ⊗ (v • · · · • v))(z)

∥∥MN (B(H))

.

Let (pt)t ⊂ B(H) be a bounded net of finite rank projections converging to theidentity in the strong operator topology. Then for any T1, . . . , Tr ∈ B(H), thenet of operators ptT1pt · · · ptTrpt converges to T1 · · ·Tr in the WOT. Thus lettingvt = ptv(·)pt for any t, we see that

‖z‖h = supt

∥∥(IMN ⊗ (vt • · · · • vt))(z)∥∥

MN (B(H))

.

Since each vt is completely contractive and acts on a finite-dimensional (henceseparable) Hilbert space, this yields the result.


Theorem 5.2.8 Let A be an algebra, with r-fold multiplication mapping denotedby mr : A ⊗ · · · ⊗ A → A for any r ≥ 1. As usual, we let m = m2. We assumethat A is equipped with two operator space structures denoted by A1 and A2.(1) Assume that there exist two constants C ≥ 1 and K > 0 such that∥∥∥mr : A1 ⊗h · · · ⊗h A1 −→ A2

∥∥∥cb

≤ CKr−1 (5.7)

for any r ≥ 1. Then there exist an operator algebra D and an algebra iso-morphism ρ : A → D such that

‖ρ : A1 −→ D‖cb ≤ 2K and ‖ρ−1 : D −→ A2‖cb ≤ CK−1.

(2) If the identity mapping IA is completely bounded from A1 into A2, and if mextends to a completely bounded map m : A1 ⊗h A2 → A2, then there existsan algebra isomorphism ρ : A → D from A onto some operator algebra D,such that ρ : A1 → D and ρ−1 : D → A2 are completely bounded.

Proof (1) Assume (5.7). By 5.2.5 (2), it suffices to show that (5.3) holds forhomogeneous polynomials with B = B(2), δ = K−1, and M = CK−1. Letn, N, r ≥ 1 and let F ∈ MN (Pn) be a homogeneous polynomial of degree r.Using notation from 5.2.4 and the identification MN(Pn) = MN ⊗Pn, we write

F =∑

α∈Λr

λα ⊗ Xα1 · · ·Xαr , with λα ∈ MN .

Let a = [aij ] ∈ Mn(A1) with ‖a‖ ≤ δ = K−1, and let a′ = Ka. Then

F (a) =∑

α∈Λr

λα ⊗ aα1 · · · aαr

=(IMN ⊗ mr

)( ∑α∈Λr

λα ⊗ aα1 ⊗ · · · ⊗ aαr

).

Hence by (5.7), we have

‖F (a)‖MN (A2) ≤ CKr−1∥∥∥ ∑

α∈Λr

λα ⊗ aα1 ⊗ · · · ⊗ aαr

∥∥∥MN (A1⊗h···⊗hA1)

≤ CK−1∥∥∥ ∑

α∈Λr

λα ⊗ a′α1 ⊗ · · · ⊗ a′

αr

∥∥∥MN (A1⊗h···⊗hA1)

.

It follows from Lemma 5.2.7 that the norm on the right is equal to

sup∥∥F (

[v(a′ij ])∥∥

MN (B(2))

∣∣∣ v : A1 → B(2), ‖v‖cb ≤ 1.

This quantity is less than or equal to ‖F‖B(2), since ‖a′‖Mn(A1) ≤ 1. Thus‖F (a)‖MN (A2) ≤ CK−1‖F‖B(2). This completes the proof of (1).


(2) If C = ‖IA : A1 → A2‖cb and K = ‖m : A1 ⊗h A2 → A2‖cb are bothfinite, then by induction and the ‘functoriality’ of the Haagerup tensor product,we see that ‖mr : A1 ⊗h · · · ⊗h A1 ⊗h A2 → A2‖cb ≤ Kr−1 for any r ≥ 2. Hence(5.7) holds for any r ≥ 1. Thus (2) follows from (1).

5.2.9 (Proof of Theorem 5.2.1) The ‘only if’ part was observed in 5.1.8. The‘if’ part is a special case of 5.2.8 (2), obtained by taking A = A1 = A2. In fact,it follows from Theorem 5.2.8 (1) that if m : A⊗h A → A is completely boundedand m = 0, then there exists an operator algebra B and a completely boundedhomomorphism ρ : A → B such that ‖ρ‖cb ≤ 2‖m‖cb and ‖ρ−1‖cb ≤ ‖m‖−1

cb .

5.2.10 (A completely isometric characterization) Let A be an operator spacewhich is also an algebra. Applying 5.2.5 (1) with A = A1 = A2 we find thatif B is an operator algebra, then A is completely isomorphic to a quotient of asubalgebra of some direct sum ∞I (B) if and only if there exist constants M, δ > 0such that ‖F‖A,δ ≤ M‖F‖B for any integers n, N ≥ 1 and any F ∈ MN (Pn).Moreover A is (completely isometrically isomorphic to) a quotient of a subalgebraof a direct sum ∞I (B) if and only if this holds with M = δ = 1, that is:

‖F‖A ≤ ‖F‖B, n, N ≥ 1, F ∈ MN(Pn). (5.8)

Thus A is an operator algebra if and only if (5.8) holds true with B = B(2).Of course this may be seen as a ‘von Neumann’s inequality’ characterization ofoperator algebras.

5.2.11 (Isomorphic characterizations) Let P [Z1, . . . , Zn] denote the algebraof all polynomials in n noncommuting variables Z1, . . . , Zn. We may obviouslyregard P [Z1, . . . , Zn] as a subalgebra of Pn by identifying Zk with Xkk for any1 ≤ k ≤ n. Let A be an algebra which is also a Banach space, and let δ > 0 be apositive number. Restricting the definitions from 5.2.2 to diagonal matrices, wedefine for any F ∈ P [Z1, . . . , Zn]:

‖F‖A,δ = sup‖F (a1, . . . , an)‖A : ak ∈ A, sup

k‖ak‖ ≤ δ

.

Arguing as in the proofs of 5.2.3 and 5.2.5, we find that if B is an operatoralgebra, then A is isomorphic to a quotient of a subalgebra of some direct sum∞I (B) if and only if there exist two constants M, δ > 0 such that

‖F‖A,δ ≤ M‖F‖B,1, n ≥ 1, F ∈ P[Z1, . . . , Zn]. (5.9)

In particular (5.9) holds with B = B(2) if and only if A is isomorphic to anoperator algebra. Moreover we see that (5.9) holds with M = δ = 1 (resp. andwith B = B(2)) if and only if A is isometrically isomorphic to a quotient of asubalgebra of some direct sum ∞I (B) (resp. to an operator algebra).

We give two more results concerning the isomorphic theory which are straight-forward applications of our operator space techniques.


Corollary 5.2.12 (Varopoulos) Let A be a Banach space which is also an al-gebra. Then A is isomorphic to an operator algebra if and only if there is aconstant K > 0 such that for any ϕ ∈ Ball(A∗) and any r ≥ 2, there ex-ist Hilbert spaces C = H0, H1, . . . , Hr−1, Hr = C and bounded linear mapsvk : A → B(Hk, Hk−1), 1 ≤ k ≤ r, such that ‖v1‖ · · · ‖vr‖ ≤ Kr and

〈ϕ, a1a2 · · · ar〉 = v1(a1) v2(a2) · · · vr(ar), a1, . . . , ar ∈ A.

Proof Indeed according to (1.10) and (1.12), A is isomorphic to an operatoralgebra if and only if it satisfies the conclusion of 5.2.8 (1), with A1 = Max(A)and A2 = Min(A). By the latter proposition together with (1.9), this holds trueif and only if, for some constant K, we have∥∥ϕ mr : Max(A) ⊗h · · · ⊗h Max(A) −→ C

∥∥cb

≤ Kr

for any r ≥ 2 and ϕ ∈ Ball(A∗). The result therefore follows by applying theCSPS theorem (see 1.5.8) to ϕ mr.

Corollary 5.2.13 (Tonge) Let A be a Banach space which is also an algebraand assume that the multiplication m : A⊗A → A is bounded from A⊗g2 A intoA (see (A.6)). Then A is isomorphic to an operator algebra.

Proof According to (1.10) and (1.46), our assumption is equivalent to the mapm : Max(A) ⊗h Min(A) → Min(A) being completely bounded. Hence the resultfollows by applying 5.2.8 (2), with A1 = Max(A) and with A2 = Min(A).

5.2.14 As a variant of the above statement, we note that if A is a Banach spacewhich is also an algebra, then Min(A) is completely isomorphic to an operatoralgebra if and only if its multiplication m is bounded from A ⊗γ2 A into A (see(A.5) for a definition). Indeed by (1.44), the boundedness of m : A ⊗γ2 A → Ais equivalent to the complete boundedness of m : Min(A) ⊗h Min(A) → Min(A).Hence the result follows from 5.2.1. See also 5.4.11 below for a related result.

5.2.15 (Multiplication and interpolation) As in 5.1.10 (2), let (A0, A1) be acompatible couple of Banach algebras with operator space structures, and assumethat A0 and A1 are both completely isomorphic to operator algebras. For any0 ≤ θ ≤ 1, let mθ : Aθ ⊗h Aθ → Aθ be the product map on Aθ. Then

‖mθ‖cb ≤ ‖m0‖1−θcb ‖m1‖θ

cb, θ ∈ (0, 1).

To see this, let n ≥ 1 be any integer and let

σn,θ : Mn(Aθ) × Mn(Aθ) −→ Mn(Aθ)

be the multiplication mapping on n × n matrices, regarded as a bilinear map.Then by (1.24) and the interpolation theorem for multilinear maps [33, Section4.4], we have ‖σn,θ‖ ≤ ‖σn,θ‖1−θ ‖σn,θ‖θ. Passing to the limit when n → ∞yields the result.


Theorem 5.2.16 Let A be an algebra which is also a dual operator space andassume that A is completely isomorphic to an operator algebra. If the producton A is separately w∗-continuous, then there exist a dual operator algebra B anda w∗-continuous invertible homomorphism ρ : A → B such that ρ and ρ−1 arecompletely bounded.

Proof This is a variant of the proof of Theorem 2.7.9. By hypothesis there isan operator algebra C and a complete algebra isomorphism σ : C → A. By 2.5.4,C is Arens regular, and hence A is Arens regular as well. Passing to second du-als, σ∗∗ : C∗∗ → A∗∗ is a complete isomorphism and also a homomorphism (thelatter by simple facts in Section 2.5). By assumption, A = X∗ completely iso-metrically for some operator space X . Let Q : A∗∗ → A be the adjoint mappingof the canonical embedding iX : X → X∗∗. Then Lemma 2.7.12 ensures thatQ is a homomorphism. Thus the mapping Qσ∗∗ : C∗∗ → A is a w∗-continuoushomomorphism. Consequently, its kernel J is a w∗-closed ideal of C∗∗, and hencethe resulting quotient algebra B = C∗∗/J is a dual operator algebra by 2.7.3and 2.7.11. The mapping τ : B → A induced by Qσ∗∗ is a w∗-continuous isomor-phism. Since Q is a complete quotient map, it is clear that τ and its inverse arecompletely bounded, which concludes the proof.

We end this section with a few words on operator modules. In Chapter 3, weonly considered the ‘completely isometric’ version of the theory. As mentioned inthe introduction to that chapter one may also define a ‘completely isomorphic’version of operator bimodules (and matrix normed bimodules). In this direction,the appropriate version of the ‘CES theorem’ 3.3.1 is the following result.

Theorem 5.2.17 Let A and B be operator algebras, and suppose that X is anA-B-bimodule. The module actions on X extend to completely bounded maps

A ⊗h X −→ X and X ⊗h B −→ X

if and only if there exists a Hilbert space H, a completely bounded linear mapΦ: X → B(H), and completely bounded homomorphisms θ : A → B(H) andπ : B → B(H) which all are complete isomorphisms onto their respective ranges,and such that

θ(a)Φ(x) = Φ(ax) and Φ(x)π(b) = Φ(xb),

for all a ∈ A, b ∈ B and x ∈ X.

Proof The ‘if’ part is easy, as in 3.3.1; we only prove the more difficult ‘only if’implication. We let u : A ⊗h X → X and v : X ⊗h B → X denote the mappingsinduced by the module actions. Consider the algebra D of matrices[

a x0 b

]for a ∈ A, b ∈ B, x ∈ X , as discussed in 3.3.4. One may give the algebra D anyoperator space structure which retains the original matrix norms on the three


nonzero corners, but for specificity we will use the structure A ⊕∞ X ⊕∞ B. Itis a simple matter to check that the multiplication on D is completely bounded.This follows because, for example,∣∣∣∣∣∣∣∣[a1 x1

0 b1

] [a2 x2

0 b2

]∣∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣[a1a2 a1x2 + x1b2

0 b1b2

]∣∣∣∣∣∣∣∣= max‖a1a2‖, ‖a1x2 + x1b2‖, ‖b1b2‖,

which is clearly dominated by

2 max‖a1‖, ‖v‖cb‖x1‖, ‖b1‖ max‖a2‖, ‖u‖cb‖x2‖, ‖b2‖

≤ K

∣∣∣∣∣∣∣∣[a1 x1

0 b1

]∣∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣∣[a2 x2

0 b2

]∣∣∣∣∣∣∣∣ ,for the constant K = 2 max1, ‖v‖cb max1, ‖u‖cb. Thus we may appeal toTheorem 5.2.1 to obtain a complete isomorphism ρ of D onto some concreteoperator subalgebra of B(H) say. Letting

θ(a) = ρ

([a 00 0

]); π(b) = ρ

([0 00 b

]); Φ(x) = ρ

([0 x0 0

]),

we obtain the desired result.

5.2.18 (Dual modules) Assume that A, B are dual operator algebras, thatX is a dual operator space and that the module actions A × X → X andX × B → X are both completely bounded and separately w∗-continuous. Thenthe conclusions of Theorem 5.2.17 hold with the additional property that themappings θ, π and Φ are w∗-continuous. Indeed with these assumptions, thealgebra D in the above proof is a dual operator space and the product on Dis separately w∗-continuous. Hence Theorem 5.2.16 ensures that the homomor-phism ρ : D → B(H) in the proof of 5.2.17 can be taken to be w∗-continuous.Then the mappings θ, π and Φ defined from ρ are clearly w∗-continuous. Notethat in that case, their ranges are w∗-closed, so that θ(A), π(B) are dual opera-tor algebras and Φ(X) is a dual operator space. Moreover the inverse mappingsθ−1 : θ(A) → A, π−1 : π(B) → B and Φ−1 : Φ(X) → X are all w∗-continuous.These facts follow from a simple variant of A.2.5.

5.3 EXAMPLES OF OPERATOR ALGEBRA STRUCTURES

The aim of this section (and of Section 5.5) is to ‘get our hands dirty’ withseveral very concrete examples; and thus to give the reader a feel for the kind ofgeometrical features of the matrix norms which determine whether the algebrais completely isomorphic to an operator algebra or not.

5.3.1 (Multiplication on p) For any 1 ≤ p ≤ ∞, we equip the Banach spacep with the pointwise product. Namely, if x = (xk)k≥1 and y = (yk)k≥1 are

210 Examples of operator algebra structures

arbitrary elements of p, we let xy = (xkyk)k≥1. It is clear that this product iswell-defined and makes p a Banach algebra. It was proved by Davie [116] andVaropoulos [415] in the seventies that p is isomorphic to an operator algebrafor every 1 ≤ p ≤ ∞. We consider here the question of determining some, orall, corresponding operator algebra structures on p. We will obtain the Davie–Varopoulos result en route (e.g. see 5.3.5 below).

Let (en)n≥1 denote the canonical basis of p, and for any integer n ≥ 1 identifypn with the subspace of p spanned by e1, . . . , en. Then p

n is a subalgebra of p.Let mn : p

n ⊗ pn → p

n denote the corresponding product. We have

mn

( ∑1≤i,j≤n

aij ei ⊗ ej

)=

∑1≤i≤n

aiiei, aij ∈ C. (5.10)

If p is equipped with an operator space structure then we regard pn as equipped

with the induced structure. It clearly follows from Theorem 5.2.1 that the givenstructure on p is an operator algebra structure if and only if

supn

∥∥mn : pn ⊗h p

n −→ pn

∥∥cb

< ∞.

5.3.2 (The diagonal projection) We will often use the facts that Mn = Cn⊗hRn

and that S1n = M∗

n = Rn ⊗h Cn (see 1.5.14). If we think of Mn and S1n both as

n× n matrices in the usual way, then the identifications in the last sentence aregiven by the purely algebraic isomorphism between Mn and C

n⊗Cn taking Eij to

ei⊗ej. Via this last isomorphism, if Dn ⊂ Mn is the subspace of diagonal matrices(which we may also identify algebraically with C

n), (5.10) simply says that mn

corresponds to the canonical projection from Mn onto Dn. This projection isobviously completely contractive if Mn is given its usual structure Cn⊗hRn, andDn has the corresponding subspace structure ∞n . On the other hand, supposethat Mn is interpreted as the trace class matrices S1

n = Rn ⊗h Cn. In this case,in the predual we have the canonical diagonal embedding ∞n → Mn, which is acomplete isometry, and the adjoint of this embedding is the canonical projectionwhich we have identified with mn above. Thus in this case it follows that againmn is completely contractive, but now as a map Rn ⊗h Cn → (∞n )∗ = Max(1

n).

Proposition 5.1.7 shows that Min(∞) is, up to complete isomorphism, theonly operator algebra structure on ∞. It turns out that the situation for p = 1is radically different.

Proposition 5.3.3 Any operator space structure on 1 is an operator algebrastructure.

Proof We let Id denote the canonical embedding of 1 into 2. We shall firstprove that ∥∥Id : Min(1) −→ R ∩ C

∥∥cb

= 1, (5.11)

where R∩C is the Hilbert operator space defined in 1.2.24. Indeed let a1, . . . , an

be in Mm, for some m ≥ 1. Then by 1.2.21 we have

Completely isomorphic theory of operator algebras 211∥∥∥∑k

ak ⊗ ek

∥∥∥Mm(Min(1))

= sup∥∥∥∑

k

θkak

∥∥∥Mm

: θk ∈ C, |θk| ≤ 1

,

whereas ∑k

a∗kak =

12π

∫ 2π

0

(∑k

akeikt)∗(∑

k

akeikt)

dt .

Therefore,∥∥∥∑k

a∗kak

∥∥∥Mm

≤ 12π

∫ 2π

0

∥∥∥∑k

akeikt∥∥∥2

dt ≤∥∥∥∑

k

ak ⊗ ek

∥∥∥2

Mm(Min(1)).

Similarly we have ∥∥∥∑k

aka∗k

∥∥∥Mm

≤∥∥∥∑

k

ak ⊗ ek

∥∥∥2

Mm(Min(1)).

Hence (5.11) follows from (1.16).Let n ≥ 1 be an integer. By the discussion at the end of 5.3.2, we have∥∥mn : Rn ⊗h Cn −→ Max(1

n)∥∥

cb≤ 1. (5.12)

Combining with (5.11) applied twice, we deduce that∥∥mn : Min(1n) ⊗h Min(1

n) −→ Max(1n)∥∥

cb≤ 1.

This yields the result, by the discussion at the end of 5.3.1.

5.3.4 (The operator space Op) The Banach space p has a natural operatorspace structure, denoted by Op, which was introduced by Pisier (see [331] and[328]). Its definition is based on complex interpolation and on the fact thatp = [∞, 1] 1

pisometrically for any 1 ≤ p ≤ ∞. We let O∞ = Min(∞),

O1 = Max(1) and according to 1.2.30, we define

Op = [Min(∞), Max(1)] 1p

(5.13)

for 1 < p < ∞. The operator space O2 is usually denoted by OH . This operatorHilbert space has various remarkable properties regarding duality and interpola-tion, which are studied in great detail in [328]. In the sequel, the n-dimensionalversions of the above spaces will be denoted by Op

n (resp. OHn).

Corollary 5.3.5 For any 1 ≤ p ≤ ∞, Op is completely isomorphic to an oper-ator algebra.

Proof This clearly follows from Proposition 5.3.3 and 5.1.10 (2). An alternateproof is obtained by combining (5.14) below with Theorem 5.2.1.

212 Examples of operator algebra structures

5.3.6 (A counterexample) We show here that the BRS theorem 2.3.2 is notvalid if we remove the hypothesis that A has an identity of norm 1 or a cai.Indeed it follows from the proof of 5.3.3 that mn is completely contractive fromMax(1

n) ⊗h Max(1n) to Max(1

n). Since it is also completely contractive fromMin(∞n ) ⊗h Min(∞n ) to Min(∞n ), it follows from 5.2.15 that∥∥mn : Op

n ⊗h Opn −→ Op

n

∥∥cb

≤ 1. (5.14)

However if p is finite and n ≥ 2, then pn is not isometrically isomorphic to

an operator algebra. Indeed take n = 2 and assume that ρ : p2 → B(H) is an

isometric homomorphism, for some Hilbert space H . Then ρ(e1) and ρ(e2) aredisjoint contractive (and hence orthogonal) projections. Thus for any complexnumbers θ1, θ2, we have

‖θ1e1 + θ2e2‖ = ‖θ1ρ(e1) + θ2ρ(e2)‖ = max|θ1|, |θ2|,a contradiction. Note that e1 + e2 is an identity for Op

2, with norm 21/p.

Proposition 5.3.7 Let 1 ≤ p ≤ ∞, then:(1) Min(p) is completely isomorphic to an operator algebra if and only if p = 1

or p = ∞.(2) Max(p) is completely isomorphic to an operator algebra if and only if we

have 1 ≤ p ≤ 2.

Proof Suppose that Min(p) is an operator algebra structure on p, where1 < p < ∞. Let n ≥ 1 be an integer, and let un =

∑1≤k≤n ek ⊗ ek, regarded as

an element of pn ⊗ p

n. Then mn(un) =∑

1≤k≤n ek , whose norm in pn is equal to

n1/p. We deduce that there is a constant C > 0 (not depending on n) such that

n1/p ≤ C∥∥∥ n∑

k=1

ek ⊗ ek

∥∥∥Min(p

n)⊗hMin(pn)

.

By (1.44) and (A.5), we have∥∥∥ n∑k=1

ek ⊗ ek

∥∥∥Min(p

n)⊗hMin(pn)

≤∥∥∥ n∑

k=1

ek ⊗ ek

∥∥∥2

2n⊗pn

=∥∥Id : 2

n → pn

∥∥2.

If p ≥ 2, the right side of the above inequality is equal to 1. If p ≤ 2, it is equalto n

2p−1. This yields a contradiction and proves the ‘only if’ part of (1). The ‘if’

part follows from 5.3.3.We now turn to the proof of (2). Assume that 1 ≤ p ≤ 2 and let n ≥ 1 be an

integer. It follows from (5.12) that∥∥mn : Max(2n) ⊗h Max(2

n) −→ Max(1n)∥∥

cb≤ 1.

Since the identity mappings 1n → p

n and pn → 2

n are contractive, we deduceusing (1.12) that∥∥mn : Max(p

n) ⊗h Max(pn) −→ Max(p

n)∥∥

cb≤ 1.

Hence Max(p) is completely isomorphic to an operator algebra by 5.2.1.


Assume now that 2 < p ≤ ∞ and let q be the conjugate of p (p−1 + q−1 = 1).For any n ≥ 1, the norm of the identity mapping q

n → 1n is equal to n1/p. Hence

by (5.11) and (1.10) we have∥∥Id : Min(qn) −→ Rn ∩ Cn

∥∥cb

≤ n1/p.

By duality, we obtain that∥∥Id : Cn −→ Max(pn)∥∥

cb≤ n1/p and

∥∥Id : Rn −→ Max(pn)∥∥

cb≤ n1/p.

Here we used (1.30), (1.15), and (1.17). Thus∥∥Id ⊗ Id : Cn ⊗h Rn −→ Max(pn) ⊗h Max(p

n)∥∥

cb≤ n2/p.

Hence if Max(p) is an operator algebra structure on p, then there is a constantC > 0 such that ∥∥mn : Cn ⊗h Rn −→ Max(p

n)∥∥

cb≤ Cn2/p

for any n ≥ 1. Restricting to the diagonal of Cn ⊗h Rn = Mn (see 5.3.2), wereadily obtain that ∥∥Id : Min(∞n ) −→ Max(p

n)∥∥

cb≤ Cn2/p (5.15)

for any n ≥ 1. To reach a contradiction, we will use a ‘spin system’ in M2n ,that is, a finite sequence w1, . . . , wn of selfadjoint unitaries in M2n such thatwiwj + wjwi = 0 for any i = j. Such a sequence satisfies

∥∥∥ n∑k=1

tkwk

∥∥∥ ≤√

2( n∑

k=1

|tk|2)1/2

, t1, . . . , tn ∈ C; (5.16)

∥∥∥ n∑k=1

wk ⊗ wtk

∥∥∥M2n⊗minM2n

= n. (5.17)

E.g. see [337, p. 76] for the existence of such a sequence and a proof of (5.16)and (5.17). According to 1.2.21, the inequality (5.16) may be rephrased as

∥∥∥ n∑k=1

wk ⊗ ek

∥∥∥M2n (Min(2n))

≤√

2.

Hence ∥∥∥ n∑k=1

wk ⊗ ek

∥∥∥M2n (Min(q

n))≤

√2n

12− 1

p .

Let vn : Min(qn) → M2n be the linear mapping taking ek to wt

k for any k. Then

214 Examples of operator algebra structuresn∑

k=1

wk ⊗ wtk =

(IM2n ⊗ vn

)( n∑k=1

wk ⊗ ek

),

and so by (5.17) we have

n ≤ ‖vn‖cb

∥∥∥ n∑k=1

wk ⊗ ek

∥∥∥M2n (Min(q

n))≤

√2n

12− 1

p ‖vn‖cb.

On the other hand, each wk has norm 1, and so∑n

k=1 wk ⊗ ek has norm 1 inM2n(∞n ). By (5.15), this implies that∥∥∥ n∑

k=1

wk ⊗ ek

∥∥∥M2n (Max(p

n))≤ Cn2/p.

However the last norm is equal to ‖vn‖cb by (1.30), hence we finally obtain thatn ≤ C

√2n

12+ 1

p , which is impossible for p > 2.

Proposition 5.3.8 With pointwise product, R, C, and R ∩ C are completelyisomorphic to operator algebras.

Proof In the identification discussed in 5.3.2, we have Rn ⊗h Rn = (S2n)r by

1.5.14 (8). Since the projection onto diagonal matrices is contractive on S2n, it is

automatically completely contractive from (S2n)r onto Rn, and so it follows that

‖mn : Rn ⊗h Rn → Rn‖cb ≤ 1. This shows that R is completely isomorphic toan operator algebra. The argument for C is identical, and the result for R ∩ Cfollows at once.

5.3.9 (Schur product of matrices) Let H = 2 and let (ek)k≥1 denote itsusual basis. We let V : H → H ⊗2 H be the linear isometry defined by lettingV (ek) = ek ⊗ ek for any k ≥ 1. Given a, b ∈ B(H), we regard a⊗ b as an elementof B(H ⊗2 H) and we define the Schur product of a and b as

a ∗ b = V ∗(a ⊗ b)V. (5.18)

If we represent a and b by their (infinite) matrices [aij ]i,j≥1 and [bij ]i,j≥1 withrespect to the basis (ek)k≥1, then (5.18) can be written as

[aij ]i,j≥1 ∗ [bij ]i,j≥1 = [aijbij ]i,j≥1.

Indeed for any i, j ≥ 1, we have

〈(a ∗ b)ej , ei〉 =〈(a ⊗ b)V ej, V ei〉 = 〈(a ⊗ b)(ej ⊗ ej), (ei ⊗ ei)〉=〈aej , ei〉〈bej , ei〉 = aijbij .

It is clear from (5.18) that for any a, b ∈ B(H), we have

‖a ∗ b‖ ≤ ‖V ‖2‖a ⊗ b‖ = ‖a‖‖b‖.Thus equipped with the Schur product ∗, B(H) is a commutative Banach algebra.If the matrix [aij ]i,j≥1 representing a has a finite number of nonzero elements,then the same holds for the matrix [aijbij ]i,j≥1 representing a∗b. Hence compactoperators S∞(H) form an ideal of B(H) for the Schur product.


5.3.10 (Abstract Schur product) It is worthwhile to observe that the aboveconstruction is a special case of joint multiplication. If A, B are two algebras,then their algebraic tensor product A⊗B is an algebra with the product definedin (2.6). We showed in 2.2.2 that A ⊗min B with this product is an operatoralgebra if A and B are operator algebras. It follows from this fact, and fromthe functoriality of the minimal tensor product, that if A and B are operatorspaces and are completely isomorphic to operator algebras, then A ⊗min B withthe joint multiplication is completely isomorphic to an operator algebra.

It is easy to check that if we let A and B be equal to 2 equipped with itspointwise product (see 5.3.1), and if we identify 2⊗2 with finite rank operatorson H = 2 in the usual way, then the joint multiplication on 2 ⊗ 2 coincideswith the Schur product introduced in 5.3.9.

Theorem 5.3.11 Equipped with their natural operator space structure and theSchur product, B(H) and S∞(H) are completely isomorphic to operator algebras.

Proof The result for S∞(H) immediately follows from the discussion in 5.3.10.Indeed recall from 1.5.14 (5) that S∞(H) = C ⊗min R completely isometrically.Since C and R are both operator algebra structures on 2 by 5.3.8, S∞(H) iscompletely isomorphic to an operator algebra for the joint multiplication, whichturns out to be the Schur product.

We now deduce the result for B(H). It is clear from (5.18) that the Schurproduct ∗ on B(H) is separately w∗-continuous. Hence by 2.5.3, (B(H), ∗) is thesecond dual of (S∞(H), ∗) equipped with the Arens product. It is therefore iso-morphic to an operator algebra by 2.7.3. Indeed (B(H), ∗) is w∗-homeomorphicand completely isomorphic to a dual operator algebra by 5.2.16.

5.4 Q-ALGEBRAS

5.4.1 (Introduction to Q-algebras) By definition, a (concrete) Q-algebra isa Banach algebra A of the form C/J , where C is a function algebra (that is, asubalgebra of a commutative C∗-algebra), and J ⊂ C is a closed, two-sided ideal.We will regard such an algebra as an operator space, with the quotient structureA = Min(C)/Min(J). Then A is an operator algebra by 2.3.4. Conversely, wewill say that an operator algebra is an (abstract) Q-algebra if it is completelyisometrically isomorphic to a concrete Q-algebra. Any Q-algebra is commutativebut there exist commutative operator algebras which are not Q-algebras (see theNotes to Section 5.4).

The definitions and properties from 5.1.4 and 5.1.6 may be considered for thespecial class of Q-algebras; and thus we may talk about algebras isomorphic orcompletely isomorphic to a Q-algebra. We observe that subalgebras and quotientsof Q-algebras are again Q-algebras. Indeed let A = C/J be a Q-algebra and letB ⊂ A be a subalgebra. Then B = D/J , where D ⊂ C is the subalgebra ofall c ∈ C such that c ∈ B. If further B is an ideal of A, then D is an ideal ofC and we see that A/B = (C/J)

/(D/J) C/D (using standard facts about

216 Q-algebras

‘quotients of quotients’). On the other hand, any direct sum ⊕∞λ Aλ of a family

Aλ : λ ∈ I of Q-algebras is a Q-algebra. Indeed if we write Aλ = Cλ/Jλ foreach λ ∈ I, then ⊕∞

λ Aλ is a quotient of ⊕∞λ Cλ. It follows from these observations

that if A is an algebra which is also a Banach space, and if A is isomorphic toa Q-algebra, then any subalgebra of A, any quotient of A, and any direct sum∞I (A) also is isomorphic to a Q-algebra. Similarly, if further A is an operatorspace which is completely isomorphic to a Q-algebra, then the same holds forquotients, subalgebras, and direct sums of A.

5.4.2 (Interpolation of Q-algebras) Let (A0, A1) be a compatible couple ofBanach algebras. If A0 and A1 are both Q-algebras, then we claim that for anyθ ∈ (0, 1), the operator algebra Aθ = [A0, A1]θ is a Q-algebra. Indeed it fol-lows from 2.3.6 and the proof of 2.3.7 that Aθ is a quotient of a subalgebra ofC0(R; A0) ⊕∞ C0(R; A1). The claim follows from this and the stability proper-ties of Q-algebras discussed in the second paragraph of 5.4.1. As in 5.1.10, thefunctorial properties of interpolation ensure that if A0, A1 are isomorphic to Q-algebras, then Aθ is isomorphic to a Q-algebra. Similarly, if A0 and A1 are alsooperator spaces and if A0, A1 are completely isomorphic to Q-algebras, then Aθ

is completely isomorphic to a Q-algebra.

5.4.3 (Polynomial inequalities) An operator algebra is a Q-algebra if and onlyif it is a quotient of a subalgebra of some direct sum ∞I of the one-dimensionalalgebra C. Applying 5.2.10 and 5.2.11, we obtain the following polynomial char-acterizations:

(1) If A is an operator space which is also an algebra, then A is completelyisomorphic to a Q-algebra if and only if there exist M, δ > 0 such that

‖F‖A,δ ≤ M‖F‖C, n, N ≥ 1, F ∈ MN (Pn). (5.19)

In fact it suffices to prove (5.19) for homogeneous polynomials. Moreover A is(completely isometrically isomorphic to) a Q-algebra if and only if the latterholds with M = δ = 1.

(2) If A is a Banach space which is also an algebra, then A is isomorphicto a Q-algebra if and only if there exist M, δ > 0 such that ‖F‖A,δ ≤ M‖F‖C

for any integer n ≥ 1 and any F ∈ P [Z1, . . . , Zn]. Moreover A is isometricallyisomorphic to a Q-algebra if and only if the latter holds with M = δ = 1.

5.4.4 The next result is a handy criterion for when an algebra is completelyisomorphic to a Q-algebra. We will need the following notation. Let n, N, r ≥ 1be three integers and let u : Mn × · · · × Mn → MN be an r-linear map. For anyalgebra A, denote by

uA : Mn(A) × · · · × Mn(A) −→ MN (A)

the unique r-linear map defined by letting

uA(x1 ⊗ a1, . . . , xr ⊗ ar) = u(x1, . . . , xr) ⊗ a1 · · · ar, xk ∈ Mn, ak ∈ A.


Theorem 5.4.5 Let A be a commutative algebra which is also an operator space.Then A is completely isomorphic to a Q-algebra if and only if there is a positiveconstant K > 0 such that for any integers n, N, r ≥ 1 and any r-linear mapu : Mn × · · · × Mn → MN , we have ‖uA‖ ≤ Kr‖u‖.Proof We assume that ‖uA‖ ≤ Kr‖u‖ for all u as above. Let F ∈ MN (Pn)be a homogeneous polynomial of degree r ≥ 1, the integers n, N, r being arbi-trary. According to 5.4.3 (1), it will suffice to show that F satisfies (5.19). Usingnotation from 5.2.4 as in the proof of Theorem 5.2.8, we write

F =∑

α∈Λr

λα ⊗ Xα1 · · ·Xαr ,

with λα ∈ MN . Then we let v : Mn×· · ·×Mn → MN be the r-linear map definedby v(Eα1 , . . . , Eαr ) = λα for any α = (α1, . . . , αr) ∈ Λr, so that

F (a) = vA(a, . . . , a), a ∈ Mn(A). (5.20)

Denote by Sr the permutation group on 1, . . . , r. We introduce the symmetricr-linear map u : Mn × · · · × Mn → MN associated to v by letting

u(x1, . . . , xr) =1r!

∑σ∈Sr

v(xσ(1), . . . , xσ(r)), xp ∈ Mn.

We shall now prove that

vA(a, . . . , a) = uA(a, . . . , a), a ∈ Mn(A); (5.21)

and‖u‖ ≤ er‖F‖C. (5.22)

Let a =∑k

p=1 xp ⊗ ap be an arbitrary element of Mn(A). Then vA(a, . . . , a) isequal to

∑v(xp1 , . . . , xpr ) ⊗ ap1 · · · apr , where the sum is over all the r-tuples

(p1, . . . , pr) valued in 1, . . . , k (kr terms). Permuting the indices we see thatfor σ ∈ Sr we have

vA(a, . . . , a) =∑

1≤pj≤k

v(xpσ(1) , . . . , xpσ(r)

)⊗ apσ(1) · · · apσ(r) . (5.23)

Then we obtain

uA(a, . . . , a) =∑

1≤pj≤k

u(xp1 , . . . , xpr

)⊗ ap1 · · · apr

=∑

1≤pj≤k

(1r!

∑σ∈Sr

v(xpσ(1) , . . . , xpσ(r)

))⊗ ap1 · · · apr

=1r!

∑σ∈Sr

∑1≤pj≤k

v(xpσ(1) , . . . , xpσ(r)

)⊗ apσ(1) · · ·apσ(r)

because A is commutative. Then (5.21) follows from (5.23).

218 Q-algebras

To prove (5.22), we choose any x1, . . . , xr ∈ Ball(Mn), and let (ε1, . . . , εr)be an r-tuple of independent ±1-valued random variables on a probability space(Ω, µ), with µεp = 1 = µεp = −1 = 1/2. Then

v( r∑

p=1

εpxp, . . . ,

r∑p=1

εpxp

)=

∑1≤pj≤r

εp1 · · · εpr v(xp1 , . . . , xpr );

and ∫Ω

ε1 · · · εr εp1 · · · εpr dµ = 1 if p1, . . . , pr = 1, . . . , r,

= 0 if p1, . . . , pr = 1, . . . , r.

Consequently, the following integral representation formula holds:

u(x1, . . . , xr) =1r!

∫Ω

ε1 · · · εr v( r∑

p=1

εpxp, . . . ,

r∑p=1

εpxp

)dµ .

Since ‖xp‖ ≤ 1 for any p, we have ‖∑rp=1 εpxp‖ ≤ r. Hence

‖u(x1, . . . , xr)‖ ≤ rr

r!sup

‖v(x, . . . , x)‖ : x ∈ Mn, ‖x‖ ≤ 1

.

By (5.20) (taking A = C) the latter supremum equals ‖F‖C. Also, rr

r! ≤ er. Thisyields (5.22).

Combining (5.20) and (5.21), we obtain that F (a) = uA(a, . . . , a) for anya ∈ Mn(A). Hence

‖F (a)‖ ≤ ‖uA‖‖a‖r ≤ Kr‖u‖‖a‖r ≤ (Ke)r‖F‖C‖a‖r

by (5.22). This shows that F satisfies (5.19) with δ = (Ke)−1 and M = 1, whichconcludes the proof of the ‘if’ part.

The converse direction is easier. Clearly we may assume that A is a Q-algebra.Thus A = C/J , where C ⊂ C(Ω) is a subalgebra of C(Ω) for some compact spaceΩ, and J ⊂ C is an ideal. Let u : Mn × · · · × Mn → MN be any r-linear map.We let q : C → A denote the canonical quotient map. Given any a1, . . . , ar inMn(A), with ‖ai‖ < 1, let c1, . . . , cr in Mn(C) be such that qn(ci) = ai and‖ci‖ < 1 for any 1 ≤ i ≤ r. We have

uA(a1, . . . , ar) = qN

(uC(c1, . . . , cr)

).

Using the isometric identifications Mm(C(Ω)) = C(Ω; Mm) (e.g. from 1.2.3), wemay regard c1, . . . , cr and uC(c1, . . . , cr) as elements of C(Ω; Mn) and C(Ω; MN )respectively. Then

uC(c1, . . . , cr)(ω) = u(c1(ω), . . . , cr(ω)), ω ∈ Ω.

We deduce that ‖uC(c1, . . . , cr)‖ ≤ ‖u‖, whence ‖uA(a1, . . . , ar)‖ ≤ ‖u‖. Thisshows that ‖uA‖ ≤ ‖u‖.


5.4.6 (Q-spaces) We say that an operator space X is a Q-space if there existBanach spaces F ⊂ E such that X = Min(E)/Min(F ) completely isometrically.If X is merely completely isomorphic to a Q-space, we let

dQ(X) = inf‖v : X −→ Y ‖cb ‖v−1 : Y −→ X‖cb

,

where the infimum runs over all possible v : X → Y such that Y is a Q-space andv is a complete isomorphism. The parameter dQ(X) measures the ‘completelybounded distance’ from X to a Q-space.

If X is a Q-space, and if u : Mn → MN is a linear mapping, then we have‖u⊗IX : Mn(X) → MN(X)‖ ≤ ‖u‖. Indeed, assume that X = Min(E)/Min(F ),let q : E → X be the associated quotient map, and let x ∈ Mn(X) with norm< 1. Then there exists a z ∈ Mn(Min(E)) with ‖z‖ < 1 and qn(z) = x. SinceMn(Min(E)) = Mn⊗E and MN(Min(E)) = Mn⊗E isometrically (see (1.11)),we have ‖(u ⊗ IE)z‖ ≤ ‖u‖‖z‖ ≤ ‖u‖ by the ‘functoriality’ of the Banach spaceinjective tensor product A.3.1. Since

(u ⊗ IX)x = qN (u ⊗ IE)z,

we see that ‖(u ⊗ IX)x‖ ≤ ‖u‖, which proves the announced estimate.

5.4.7 (Comparing Q-spaces and Q-algebras) Obviously any Q-algebra is a Q-space. Conversely, let X be a Q-space and let A be X with the ‘zero product’(see 2.2.9). Then A is a Q-algebra. To see this, let F be any element of MN (Pn)and consider its decomposition F =

∑r≥1 Fr, with Fr ∈ MN (Pn) homogeneous

of degree r (see 5.2.4). Since the product on A is zero, we have ‖F‖A = ‖F1‖A.Moreover we may write F1 =

∑1≤i,j≤n λij ⊗ Xij with λij ∈ MN , and we have

‖F1‖A = sup∥∥∥∑

i,j

λij ⊗ xij

∥∥∥MN (X)

: xij ∈ X, ‖[xij ]‖Mn(X) ≤ 1.

Thus if we let u : Mn → MN be the linear mapping defined by u(Eij) = λij

for any 1 ≤ i, j ≤ n, then ‖F1‖A coincides with ‖u ⊗ IX : Mn(X) → MN (X)‖whereas ‖F1‖C coincides with ‖u‖. Applying 5.4.6 and 5.2.4, we obtain that

‖F‖A = ‖F1‖A ≤ ‖F1‖C ≤ ‖F‖C.

By 5.4.3 (1), this shows that A is a Q-algebra.

Proposition 5.4.8 (Junge) Let X be an operator space. Then X is completelyisomorphic to a Q-space if and only if there exists a constant K ≥ 1 such thatfor any n ≥ 1 and any linear map u : Mn → Mn, we have∥∥u ⊗ IX : Mn(X) −→ Mn(X)

∥∥ ≤ K‖u‖. (5.24)

Moreover dQ(X) coincides with the smallest K having that property.

220 Q-algebras

Proof The ‘only if’ direction follows from 5.4.6. Conversely if X satisfies (5.24),then arguing as in 5.4.7 with A = X with ‘zero product’, we obtain that‖F‖A ≤ K‖F‖C for any F ∈ MN (Pn). As in the last line of 5.4.7 we deducethat there is a Q-algebra B and an algebra isomorphism ρ : A → B such that‖ρ‖cb‖ρ−1‖cb ≤ K. Since X = A completely isometrically, this is equivalent tosaying that dQ(X) ≤ K.

5.4.9 (Interpolation of Q-spaces) Let (X0, X1) be a compatible couple of opera-tor spaces (see 1.2.30). If E0 and E1 are both completely isomorphic to Q-spaces,then the interpolation space Xθ = [X0, X1]θ is completely isomorphic to a Q-space as well. This can be proved by the argument in 5.4.2. Alternatively, onemay appeal to 5.4.8, which yields the estimate

dQ(Xθ) ≤ dQ(X0)1−θ dQ(X1)θ.

Indeed this follows by interpolation from (1.24).

Theorem 5.4.10 Let A be a commutative algebra which is also an operatorspace, and assume that A is completely isomorphic to a Q-space. If the multipli-cation on A extends to a completely bounded map

m : A ⊗min A −→ A,

then A is completely isomorphic to a Q-algebra.

Proof We shall prove that A satisfies the sufficient condition of Theorem 5.4.5.By assumption, there exist a Banach space E and a completely bounded mapq : Min(E) → A inducing a complete isomorphism from Min(E)/Min(Ker(q))onto A. We let K > 0 be such that for any n ≥ 1 and any a ∈ Mn(A) with‖a‖ < 1, there exists z ∈ Mn(Min(E)) such that qn(z) = a and ‖z‖ < K. We fixan r-linear map u : Mn×· · ·×Mn → MN and shall estimate the norm of uA. Forthat purpose we shall first estimate the norm of the auxiliary r-linear mapping

u : Mn(Min(E)) × · · · × Mn(Min(E)) −→ MN ⊗E⊗ · · · ⊗E

defined by letting u(x1 ⊗ z1, . . . , xr ⊗ zr) = u(x1, . . . , xr) ⊗ z1 ⊗ · · · ⊗ zr for anyxp in Mn and any zp in E. Let z1 = [z1

ij ], . . . , zr = [zr

ij ] ∈ Ball(Mn(Min(E))).Using notation from 5.2.4, we set λα = u(Eα1 , . . . , Eαr ) for any α ∈ Λr. Then

u(z1, . . . , zr) =∑

α∈Λr

λα ⊗ z1α1

⊗ · · · ⊗ zrαr

.

Given any functionals ϕ1, . . . , ϕr ∈ Ball(E∗), we have⟨u(z1, . . . , zr), ϕ1 ⊗ · · · ⊗ ϕr

⟩=

∑α∈Λr

λα 〈ϕ1, z1α1〉 · · · 〈ϕr , z

rαr〉

= u([

〈ϕ1, z1α1〉], . . . ,

[〈ϕr, z

rαr〉])

.


Hence∥∥⟨u(z1, . . . , zr), ϕ1 ⊗ · · · ⊗ ϕr

⟩∥∥MN

≤ ‖u‖∥∥[〈ϕ1, z

1α1〉]∥∥ · · · ∥∥[〈ϕr, z

rαr〉]∥∥ ≤ ‖u‖.

By A.3.1, this shows that ‖u‖ ≤ ‖u‖. On the other hand,

Min(E) ⊗min · · · ⊗min Min(E) = Min(E⊗ · · · ⊗E)

completely isometrically by 1.5.3 (2). From this and the ‘functoriality’ of ⊗min,for any r ≥ 1, the r-fold tensor product of q satisfies∥∥q ⊗ · · · ⊗ q : Min(E⊗ · · · ⊗E) −→ A ⊗min · · · ⊗min A

∥∥cb

≤ ‖q‖rcb. (5.25)

Next, we denote as usual by mr : A ⊗min · · · ⊗min A → A the r-fold multipli-cation on A. Let a1, . . . , ar in Mn(A), with ‖ai‖ < 1, and choose z1, . . . , zr inMn(Min(E)) such that qn(zp) = ap and ‖zp‖ < K for any 1 ≤ p ≤ r. Then itfollows from the respective definitions of uA and u that

uA(a1, . . . , ar) =[IMN ⊗

(mr (q ⊗ · · · ⊗ q)

)] (u(z1, . . . , zr)

).

Hence combining (5.25) with the inequality ‖u‖ ≤ ‖u‖, we find that

‖uA(a1, . . . , ar)‖ ≤ ‖q‖rcb ‖mr‖cb Kr ‖u‖ ≤ ‖q‖r

cb ‖m‖r−1cb Kr ‖u‖.

By taking the supremum over all possible such a1, . . . , ar, we may deduce that‖uA‖ ≤ ‖q‖r

cb ‖m‖r−1cb Kr ‖u‖. This yields the result by Theorem 5.4.5.

Corollary 5.4.11 Let A be a commutative algebra which is also a Banach spaceand assume that the multiplication on A extends to a bounded map

m : A ⊗A −→ A.

Then Min(A) is completely isomorphic to a Q-algebra.

Proof A minimal operator space is a Q-space, and by (1.10) and 1.5.3 (2), themap m : A⊗A → A is bounded if and only if m : Min(A)⊗min Min(A) → Min(A)is completely bounded. Hence Min(A) satisfies 5.4.10, whence the result.

5.4.12 We now come back to the example of p with pointwise product. Thiscommutative Banach algebra is isomorphic to an operator algebra. In Section5.3 we gave various examples of corresponding operator algebra structures onp. In fact Davie [116] and Varopoulos [415] showed that p is isomorphic to aQ-algebra. This will be established in 5.4.13 below using operator spaces. Alsowe wish to consider the question of whether the operator algebra structureson p exhibited in the previous section make p completely isomorphic to a Q-algebra. It follows from a deep result of Junge–Pisier (see [209, Theorem 3.2])that Max(E) cannot be a Q-space if E is an infinite-dimensional Banach space.The other cases will be settled in 5.4.15 and 5.4.16.

222 Q-algebras

Corollary 5.4.13 For any 1 ≤ p ≤ ∞, p with pointwise product is isomorphicto a Q-algebra. Indeed the operator space [Min(∞), Min(1)] 1

p(whose underlying

Banach space is p) is completely isomorphic to a Q-algebra.

Proof The result is obvious for p = ∞. Hence by interpolation, (see 5.4.2) itsuffices to prove that Min(1) is completely isomorphic to a Q-algebra. Accordingto Theorem 5.3.3 and 5.2.14, the multiplication on 1 extends to a bounded mapm : 1 ⊗γ2 1 → 1. However the injective tensor norm and γ2 are equivalent on1 ⊗ 1. Indeed, this is one of the equivalent forms of Grothendieck’s inequality(see [324, Chapters 4–6]). Thus m : 1⊗1 → 1 is bounded, and the result followsfrom 5.4.11.

5.4.14 (A noncommutative Grothendieck inequality) In the next proof, wewill use a remarkable inequality due to Pisier, which is a consequence of theso-called ‘noncommutative Grothendieck inequality’ (see [324, Corollary 9.5]): Ifu : A → B is a bounded operator between C∗-algebras, then for any x1, . . . , xn

in A, we have

max∥∥∑u(xk)∗u(xk)

∥∥ ,∥∥∑u(xk)u(xk)∗

∥∥≤ K2‖u‖2 max

∥∥∑x∗kxk

∥∥,∥∥∑xkx∗

k

∥∥,

where K is an absolute constant. Assume that A = B = MN for some N ≥ 1.Then according to (1.16), this is equivalent to∥∥u ⊗ IR∩C : MN(R ∩ C) −→ MN (R ∩ C)

∥∥ ≤ K‖u‖.

Corollary 5.4.15 R ∩ C is completely isomorphic to a Q-algebra.

Proof First, the operator space R ∩ C is completely isomorphic to a Q-space.This follows by applying 5.4.14 above, and Junge’s characterization 5.4.8. Let mdenote the multiplication mapping on 2. We know from 5.3.8 and its proof thatm : R⊗h R → R is completely contractive. Since R⊗h R = R⊗min R completelyisometrically (see 1.5.14 (2)), we deduce that m : (R ∩ C) ⊗min (R ∩ C) → R iscompletely contractive. Likewise, the product m : (R ∩ C) ⊗min (R ∩ C) → C iscompletely contractive, and so∥∥m : (R ∩ C) ⊗min (R ∩ C) −→ R ∩ C

∥∥cb

≤ 1.

The result therefore follows from Theorem 5.4.10.

Proposition 5.4.16(1) Up to complete isomorphism, the minimal operator space structure is the

only one for which 1 is completely isomorphic to a Q-space.(2) R and C are not completely isomorphic to Q-spaces. Indeed for any n ≥ 1,

dQ(Rn) = dQ(Cn) =√

n.


(3) For any 1 ≤ p < ∞, Op is not completely isomorphic to a Q-space. Indeedfor any n ≥ 1, we have

n12p

√2

≤ dQ(Opn) ≤ n

12p .

Proof (1) Assume that 1 is equipped with an operator space structure suchthat there exist a Banach space E, an operator space X , a complete quotient mapq : Min(E) → X , and an isomorphism u : 1 → X such that u−1 is completelybounded. By the lifting property of 1, there exists a bounded map σ : 1 → Esuch that qσ = u. This map is automatically completely bounded from Min(1)into Min(E), and so I1 = u−1qσ : Min(1) → 1 is completely bounded.

(2) Let n, m be two positive integers and let u : Mm → Mm be a linear map.

If a1, . . . , an in Mm, then∥∥∥∑n

k=1 u(ai)∗u(ai)∥∥∥1/2

is dominated by

( n∑k=1

‖u(ai)‖2)1/2

≤ ‖u‖( n∑

k=1

‖ai‖2)1/2

≤√

n ‖u‖∥∥∥ n∑

k=1

a∗i ai

∥∥∥1/2

.

This shows that u ⊗ ICn : Mm(Cn) → Mm(Cn) has norm less than or equal to√n ‖u‖. Thus by 5.4.8 we have dQ(Cn) ≤ √

n. Conversely, let u : Mn → Mn

be equal to the transposition map. Under the identifications Rn(Cn) = Mn andRn(Rn) = S2

n (see 1.5.14), the restriction of u ⊗ ICn to Rn(Cn) coincides withthe identity mapping Mn → S2

n. The latter has norm equal to√

n, giving theequality dQ(Cn) =

√n. The same argument works as well for Rn.

(3) Let n, N ≥ 1 be integers, let a1, . . . , an in MN , and let u be the linearmap from Min(∞n ) to MN taking ek to ak for any k. Then u(θ) =

∑k akθk for

any θ = (θ1, . . . , θn) ∈ ∞n , hence

‖u‖cb ≤∥∥∥ n∑

k=1

aka∗k

∥∥∥ 12 √

n

by the (easy) converse of the representation theorem of completely bounded maps1.2.8. Since CB(Min(∞n ), MN ) = MN(Max(1

n)) isometrically, this shows thatthe identity mapping Id : 2

n → 1n satisfies

‖Id : Rn −→ Max(1n)‖cb ≤

√n. (5.26)

Combining with (5.11), this implies that∥∥I1n: Min(1

n) −→ Max(1n)∥∥

cb≤

√n.

Since Min(1n) is a Q-space, this yields dQ(Max(1

n)) ≤ √n. By interpolation

(see 5.3.4 and 5.4.9), we deduce that dQ(Opn) ≤ n

12p . We now turn to the lower

estimates (which show that Op is not completely isomorphic to a Q-space). Let

224 Applications to the isomorphic theory

w1, . . . , wn ∈ M2n be unitaries satisfying (5.16) and (5.17), and let u : Cn → M2n

be the linear map taking the basis vector ek to wk, for any 1 ≤ k ≤ n. We clearlyhave ‖u : Min(∞n ) → M2n‖cb ≤ n, and ‖u : Max(1

n) → M2n‖cb ≤ 1 (see (1.12)).By interpolation, this yields∥∥u : Op

n −→ M2n

∥∥cb

≤ n1− 1p .

By (5.17) we have

n ≤∥∥∥ n∑

k=1

ek ⊗ wtk

∥∥∥M2n (Op

n)

∥∥u : Opn −→ M2n

∥∥cb

,

hence we obtain that

n1p ≤

∥∥∥ n∑k=1

ek ⊗ wtk

∥∥∥M2n (Op

n). (5.27)

Let q be the conjugate of p (so p−1 + q−1 = 1). Passing to the adjoint in (5.26),we have ‖Id : Min(∞n ) → Cn‖cb =

√n. Since ‖Id : Max(1

n) → Cn‖cb = 1, weobtain by interpolation that ‖Id : Oq

n → Cn‖cb ≤ n12p . Equivalently,∥∥∥ n∑

k=1

ek ⊗ ek

∥∥∥Cn(Op

n)≤ n

12p . (5.28)

Clearly∑n

k=1 ek ⊗wtk =

(I ⊗ τ u

) (∑k ek ⊗ ek

), if τ denotes the transposition

map on M2n . Moreover ‖u : Cn → M2n‖ ≤√

2 by (5.16). Thus we deduce fromthe easy direction of 5.4.8 that∥∥∥ n∑

k=1

ek ⊗ wtk

∥∥∥M2n (Op

n)≤

√2 dQ(Op

n)∥∥∥ n∑

k=1

ek ⊗ ek

∥∥∥Cn(Op

n).

Combining with (5.27) and (5.28), we see that n12p ≤

√2 dQ(Op

n).

5.5 APPLICATIONS TO THE ISOMORPHIC THEORY

5.5.1 (Strategy) In this section we will see that operator space theory canbe sometimes used as a simple but efficient tool in the ‘isomorphic theory’ ofoperator algebras. Assume that we are given a Banach algebra A and supposethat we wish to show that A is isomorphic to an operator algebra. Accordingto Theorem 5.2.1, it suffices to find one operator space structure on A (notnecessarily the ‘natural one’), such that m : A⊗h A → A is completely bounded.This reduces the task to creating the appropriate operator space structure on A.To illustrate this principle, consider A = S1(2) equipped with the usual product.We will show in the proof of 5.5.7, that Max(2)⊗hMax(2) is an operator algebrastructure on S1(2). Hence indeed S1(2) is isomorphic to an operator algebra.However we show in 5.5.8 that S1(2) is not completely isomorphic to an operatoralgebra when equipped with its usual operator space structure.


5.5.2 (Schatten spaces) For any 1 ≤ p < ∞, we let Sp = Sp(2) denote theSchatten p-class on 2 (see A.1.3), considered as consisting of infinite matriceswith scalar entries. We equip this space with the Schur product. If a, b ∈ Sp(2),then a⊗ b ∈ Sp(2 ⊗2 2), with ‖a⊗ b‖p = ‖a‖p‖b‖p. Indeed |a⊗ b|p = |a|p ⊗ |b|p.Hence (5.18) shows that a ∗ b belongs to Sp, with

‖a ∗ b‖p ≤ ‖V ‖2 ‖a ⊗ b‖p = ‖a‖p‖b‖p

Thus (Sp, ∗) is a Banach algebra. On the other hand we observe that Sp is alsoa Banach algebra for the usual product of operators. Indeed this multiplicationsatisfies ‖ab‖p ≤ ‖a‖p‖b‖∞ if a ∈ Sp and b ∈ S∞, and hence ‖ab‖p ≤ ‖a‖p‖b‖p

if a, b ∈ Sp. We shall show below that Sp is isomorphic to an operator algebra,either for the Schur or the usual product.

We will need the well-known fact that the Schatten spaces form an interpo-lation scale. Namely for any 1 < p < ∞ we have

Sp = [S∞, S1] 1p. (5.29)

Theorem 5.5.3 Let A be an algebra and a Banach space, and assume that A isisomorphic to an operator algebra. Equip S1 with the Schur product. Then S1⊗Ais isomorphic to an operator algebra for the joint multiplication (see 5.3.10).

Proof We let m1, m2, and m denote the multiplications on S1, A, and S1 ⊗ Arespectively. It is clear that if A and B are isomorphic as normed algebras, thenS1⊗A and S1⊗B are isomorphic as well, and so we may assume that A is anoperator algebra. Thus m2 : A ⊗h A → A is completely contractive (see 2.3.1).By 1.5.12 (1), Max(S1)

⊗ A is an operator space structure on S1⊗A. Thus by

Theorem 5.2.1, S1⊗A is isomorphic to an operator algebra provided∥∥m : (Max(S1)⊗ A) ⊗h (Max(S1)

⊗ A) −→ Max(S1)

⊗ A

∥∥cb

≤ 1. (5.30)

As Banach algebras, S2 with Schur product equals 2N

2 with pointwise multiplica-tion. Hence (5.12) yields a complete contraction from (S2)r⊗h(S2)c → Max(1

N2).

Moreover the ‘identity mapping’ from 1N

2 into S1 is contractive, and hence com-pletely contractive with respect to the maximal operator space structures. Thuswe obtain a complete contraction

s : (S2)r ⊗h (S2)c −→ Max(S1),

whose restriction to S1 ⊗ S1 coincides with the Schur product m1. By 1.5.14,((S2)r ⊗h (S2)c

) ⊗ A = (S2)r ⊗h A ⊗h (S2)c

completely isometrically. Thus s ⊗ IA extends to a complete contraction from(S2)r ⊗h A ⊗h (S2)c into Max(S1)

⊗ A. Since the ‘identity mapping’ from S1

into S2 is contractive, we finally obtain that

226 Applications to the isomorphic theory∥∥m1 ⊗ IA : Max(S1) ⊗h A ⊗h Max(S1) −→ Max(S1)⊗ A

∥∥cb

≤ 1.

Since⊗ is commutative and dominates ⊗h (see 1.5.13), and since ⊗h is associa-

tive, the ‘identity mapping’ induces a complete contraction

(Max(S1)⊗ A) ⊗h (Max(S1)

⊗ A) −→ Max(S1) ⊗h (A ⊗h A) ⊗h Max(S1).

We deduce that the following are complete contractions:

(Max(S1)⊗ A) ⊗h (Max(S1)

⊗ A)

IS1⊗m2⊗IS1−→ Max(S1) ⊗h A ⊗h Max(S1)m1⊗IA−→ Max(S1)

⊗ A.

Thus (5.30) holds, which completes our proof.

We shall now derive a few consequences of Theorem 5.5.3.

Corollary 5.5.4 For every 1 ≤ p ≤ ∞, the Schatten space Sp equipped with theSchur product is isomorphic to an operator algebra.

Proof The result holds for p = 1 by applying 5.5.3 with A = C. It holdstrue as well for p = ∞ by Theorem 5.3.11. Thus the result holds for any p byinterpolation, using (5.29) and 5.1.10 (1).

Corollary 5.5.5 Let A be isomorphic to an operator algebra and let 1 ≤ p ≤ ∞be a number. Equip p(A) with the ‘pointwise A-multiplication’ (that is, defineαβ = (akbk)k≥1, for α = (ak)k≥1 and β = (bk)k≥1 in p(A)). Then p(A) isisomorphic to an operator algebra as well.

Proof The pointwise A-multiplication on p⊗A is just the joint multiplication,if p is equipped with its usual pointwise multiplication. Regard 1 ⊂ S1 as thesubspace consisting of all diagonal operators. This subspace is the range of acontractive idempotent S1 → S1 (namely, the ‘diagonal projection’), and it easilyfollows that 1⊗A ⊂ S1⊗A isometrically. Since 1(A) = 1⊗A isometrically (e.g.see [121, VIII, Example 10]), we obtain that 1(A) is a subalgebra of S1⊗A, andso Theorem 5.5.3 ensures that 1(A) is isomorphic to an operator algebra if Ais. The result thus holds for p = 1. It is plain that it holds as well for p = ∞,and the general case follows again by interpolation.

5.5.6 (The algebra 2⊗ · · · ⊗2) Regard 2 ⊂ S1 as the subspace of all matriceswith non-zero entries only on the first row. Then this subspace is the range of acontractive idempotent S1 → S1. Arguing as in 5.5.5, we find that 2⊗A is iso-morphic to an operator algebra provided that A is. By induction we deduce thatfor any integer n ≥ 1, the Banach space projective tensor product 2⊗ · · · ⊗2 ofn copies of 2 is isomorphic to an operator algebra.

Theorem 5.5.7 For every 1 ≤ p ≤ ∞, the Schatten space Sp equipped with theusual product is isomorphic to an operator algebra.


Proof The result is obvious for p = ∞, and hence by interpolation again itsuffices to prove the result for p = 1. From (A.8) and (A.7), Max(2)⊗hMax(2) isan operator space structure on S1. It therefore suffices to show that this operatorspace satisfies Theorem 5.2.1. Indeed we will show that the usual product on S1

induces a complete contraction

m :(Max(2) ⊗h Max(2)

)⊗h

(Max(2) ⊗h Max(2)

)→ Max(2) ⊗h Max(2).

Note that the domain of m reordered by associativity is

Max(2) ⊗h

(Max(2) ⊗h Max(2)

)⊗h Max(2).

For any x, y, z, t in 2, we have m((x ⊗ y) ⊗ (z ⊗ t)

)= 〈y, z〉2 x ⊗ t. Letting

tr : 2 ⊗ 2 → C denote the usual trace, we have

m = I2 ⊗ tr ⊗ I2 on 2 ⊗ 2 ⊗ 2 ⊗ 2. (5.31)

Since Max(2) ⊗h Max(2) = S1, the trace tr extends to a contractive (hencecompletely contractive) functional on Max(2)⊗hMax(2). Hence m is completelycontractive on Max(2) ⊗h

(Max(2) ⊗h Max(2)

)⊗h Max(2).

Proposition 5.5.8 With either the usual or the Schur product, S1 ∼= R⊗h C isnot completely isomorphic to an operator algebra.

Proof Assume, by way of contradiction, that R ⊗h C, with the usual product,is completely isomorphic to an operator algebra. Writing

(R ⊗h C) ⊗h (R ⊗h C) = R ⊗h (C ⊗h R) ⊗h C, (5.32)

we see from (5.31) that the trace functional has to be bounded on C⊗R equippedwith the Haagerup tensor norm. Since C ⊗h R = S∞, this is a contradiction.

Now let m denote the Schur product and assume that

m : (R ⊗h C) ⊗h (R ⊗h C) −→ (R ⊗h C)

is completely bounded. Consider the following isometric identifications, whichfollow successively from (5.32), (1.51), and 1.5.14 (4):

CB((R ⊗h C) ⊗h (R ⊗h C),(R ⊗h C)) = CB(R ⊗h S∞ ⊗h C, S1)= CB(S∞, (R ⊗h S∞ ⊗h C)∗)

= CB(S∞, CB(S∞, B(2))).

Let u : S∞ → CB(S∞, B(2)) be the completely bounded map corresponding tom in this identification. Then it is easy to check (we leave this to the reader)that for any a = [aij ]i,j≥1 and b = [bij ]i,j≥1 in S∞, we have(

u(a))(b) = a ∗ bt = [aijbji]i,j≥1.


The desired contradiction follows since u is not bounded. To see this, given aninteger n ≥ 1, let

a =1√n

[θjk]1≤j,k≤n ∈ Mn ⊂ S∞,

where θjk ∈ C with |θjk| = 1 and ‖a‖ = 1. (Take, for example, θjk = eiπ(

j−kn

).)

For any 1 ≤ j, k ≤ n, we have(u(a)

)(Ekj) = 1√

nθjkEjk. Thus

(IMn ⊗ u(a)

)([θjk Ekj

])=

1√n

[Ejk ] ∈ Mn(Mn).

However∥∥[θjkEkj

]∥∥ = 1, whereas∥∥[Ejk]

∥∥Mn(Mn)

= n. Hence ‖u(a)‖cb ≥ √n‖a‖,

which proves the result.


The study of Banach algebras which are/are not isomorphic to operator algebrasor Q-algebras was initiated in the seventies. Among the many contributionsfrom that time, we mention the works of Carne [81, 82], Craw and Davie [116],Dixon [123, 124], Tonge [411] and Varopoulos [414–417], which played a role inthe development of the completely isomorphic theory presented in this chapter. IfA is a Banach space which is also an algebra, and if α is a tensor norm on A⊗A,we say that A is an α-algebra if the multiplication on A extends to a boundedmap m : A ⊗α A → A. With this terminology, Tonge’s result 5.2.13 says thatany g2-algebra is isomorphic to an operator algebra. Indeed several authors triedto find a specific tensor norm α such that the class of α-algebras coincides withthe class of operator algebras up to isomorphism. However Carne [81] showedthat no such α exists by constructing a γ∗

2 -algebra (see (A.7)) which is notisomorphic to an operator algebra. Switching from Banach spaces to operatorspaces, the completely isomorphic characterization of operator algebras providedby Theorem 5.2.1 allows to find one operator space tensor norm, namely theHaagerup norm, characterizing operator algebras.

5.1: Paulsen’s similarity result 5.1.2 is from [305, 306], and was originallymotivated by the case when A = A(D) is the disc algebra. Theorem 5.1.2 forA(D) says that an operator T on a Hilbert space is similar to a contraction if (andonly if) it is completely polynomially bounded, i.e. there exists a constant K ≥ 1such that ‖[fij(T )‖ ≤ K sup‖[fij(z)]‖ : z ∈ C, |z| ≤ 1 for any matrix [fij ] ofpolynomial. We recall here that T is said to be polynomially bounded if we merelyhave ‖f(T )‖ ≤ K sup|f(z)| : z ∈ C, |z| ≤ 1 for any polynomial f . Theorem5.1.2 in the case when A is a C∗-algebra had been previously established byHaagerup [178]; thus a bounded homomorphism π : A → B(H) is similar to a∗-representation if (and only if) it is completely bounded. Two famous similarityproblems are attached to the situations considered above. The first one, known asthe Halmos problem, asks whether any polynomially bounded operator is similarto a contraction. This was solved in the negative by Pisier in 1997 [330]. The


second one, known as the Kadison similarity problem, asks whether any boundedrepresentation on a C∗-algebra A is similar to a ∗-representation. This is stillopen in general but has been settled in several important cases. For example,the answer is positive if A = B(H) [178], or if A is nuclear [78, 91] (as notedin 5.1.3). The main references for similarity problems and their relationshipswith operator spaces are [335] (see also [332] and Pisier’s series of papers on thesimilarity degree of an operator algebra [333,334,336]), and [314] (and referencestherein). Other developments may be found in [250,251].

The result 5.1.9 is from [44]. Its proof also shows that CB(X) is completelyisometrically isomorphic to an operator algebra if and only if X is a (completelyisometric to) a column Hilbert space. Similar proofs show that given a Banachspace E, the Banach algebra B(E) is isomorphic (resp. isometrically isomor-phic) to an operator algebra if and only if E is isomorphic (resp. isometricallyisomorphic) to a Hilbert space. Alternatively, note that if B(E) is isometric to anoperator algebra then it satisfies von Neumann’s inequality. But this was shownby Foias to imply that X is Hilbertian [29, p. 230].

5.2: Theorem 5.2.1 is due to Blecher [44]. Some of its variants discussedhere (e.g. 5.2.5) are taken from [62]. The proof presented here emphasizes theanalogy with techniques developed in the seventies for the study of operatoralgebras up to isomorphism. A breakthrough result from that period is Craw’sLemma (from [116]) characterizing Q-algebras up to isomorphism by polyno-mial inequalities. That lemma formally corresponds to 5.4.3 (2). Its extension tooperator algebras presented in 5.2.11 was obtained by Dixon [124] and Varopou-los [417] independently. In this respect, Proposition 5.2.5 should be regarded asan operator space version of Craw’s Lemma. The idea of reducing computationsto homogeneous polynomials goes back to Davie [116]. Lemma 5.2.7 was inspiredby [417], and 5.2.12 is taken from the same paper. It is remarkable that the lat-ter result was proved before operator spaces existed. Another key result usedthroughout this section is the fact that quotients of operator algebras are againoperator algebras—see the Notes to Section 2.3 for historical comments on thatresult. Pisier presents in [337] a variant on the proof of 5.2.1 that fits within thecontext of his ‘universal operator algebra of an operator space’.

The dual version 5.2.16 is due to Le Merdy [248], whereas the completelyisomorphic characterization of operator bimodules in 5.2.17 is due to Blecher [46].

Any finite-dimensional algebra (resp. finite-dimensional algebra with an oper-ator space structure) is isomorphic (resp. completely isomorphic) to an operatoralgebra, and indeed to a subalgebra of Mn. This follows easily from 5.2.1, but infact has a simple direct proof (e.g. see the proof of 2.11 in [155]).

5.3: The examples and results that we present in this section (except 5.3.11)are mostly from [62]. Other examples of operator algebra structures can be foundin that paper. That p equipped with the pointwise multiplication is isomorphicto a Q-algebra (and hence to an operator algebra) was obtained independentlyby Davie [116] in the case when p ≤ 2, and Varopoulos [415] in the general case.Both proofs use Craw’s Lemma; however the idea of using interpolation as in


the proof of 5.3.5 is from [415].As a consequence of his isomorphic characterization of operator algebras (see

5.2.12), Varopoulos proved in [417] that B(H) with the Schur product is isomor-phic to an operator algebra. In [69] it was checked that the Schur multiplication∗ on B(H) is completely contractive, and it was shown that (B(H), ∗) cannotbe isometrically isomorphic to an operator algebra. The proof is similar to theone in 5.3.6.

5.4: There exist commutative operator algebras which are not isomorphic toa Q-algebra. This follows from some remarkable work of Varopoulos [416] on vonNeumann’s inequality in more than three variables. Namely Varopoulos showedthat for any constant K ≥ 1, there exist an integer n = n(K), a homogeneouspolynomial F of degree 3 in n variables, and an n-tuple (a1, . . . , an) of commutingcontractions on some Hilbert space, such that

‖F (a1, . . . , an)‖ ≥ K sup|F (z1, . . . , zn)| : zk ∈ C, |zk| ≤ 1

= K ‖F‖C.

Apply the above result for each integer K ≥ 1, and let AK be the operator algebragenerated by a1, . . . , an. Then their direct sum A = ⊕KAK is a commutativeoperator algebra, which cannot be isomorphic to a Q-algebra. Indeed if A wereisomorphic to a Q-algebra, there would exist by 5.4.3 (2) two constants δ, M > 0such that ‖F‖A,δ ≤ M‖F‖C for any n ≥ 1 and any F ∈ P [Z1, . . . , Zn]. If F ishomogeneous of degree 3, then ‖F‖A,δ = δ3‖F‖A,1. Hence we find that for anyK ≥ 1, for any n ≥ 1, and for any homogeneous polynomial F of degree 3, wehave ‖F‖A,1 ≤ Mδ−3 ‖F‖C. This contradicts the above estimate.

Our main characterization of Q-algebras (Theorem 5.4.5) is taken from [62].Again the proof borrows some ideas from the isomorphic theory developed inthe 1970s. In particular, the idea of symmetrization is from [116].

Recent interest in Q-spaces apparently started from the noncommutativeKhintchine inequalities of Lust-Piquard and Pisier [260], on the Schatten spaceS1. Let (gk)k≥1 be a sequence of complex valued independent standard Gaus-sian random variables on a probability space (Ω, µ), and let G ⊂ L1(Ω) be theclosed linear span of the gk’s. As explained in [331, Section 8.3], the main resultfrom [260] has the following consequence: if we regard G as an operator subspaceof Max(L1(Ω)), then the linear mapping taking ek to gk for each k ≥ 1 extendsto a complete isomorphism G ≈ R + C, where R + C denotes the dual operatorspace of R ∩ C. By duality, one obtains that R ∩ C ≈ L∞(Ω)/G⊥ is a Q-space.Shortly after this nontrivial example was discovered, Pisier conjectured the char-acterization of Q-spaces stated as 5.4.8. This question was eventually settled byJunge, but published in [327]. The latter paper also characterizes quotients ofsubspaces of C∗-algebras of the form MN (C(Ω)) for a fixed integer N ≥ 1. Fur-ther developments may be found in [206]. The short proof of 5.4.8, based onthe relationship between Q-spaces and Q-algebras, is from [62]. The interplaybetween the noncommutative Grothendieck inequality and the representation ofR ∩ C as a Q-space, appears in [260]. New results on Q-spaces related to 5.4.5appear in Ricard’s work [357,358].


Theorem 5.4.10 and its applications to the spaces [Min(∞), Min(1)] 1p

andto R ∩ C were obtained in [62]. In 5.4.16, part (2) is a well-known consequenceof 5.4.8, whereas parts (1) and (3) are from [62]. The proof of the latter partis related to some ideas from [308]. In particular, the fact that the completelybounded norm of the identity mapping from Min(1

n) into Max(1n) is dominated

by√

n is due to Paulsen.5.5: The fact that Sp is isomorphic to an operator algebra with either the

usual or the Schur product was proved in [62], where the reader will find morediscussion on operator algebra structures on Sp. This includes the negative result5.5.8. The matrix appearing in the proof of that result, showing that u is notbounded, was provided by Smith. That S1 with its ‘natural’ operator spacestructure, and either the usual or the Schur product, is not completely isomorphicto an operator algebra may explain why 5.5.4 and 5.5.7 remained unnoticed in theseventies. As an application of Theorem 5.2.16, we show now that S1 with eitherthe usual or the Schur product is actually w∗-homeomorphic and isomorphic toa dual operator algebra. Indeed the Schur product and the usual product areboth separately w∗-continuous on S1. Hence by 5.2.16 and the proofs of 5.5.3and 5.5.7, it suffices to check that Max(S1) and Max(2) ⊗h Max(2) are bothdual operator spaces. This is clear for the first of these spaces, which is the dualof Min(S∞) by 1.4.12. The second case is a bit more delicate. Arguing as in theproof of (1.60), one can show that Max(2) ⊗w∗h X = Max(2) ⊗h X for anyoperator space X . In particular Max(2) ⊗w∗h Max(2) = Max(2) ⊗h Max(2).Hence the latter space is the dual of Min(2) ⊗h Min(2), by (1.58).

Theorem 5.5.3 and its consequences 5.5.5 and 5.5.6 are new (but easy) ex-tensions of results from [62]. Since Sp with the Schur product is isomorphic toan operator algebra and is commutative, it is natural to ask if it is actuallyisomorphic to a Q-algebra. This problem was first raised by Varopoulos in [417]in the case p = ∞. It is proved that Sp is indeed isomorphic to a Q-algebrafor 2 ≤ p ≤ 4 in [247] and for 1 ≤ p ≤ 2 in [322]. The question for p > 4 isapparently still open at the time of this writing.

6

Tensor products of operator algebras

6.1 THE MAXIMAL AND NORMAL TENSOR PRODUCTS

Tensor products and C∗-norms play a prominent role in the theory of C∗-algebras, in particular in the study of nuclear C∗-algebras and semidiscrete (orinjective) von Neumann algebras. Our major goal in this chapter is to extendpart of that theory to nonselfadjoint operator algebras, and to give some appli-cations. Recall that if A and B are operator algebras, then their algebraic tensorproduct A ⊗ B is an algebra with the product defined by(∑

i

ai ⊗ bi

)(∑j

cj ⊗ dj

)=∑i,j

aicj ⊗ bidj ,

for finite families (ai)i, (cj)j in A, and (bi)i, (dj)j in B. It follows from 2.2.2that A ⊗min B with this product is an operator algebra. In this first section, weshall consider some other operator algebra tensor norms on A⊗B. For simplicityin this chapter, we will usually assume that our operator algebras are at leastapproximately unital.

6.1.1 (Maximal tensor product) Suppose that H is a Hilbert space, that Xand Y are operator spaces, and that Φ: X → B(H) and Ψ: Y → B(H) arecompletely bounded maps. Following 5.2.6, we write

Φ • Ψ: X ⊗ Y −→ B(H)

for the linear mapping taking x ⊗ y to Φ(x)Ψ(y), for any x ∈ X and y ∈ Y . Inthis chapter we are often concerned with the case that Φ and Ψ have commutingranges, that is, Φ(x)Ψ(y) = Ψ(y)Φ(x) for all x ∈ X, y ∈ Y .

If A and B are approximately unital operator algebras, define I = I(A, B)to be the collection of all pairs (π, ρ) of completely contractive representationsπ : A → B(H) and ρ : B → B(H), for a Hilbert space H , such that π and ρ havecommuting ranges. For any integer n ≥ 1, and any y ∈ Mn(A ⊗ B), we define

|||y|||n = sup∥∥(IMn ⊗ (π • ρ)

)(y)

∥∥Mn(B(H))

: (π, ρ) ∈ I. (6.1)

It is clear that each |||· |||n is a seminorm. To show that this is a norm, assume thatA and B are represented as operator algebras on H1 and H2 respectively. Thus

Tensor products of operator algebras 233

A⊗min B ⊂ B(H1 ⊗2 H2) completely isometrically, by 1.5.2. The two mappingsπ : A → B(H1 ⊗2 H2) and ρ : B → B(H1 ⊗2 H2) defined by π(a) = a ⊗ IH2 andρ(b) = IH1⊗b, are completely isometric homomorphisms with commuting ranges.By the above, ‖(IMn ⊗ (π • ρ))(y)‖ = ‖y‖min, for y ∈ Mn(A ⊗ B). Hence |||· |||ndominates the minimal norm, and is therefore also a norm on Mn(A⊗B). We letA⊗maxB be the completion of A⊗B in the norm |||· |||1, and we call A⊗maxB themaximal tensor product of A and B. By 2.3.3, A⊗max B is an operator algebra.The matrix norms on A ⊗max B will be often denoted by ‖· ‖max.

It is easily seen that A ⊗max B = B ⊗max A as operator algebras.

Proposition 6.1.2 Let A and B be approximately unital operator algebras. ThenA⊗max B is an approximately unital operator algebra, and the identity mappingon A ⊗ B extends to complete contractions

A ⊗h B −→ A ⊗max B −→ A ⊗min B.

If further A and B are unital, then A⊗maxB is unital as well (with unit 1A⊗1B).

Proof Most of this was shown in 6.1.1. Arguing as in 2.2.2 shows that A⊗maxBis approximately unital. By the easy direction of the CSPS theorem (see 1.5.7),together with (6.1), the canonical map from A × B to A ⊗max B is completelycontractive. Thus by 1.5.4, we see that the identity mapping on A ⊗ B extendsto a completely contractive linear map from A ⊗h B to A ⊗max B.

6.1.3 (Normal tensor product) Let A be an approximately unital operatoralgebra and let M be a unital dual operator algebra. We let Inor = Inor(A, M)be the collection of all pairs (π, ρ) ∈ I(A, M) such that ρ is w∗-continuous. Forany n ≥ 1, and any y ∈ Mn(A ⊗ M), we define

‖y‖nor = sup∥∥(IMn ⊗ (π • ρ)

)(y)

∥∥Mn(B(H))

: (π, ρ) ∈ Inor

.

As in 6.1.1 we see that these are matrix norms on A ⊗ B. We let A ⊗nor Mbe the resulting completion. Arguing as in 6.1.1, we see that A ⊗nor M is anapproximately unital operator algebra, which is unital if A is unital. Moreoverwe have a canonical sequence of completely contractive homomorphisms

A ⊗max M −→ A ⊗nor M −→ A ⊗min M. (6.2)

For an approximately unital operator algebra B, it easily follows from the ex-tension principle 2.5.5 that A ⊗max B ⊂ A ⊗nor B∗∗ completely isometrically.Consequently we also have A ⊗max B ⊂ A ⊗max B∗∗ completely isometrically.

6.1.4 (The selfadjoint case) Assume that A and B are C∗-algebras. Thenaccording to 1.2.4, I(A, B) is the collection of all pairs of commuting ∗-represen-tations π : A → B(H) and ρ : B → B(H). Hence in this case, A ⊗max B is theclassical maximal tensor product from the C∗-theory (e.g. see [407, IV.4]). Indeedin this case, ‖· ‖max is the maximal C∗-norm on A ⊗ B, and A ⊗max B is a C∗-algebra. Likewise, if A is a C∗-algebra and M is a W ∗-algebra, then A ⊗nor Mis the normal tensor product considered in [140].

234 The maximal and normal tensor products

Lemma 6.1.5 Suppose that A and B are closed approximately unital subalge-bras of an operator algebra D, such that ab = ba for all a ∈ A, b ∈ B. LetC = AB, also a closed subalgebra of D. If θ : C → B(H) is a completely con-tractive homomorphism, then there exist two commuting completely contractivehomomorphisms π : A → B(H) and ρ : B → B(H), such that θ(ab) = π(a)ρ(b)for a ∈ A and b ∈ B, and [θ(C)]′ = [π(A)]′ ∩ [ρ(B)]′. If, further, θ is nondegen-erate and completely isometric (see 2.1.5), then π and ρ map into the canonicalcopy (see Section 2.6) of the multiplier algebra M(C) inside B(H).

Proof If A and B are unital with identities e and f respectively, then ef isan identity for C. In this case this result is then obvious, by simple algebra.In the general case we will use second duals and their units, viewing A∗∗ andB∗∗ as subalgebras of D∗∗. The canonical separately w∗-continuous product onD∗∗ restricts to a separately w∗-continuous map on A∗∗ × B∗∗. This map hasrange inside C∗∗ by a routine density argument using (2.21). For θ as above,let θ : C∗∗ → B(H) be the canonical w∗-continuous extension of θ. We definew : A∗∗×B∗∗ → B(H) to be the composition of the last two maps. Let e and f bethe units of A∗∗ and B∗∗ respectively. Define π : A → B(H) and ρ : B → B(H)by π(a) = w(a, f) and ρ(b) = w(e, b), for a ∈ A and b ∈ B.

Let (et)t and (fs)s be cai’s for A and B respectively. By Proposition 2.5.8,et → e in the w∗-topology of A∗∗. If a ∈ A and b ∈ B, then ρ(b) is the w∗-limit of w(et, b) = θ(etb), and π(a) is the w∗-limit of w(a, fs) = θ(afs). Sinceθ(aetfsb) = θ(afs)θ(etb), we deduce by taking iterated w∗-limits over s and t,that θ(ab) = π(a)ρ(b). A similar argument, beginning with θ(bfseta), shows thatθ(ab) = ρ(b)π(a). Thus π and ρ have commuting ranges.

Suppose that θ is a nondegenerate embedding. For a′ ∈ A we have

π(a′)θ(ab) = w∗ − lim θ(a′fs)θ(ab) = w∗ − lim θ(a′afsb) = θ(a′ab).

Thus π(a′)θ(C) ⊂ θ(C), and a similar argument shows that θ(C)π(a′) ⊂ θ(C).Thus π maps into M(C). Similarly, ρ maps into M(C). The arguments for theremaining assertions of the lemma are similar, using iterated limits involving thecai’s, and are left as an exercise.

6.1.6 (Reduction to the unital case) Tensor products of operator algebras Aand B are a little easier to deal with if A and B are unital, mostly because inthis case there are canonical embeddings of A and B into the tensor product.In fact there are useful techniques to reduce many situations to the unital case,which are similar to what happens in the well-known selfadjoint case. We willnot take the time to develop this topic fully, in fact we will content ourselveswith proving here just a little more than is needed for this chapter. Our firstobservation is that if C1 is the unitization from Section 2.1 of an algebra C, thenit is easy to see from Meyer’s theorem from that section, together with (6.1), thatA⊗maxB ⊂ (A⊗max B)1 ⊂ A1⊗maxB1, completely isometrically as subalgebras,if A and B are nonunital. A similar result holds for the minimal tensor product.


Next we notice that if A and B are not unital, but are approximately unital,then there are useful embeddings of A and B into the multiplier algebra of thetensor product. To see this in the case of the maximal tensor product, applythe previous lemma with C = A ⊗max B, and D = A1 ⊗max B1. Here we areidentifying A with A ⊗ 1 and B with 1 ⊗ B. We obtain canonical commutingcompletely contractive homomorphisms π : A → M(C) and ρ : B → M(C). Tosee that π is isometric, for example, note that if b ∈ B with ‖b‖ = 1, and if (et)is a cai for A, then

‖π(a)‖ ≥ ‖π(a)(et ⊗ b)‖ = ‖(aet) ⊗ b‖ = ‖aet‖.

Taking the limit shows that π is isometric. A similar argument shows that π iscompletely isometric.

Again, similar arguments establish the analoguous facts for A ⊗min B.The following follows immediately from the above:

Corollary 6.1.7 Let A and B be approximately unital operator algebras, letH be a Hilbert space and let θ : A ⊗max B → B(H) be a completely contrac-tive homomorphism. Then there exist two commuting completely contractive ho-momorphisms π : A → B(H) and ρ : B → B(H) such that θ = π • ρ and[θ(A ⊗ B)]′ = [π(A)]′ ∩ [ρ(B)]′. If further A, B and θ are unital, this may beachieved with π and ρ being unital.

6.1.8 (Universal properties) Suppose that A and B are approximately unitaloperator algebras, set C = A⊗max B, and let i be the canonical map from A⊗Bto C. Let ν : A → M(C) and κ : B → M(C) be the canonical embeddings (see6.1.6), then i = ν • κ. In fact, C = A ⊗max B is uniquely determined by thefollowing universal property. The algebra C is a matrix normed algebra, thereexist completely contractive homomorphisms ν : A → M(C) and κ : B → M(C),with commuting ranges, the mapping i = ν •κ : A⊗B → M(C) has range in C,and indeed which is dense in C, and the following property holds:

Given any operator algebra D, and any completely contractive homomor-phisms π : A → D and ρ : B → D whose ranges commute, there exists a(necessarily unique) completely contractive homomorphism θ : A⊗max B → Dsuch that θ i = π • ρ.

That this property indeed characterizes the maximal tensor product up to com-pletely isometric isomorphism, follows easily using the usual type of algebraic ar-gument for proving such results. Of course if A and B are unital, then C = M(C).

Let B1, B2 be two unital matrix normed algebras, and consider the unitalalgebra B1

⊗ B2. By an argument similar to that of 3.1.5 (3), B1

⊗ B2 is a

matrix normed algebra. Combining 2.4.7 and the above paragraph, it is easy todeduce that O(B1

⊗ B2) = O(B1) ⊗max O(B2) as operator algebras. Now let

G1, G2 be discrete semigroups, and apply that result with B1 = Max(1G1

) andB2 = Max(1

G2). Using 2.4.9, we deduce that O(G1) ⊗max O(G2) is equal to the

enveloping operator algebra of Max(1G1

)⊗ Max(1

G2). The latter space is equal


to Max(1G1

⊗1G2

) by 1.5.12 (2), and hence equals Max(1G1×G2

) by [121, VIII,Example 10]. Thus we obtain that as operator algebras, we have

O(G1 × G2) = O(G1) ⊗max O(G2).

6.1.9 (Functoriality) Let (A, B) and (C, D) be two pairs of approximatelyunital operator algebras and let π : A → C and ρ : B → D be completely con-tractive homomorphisms. Then the mapping π ⊗ ρ : A ⊗ B → C ⊗ D extendsto a completely contractive homomorphism from A ⊗max B into C ⊗max D. Weleave this as an exercise using either the definition of the tensor product, or itsuniversal property above. In particular if A and B are subalgebras of C and Drespectively, then the embedding A⊗B ⊂ C ⊗D extends to a complete contrac-tion A ⊗max B → C ⊗max D. In general, this mapping is not an isometry. Thatis, the maximal tensor product is not ‘injective’. This point will be discussed inmore detail in Section 6.2.

We note however that if B is a C∗-algebra, then we have

A ⊗max B ⊂ C∗max(A) ⊗max B completely isometrically. (6.3)

Indeed let y ∈ Mn(A ⊗ B) for some n ≥ 1, and consider two commuting com-pletely contractive representations π : A → B(H) and ρ : B → B(H). We letD = [ρ(B)]′ ⊂ B(H) be the commutant of the range of ρ. Since B is a C∗-algebra, the mapping ρ is a ∗-representation (by 1.2.4), hence D is a C∗-algebra.Let C = C∗

max(A). Since π is valued in D, it follows from Proposition 2.4.2 that itextends to a ∗-representation π : C → D ⊂ B(H). Thus π and ρ are commutingrepresentations, and hence∥∥(IMn ⊗ (π • ρ)

)(y)

∥∥ =∥∥(IMn ⊗ (π • ρ)

)(y)

∥∥ ≤ ‖y‖Mn(C⊗maxB).

We deduce that ‖y‖Mn(A⊗maxB) ≤ ‖y‖Mn(C⊗maxB) by taking the supremum overall possible pairs (π, ρ). This proves (6.3).

The above results also have ‘normal’ versions. Thus if π : A → C is a com-pletely contractive homomorphism between approximately unital operator alge-bras, and if ρ : M → N is a w∗-continuous completely contractive homomorphismbetween unital dual operator algebras, then we obtain a canonical completelycontractive homomorphism π ⊗ ρ : A ⊗nor B → C ⊗nor N . Moreover arguing asabove, we see that if M is a W ∗-algebra, we have

A ⊗nor M ⊂ C∗max(A) ⊗nor M completely isometrically. (6.4)

If u : B → C is a completely bounded linear map between operator algebras,then in general IA ⊗ u need not be bounded as a map from A ⊗max B intoA ⊗max C. (See the Notes to Section 6.5 for more on this). However, we have:

Lemma 6.1.10 Let B and C be C∗-algebras, and let u : B → C be a completelypositive map. Then for any approximately unital operator algebra A, we have∥∥IA ⊗ u : A ⊗max B −→ A ⊗max C

∥∥cb

≤ ‖u‖cb = ‖u‖. (6.5)


Proof This result is well-known if A is a C∗-algebra (e.g. see [407, IV.4.23]).In that case, IA ⊗ u : A ⊗max B → A ⊗max C is actually completely positive.

Suppose now that A is an approximately unital operator algebra. By (6.3),we may regard IA ⊗ u as the restriction of IC∗

max(A) ⊗ u to A ⊗max B. Then theresult follows from the C∗-algebra case.

6.1.11 (Reduction to the nondegenerate case) Let A, B be approximately uni-tal operator algebras, and let I0 be the collection of all pairs (π, ρ) in I(A, B)such that π and ρ are nondegenerate (see 2.1.5). Then the definition (6.1) ofthe matrix norms on A ⊗max B is not changed if the supremum is taken overall (π, ρ) in I0 (instead of over all (π, ρ) in I). To prove this, it suffices tocheck that if π : A → B(H) and ρ : B → B(H) are two commuting completelycontractive homomorphisms, then there exists a subspace K of H , and commut-ing nondegenerate completely contractive homomorphisms π′ : A → B(K) andρ′ : B → B(K), such that (π • ρ)(y) = (π′ • ρ′)(y)PK , for y ∈ A ⊗ B. To seethis, let Kπ = [π(A)H ], Kρ = [ρ(B)H ], and K = Kπ ∩ Kρ. We let p and q bethe projections onto Kπ and Kρ respectively, and let (et)t and (fs)s be cai’s forA and B respectively. We have π(et)ρ(fs) = ρ(fs)π(et) for any t, s. By Lemma2.1.9, π(et) → p and ρ(fs) → q in the w∗-topology of B(H). We deduce that pand q commute. Then e = pq = qp is the projection onto K, and

e = w∗ − lims

limt

ρ(es)π(et) = w∗ − limt

lims

π(et)ρ(es). (6.6)

We define completely contractive mappings π′ : A → B(K) and ρ′ : B → B(K)by setting π′(a) = eπ(a)e and ρ′(b) = eρ(b)e, for a ∈ A and b ∈ B. Sinceπ and ρ commute, we deduce from (6.6) that eπ(a) = π(a)e. It is clear fromthis commutation property that π′ is a homomorphism. Likewise, ρ′ is a ho-momorphism, and ρ(b)e = eρ(b) for b ∈ B. Moreover, π′ and ρ′ have com-muting ranges. We now check that π′ and ρ′ are nondegenerate. For a ∈ Aand b ∈ B, we have lims limt ρ(fs)π(et)π(a)ρ(b) = π(a)ρ(b). Thus by (6.6),eπ(a)ρ(b) = π(a)ρ(b). On the other hand, π(et)ζ → ζ and ρ(es)ζ → ζ, forζ ∈ K. Hence lims limt ρ(fs)π(et)ζ = ζ. Thus we have K = [π(A)ρ(B)H ]. Sinceπ′(A)π(A)ρ(B) = π(A)ρ(B), we deduce the desired equality [π′(A)K] = K.Likewise, [ρ′(B)K] = K. Finally note that if a ∈ A, b ∈ B, then

π(a)ρ(b) = eπ(a)ρ(b) = eπ(a)eρ(b)e = π′(a)ρ′(b)e,

which is what was needed.

6.1.12 (The iterated maximal tensor product) Let A1, . . . , AN be approxi-mately unital operator algebras. Given pairwise commuting completely contrac-tive homomorphisms πk : Ak → B(H), let π1•· · ·•πN : A1⊗· · ·⊗AN → B(H) bethe map taking a1⊗· · ·⊗aN to π1(a1) · · ·πN (aN ), for any a1 ∈ A1, . . . , aN ∈ AN .Given n ∈ N, and y ∈ Mn

(A1 ⊗ · · · ⊗ AN ), define

‖y‖max = sup∥∥(IMn ⊗ (π1 • · · · • πN )(y)

∥∥Mn(B(H))

,


where the supremum runs over all such families (π1, . . . , πN ). Arguing as in 6.1.2,we see that ‖ ‖max is a norm on each Mn

(A1 ⊗ · · ·⊗AN), and that the resulting

completion is an approximately unital operator algebra. The latter is denotedby A1 ⊗max · · · ⊗max AN . The next result, states that ⊗max is associative. Weleave the proof as an exercise (using Corollary 6.1.7).

Lemma 6.1.13 For approximately unital operator algebras A, B, C, we have

(A ⊗max B) ⊗max C = A ⊗max B ⊗max C = A ⊗max (B ⊗max C).

completely isometrically.

By definition, a C∗-algebra B is nuclear if A ⊗min B = A ⊗max B for anyC∗-algebra A. Likewise, a W ∗-algebra M is semidiscrete if A⊗minM = A⊗norMfor any C∗-algebra A (see 6.6.1 for more on this ‘definition’). We shall see nextthat these properties are unchanged if we allow A to be nonselfadjoint.

Proposition 6.1.14(1) A C∗-algebra B is nuclear if and only if A ⊗max B = A ⊗min B completely

isometrically for any approximately unital operator algebra A.(2) A W ∗-algebra M is semidiscrete if and only if A ⊗nor M = A ⊗min M

completely isometrically for any approximately unital operator algebra A.

Proof Assume that B is nuclear. By the injectivity of the minimal tensorproduct, we have A ⊗min B ⊂ C∗

max(A) ⊗min B completely isometrically. SinceC∗

max(A)⊗min B = C∗max(A)⊗max B, the equality A⊗min B = A⊗max B follows

from (6.3). The proof of (2) is similar using (6.4).

Corollary 6.1.15 Let A, B be approximately unital operator algebras, let Mbe a unital dual operator algebra, and let n ≥ 1 be an integer. The followingidentities hold completely isometrically:

Mn(A ⊗max B) = Mn(A) ⊗max B; (6.7)

Mn(A ⊗nor M) = Mn(A) ⊗nor M = A ⊗nor Mn(M). (6.8)

Proof Since Mn is nuclear, (6.7) follows from 6.1.13 and 6.1.14.Let y ∈ Mn(A ⊗ M) for some n ≥ 1, and let (π, ρ) ∈ Inor(A, M) be a pair

of representations on some Hilbert space H . We define ρ : M → Mn(B(H)) byletting ρ(b) = 1 ⊗ ρ(b), for b ∈ M . Here 1 denotes the unit of Mn. Then wehave IMn ⊗ (π • ρ) = (IMn ⊗ π) • ρ on Mn(M). Since ρ is w∗-continuous, wededuce that ‖(IMn ⊗ (π • ρ))y‖ ≤ ‖y‖Mn(A)⊗norM . Taking the supremum over all(π, ρ) ∈ Inor(A, M), we deduce that ‖y‖Mn(A⊗norM) ≤ ‖y‖Mn(A)⊗norM .

Conversely, let (θ, ρ) ∈ Inor(Mn(A), M) be a pair of representations on H ,and recall the identification Mn(A) = Mn ⊗min A from 2.2.3. Since Mn is anuclear C∗-algebra, we may equally write Mn(A) = Mn ⊗max A by Proposition6.1.14. By Lemma 6.1.7 we may therefore write θ = σ •π, where σ : Mn → B(H)and π : A → B(H) are completely contractive representation such that σ, π and


ρ have commuting ranges. By definition, π • ρ : A ⊗ M → B(H) extends to acompletely contractive homomorphism on A ⊗nor M . Its range commutes withthe range of σ. Hence in turn, θ•ρ = σ•(π•ρ) extends to a completely contractivehomomorphism on Mn ⊗max (A ⊗nor M). Again since Mn is nuclear, the latterspace coincides with Mn(A ⊗nor M). Thus ‖θ • ρ(y)‖ ≤ ‖y‖Mn(A⊗norM), whichleads to ‖y‖Mn(A)⊗norM ≤ ‖y‖Mn(A⊗norM).

The above shows that Mn(A ⊗nor M) = Mn(A) ⊗nor M isometrically. Thecomplete isometry follows from the isometry by replacing n with mn, and re-placing A by Mm(A). The proof for the second identity in (6.8) is similar.

6.2 JOINT DILATIONS AND THE DISC ALGEBRA

In this section we will emphasize the interplay between dilations (or more pre-cisely, joint dilations) and the maximal tensor product. A basic illustration ofthis phenomenon is provided by Ando’s theorem reviewed in 2.4.13. As ob-served in 2.4.14, Ando’s result is equivalent to the completely isometric equalityO(N2

0) = A(D2). We also noticed that O(N) = A(D) by Nagy’s dilation theorem.Hence by the centered formulae in 6.1.8 and 2.4.13, we have

A(D) ⊗max A(D) = A(D) ⊗min A(D) (6.9)

completely isometrically. To investigate other such relationships, we will needthe following general dilation principle due to Arveson.

6.2.1 (Arveson’s dilation theorem) Let A be a unital-subalgebra of a unitaloperator algebra B. Let π : A → B(H) be a unital completely contractive repre-sentation on some Hilbert space H . If in turn B is a unital-subalgebra of a unitalC∗-algebra C, then we can regard A + A as an operator subsystem of C. ByLemma 1.3.6, π (uniquely) extends to a complete contraction φ : A+A → B(H).This extension is unital, hence completely positive (see 1.3.2). By Arveson’s ex-tension theorem 1.3.5, φ extends to a (necessarily unital) completely positive mapφ : C → B(H). Applying Stinespring’s factorization theorem (1.3.4) to this map,there is a Hilbert space K containing H as a subspace, and a ∗-representationπ : C → B(K) such that φ(· ) = PH π(· )|H . Restricting to B ⊂ C, we haveobtained a unital completely contractive representation π : B → B(K) and anisometric embedding J : H → K such that π(a) = J∗π(a)J , for a ∈ A. In thissituation, we say that π : B → B(K) is a B-dilation of π.

6.2.2 (Completely contractive representations of A(D)) In 2.4.12 we showedthat Nagy’s dilation theorem implies the ‘matricial von Neumann inequality’described there, that is, the polynomial functional calculus f ∈ P → f(T ) ex-tends to a unital completely contractive homomorphism uT : A(D) → B(H) ifT ∈ B(H) is a contraction. Using 6.2.1 we now note that these two results are infact equivalent. Indeed, if u : A(D) → B(H) is a completely contractive homo-morphism, then T = u(z) is a contraction, Also, u = uT , and [T ]′ = [uT (A(D))]′.Applying 6.2.1 with A = A(D) and B = C(T), and using the fact (obvious from

240 Joint dilations and the disc algebra

spectral theory) that unitaries in B(K) correspond to unital ∗-homomorphismsfrom C(T) to B(K), we recover Nagy’s dilation theorem.

Proposition 6.2.3 Let A and B be unital-subalgebras of unital operator algebrasC and D respectively. The following are equivalent.

(i) A ⊗max B ⊂ C ⊗max D completely isometrically.(ii) For any Hilbert space H and for any pair of commuting unital completely

contractive homomorphisms π : A → B(H) and ρ : B → B(H), there exist aHilbert space K, an isometry J : H → K, and commuting unital completelycontractive homomorphisms π : C → B(K) and ρ : D → B(K) such that

π(a)ρ(b) = J∗π(a)ρ(b)J, a ∈ A, b ∈ B. (6.10)

Proof Assume (i), and let π : A → B(H) and ρ : B → B(H) be as in (ii). Thenπ •ρ : A⊗B → B(H) extends to a unital completely contractive homomorphismθ : A ⊗max B → B(H). Since A ⊗max B is a unital subalgebra of C ⊗max D,Arveson’s theorem 6.2.1 ensures that θ admits an (C ⊗max D)-dilation θ fromC ⊗max D to B(K). By 6.1.7 we may write θ = π • ρ for a pair of commutingunital completely contractive representations π : C → B(K) and ρ : D → B(K).This pair clearly satisfies (6.10).

Conversely, assume (ii). Let π : A → B(H) and ρ : B → B(H) be two com-muting unital completely contractive homomorphisms, for some Hilbert spaceH . Let π : C → B(K) and ρ : D → B(K) be dilations of π and ρ provided by(ii). Then π • ρ(y) = J∗(π • ρ(y))J , for y ∈ A ⊗ B, and hence

‖π • ρ(y)‖ ≤ ‖π • ρ(y)‖ ≤ ‖y‖C⊗maxD.

Taking the supremum over all pairs (π, ρ) as above and applying 6.1.11, we de-duce that ‖y‖A⊗maxB ≤ ‖y‖C⊗maxD. Thus the completely contractive embeddingA ⊗max B → C ⊗max D provided by 6.1.9 is an isometry. A similar proof showsthat it is a complete isometry.

The following provides a class of embeddings satisfying the equivalent condi-tions of the last result. We use Dirichlet operator algebras (see 4.3.9 and 4.3.10).

Proposition 6.2.4 Let B be a Dirichlet operator algebra. Then for any C∗-algebra A we have A ⊗max B ⊂ A ⊗max C∗

e (B) completely isometrically.

Proof Set D = C∗e (B). By assumption, the operator system B + B is dense

in D. Let A be a C∗-algebra, and let π : A → B(H) and ρ : B → B(H) be com-muting completely contractive representations. By 6.1.11 we may assume that ρis unital. By Lemma 1.3.6, ρ admits a (unique) completely positive extension toB+B. We may extend ρ further by density to a completely positive contractionρ : D → B(H). On the other hand, let C = [π(A)]′ ⊂ B(H) be the commutant


of the range of π. Since π is a ∗-representation (see 1.2.4), C is selfadjoint. ThusRan(ρ) ⊂ C (since ρ maps into C, and ρ(b∗) = ρ(b)∗ for b ∈ B). We deduce that∥∥IA ⊗ ρ : A ⊗max D −→ A ⊗max C

∥∥cb

= 1,

by Lemma 6.1.10. Let Φ: A ⊗max C → B(H) be the contraction taking a⊗ c toπ(a)c, for any a ∈ A and c ∈ C. Then π • ρ is the restriction of Φ (IA ⊗ ρ),and so ‖(π • ρ)n(y)‖ ≤ ‖Φ((IA ⊗ ρ)(y))‖. Taking the supremum over all pairs(π, ρ) and applying 6.1.11, we obtain that ‖y‖A⊗maxB ≤ ‖y‖A⊗maxD. Thus thecomplete contraction from A⊗max B to A⊗max D is an isometry. Replacing A byMn(A), and applying Corollary 6.1.15, we conclude that it is actually a completeisometry.

6.2.5 (Dirichlet uniform algebras) Assume that B ⊂ C(Ω) is a uniform algebraon some compact space Ω. Let A be a unital-subalgebra of a unital operatoralgebra C. By 6.1.14 (1), and since commutative C∗-algebras are nuclear, wehave C ⊗max C(Ω) = C ⊗min C(Ω). Hence by the injectivity of the minimaltensor product, we have A ⊗min B ⊂ C ⊗max C(Ω) completely isometrically.Thus the canonical complete contraction from A ⊗max B to C ⊗max C(Ω) is acomplete isometry if and only if

A ⊗max B = A ⊗min B completely isometrically. (6.11)

Hence (6.11) holds true if and only if the dilation property 6.2.3 (ii) is satisfiedfor D = C(Ω). Moreover 6.2.4 shows that (6.11) is valid if B is a Dirichlet algebraand A is selfadjoint.

It is interesting to determine which unital operator algebras A have the prop-erty that A ⊗max A(D) = A ⊗min A(D) completely isometrically. Certainly thisholds if A = A(D), by (6.9), or if A is a C∗-algebra, by the previous paragraph.Indeed, A(D) is a Dirichlet uniform algebra. Thus by the observations in thelast paragraph, together with 6.2.3 and the discussion in 6.2.2, we deduce thefollowing strengthening of Nagy’s dilation theorem.

Corollary 6.2.6 Let A be a unital-subalgebra of a unital C∗-algebra C. Let Hbe a Hilbert space, let T ∈ B(H) be a contraction, and let π : A → B(H) be aunital completely contractive homomorphism such that π(a)T = Tπ(a) for a ∈ A.Then there exist a Hilbert space K, an isometry J : H → K, a ∗-representationπ : C → B(K), and a unitary operator U ∈ B(K), which satisfy π(c)U = Uπ(c)for c ∈ C, and

π(a)T n = J∗π(a)UnJ, a ∈ A, n ≥ 0.

6.3 TENSOR PRODUCTS WITH TRIANGULAR ALGEBRAS

The main goals of this section are to provide some concrete examples of tensorproducts of operator algebras; and also to indicate some remarkable connectionsto dilation theory and the well-known Sz–Nagy–Foias commutant lifting theorem.Facts in this section will not be used later in the book.

242 Tensor products with triangular algebras

6.3.1 We let T n denote the algebra of upper triangular n×n matrices. Arguingas in 6.2.5 and using the fact that Mn is nuclear, we see that whenever A is aunital-subalgebra of a unital operator algebra C, then A⊗max T n ⊂ C ⊗max Mn

completely isometrically if and only if A ⊗max T n = A ⊗min T n completelyisometrically. The latter is valid if A is a C∗-algebra, by Proposition 6.2.4, sinceT n is a Dirichlet algebra with C∗

e (T n) = Mn (see 4.3.7 (1)). Other such exampleswill be discussed in 6.3.7 and in the Notes to this section.

Let M denote the space of complex valued infinite matrices (tij)i,j≥1 withuniformly bounded truncations (see 1.2.26). Recall that M ∼= B(2). More pre-cisely, any T ∈ B(2) is identified with (tij)i,j≥1, where tij = 〈T (ej), ei〉, fori, j ≥ 1. Here (ej)j≥1 is the canonical orthonormal basis. Consider the ‘infinite’triangular algebra

T ∞ =(tij) ∈ M : tij = 0 for i > j ≥ 1

.

Clearly T ∞ is a unital dual algebra; indeed if Eij are the usual ‘matrix units’,then T ∞ is the w∗-closure of the linear span of Eij : j ≥ i ≥ 1 in M. SinceM ∼= B(2) is known to be a semidiscrete von Neumann algebra (see remarks in6.6.1), an argument similar to the one in the previous paragraph, but using 6.1.14(2), shows that for any unital-subalgebra A of C, the embedding of A ⊗nor T ∞into C ⊗nor M is a complete isometry if and only if A ⊗nor T ∞ = A ⊗min T ∞completely isometrically.

For a finite integer n ≥ 1, we may regard T n as a subalgebra of T ∞, namelyT n = SpanEij : 1 ≤ i ≤ j ≤ n.

6.3.2 (Completely contractive representations of T n) Let H be a Hilbert spaceand let w : T ∞ → B(H) be a unital and w∗-continuous completely contractivehomomorphism. For any i ≥ 1, we let pi = w(Eii). Since E2

ii = Eii and ‖Eii‖ = 1,we see that p2

i = pi and ‖pi‖ ≤ 1. Hence each pi is a projection. We let Hi ⊂ H bethe range of pi. These spaces are pairwise orthogonal since pipj = w(EiiEjj) = 0when i = j. Moreover we have

IH = w(1) = w∗ − limn

w(E11 + · · · + Enn) = w∗ − limn

(p1 + · · · + pn).

Thus H is the Hilbert space direct sum of the Hi. Clearly EiiEi(i+1)E(i+1)(i+1)

equals Ei(i+1), for i ≥ 1. Applying w, we see that piw(Ei(i+1))pi+1 = w(Ei(i+1)).This means that H⊥

i+1 ⊂ Ker(w(Ei(i+1))) and that Ran(w(Ei(i+1))) ⊂ Hi. Wemay therefore identify w(Ei(i+1)) with an operator θi from Hi+1 into Hi. Sincew is a contraction we have

‖θi : Hi+1 −→ Hi‖ ≤ 1, i ≥ 1. (6.12)

For any j > i ≥ 1, we may write Eij = Ei(i+1)E(i+1)(i+2) · · ·E(j−1)j . Applyingw, we find that

w(Eij) = θiθi+1 · · · θj−1, j > i ≥ 1. (6.13)


It turns out that (6.12) and (6.13) characterize such homomorphisms w. Namely,let (Hi)i≥1 be a sequence of Hilbert spaces, and let H be their Hilbert space di-rect sum. For any i ≥ 1, let θi : Hi+1 → Hi be a contraction. Then there exists a(necessarily unique) unital and w∗-continuous completely contractive homomor-phism w : T ∞ → B(H) satisfying (6.13), such that w(Eii) is the projection ontoHi, for i ≥ 1. This remarkable result is due to McAsey and Muhly (see [274]).In the finite-dimensional case of T n there is a similar description. Given an in-teger n ≥ 2 and a Hilbert space H , a linear mapping w : T n → B(H) is aunital completely contractive homomorphism if and only if there exist an or-thogonal decomposition H = H1 ⊕ · · · ⊕ Hn and contractions θi : Hi+1 → Hi,for 1 ≤ i ≤ n− 1, such that w(Eij) = θiθi+1 · · · θj−1 for 1 ≤ i < j ≤ n, and suchthat w(Eii) is the projection onto Hi.

Lemma 6.3.3 Let S ∈ B(2) be the ‘backwards shift’ operator, whose matrix(sij)i,j≥1 is given by si(i+1) = 1 if i ≥ 1, and sij = 0 if j = i + 1. Let uS be theassociated representation of the disc algebra A(D) on 2 (see 6.2.2). Then uS isa complete isometry, and Ran(uS) ⊂ T ∞.

Proof Since S ∈ T ∞, it is clear that uS maps into T ∞. For any f ∈ A(D),uS(f) is the so-called Toeplitz operator with matrix [f(j − i)]i,j≥1. Here f(k)denotes the kth Fourier coefficient of f (see (2.9)).

Since uS is a complete contraction and A(D) is a minimal operator space, itsuffices to show that uS is an isometry. By density it therefore suffices to showthat for any polynomial f ∈ P, we have

‖f‖∞ =∥∥[f(j − i)

]i,j≥1

∥∥ .

Let g be an arbitrary trigonometric polynomial in Ball(L2(T)). Then

‖fg‖2 =(∑

j

∣∣f g(j)∣∣2)1/2

=(∑

j

∣∣∣∑i

f(j − i)g(i)∣∣∣2)1/2

≤∥∥[f(j − i)

]i,j≥1

∥∥(∑i

∣∣g(i)∣∣2)1/2

=∥∥[f(j − i)

]i,j≥1

∥∥.The result follows by taking the supremum over all such g.

In view of the last result, we may regard A(D) as a subalgebra of T ∞.

Proposition 6.3.4 For any approximately unital operator algebra A,

A ⊗max A(D) ⊂ A ⊗nor T ∞ completely isometrically.

Proof By 6.1.15 it suffices to prove that A ⊗max A(D) ⊂ A ⊗nor T ∞ isometri-cally. This amounts to proving that for any finite sequence (ak)k≥0 in A,∥∥∥∑

k≥0

ak ⊗ zk∥∥∥

A⊗maxA(D)≤∥∥∥∑

k≥0

ak ⊗ Sk∥∥∥

A⊗norT ∞. (6.14)


To see this, note that IA ⊗ uS : A ⊗max A(D) → A ⊗nor T ∞ is a contraction, by6.3.3, (6.2), and 6.1.9. If we write an A-valued polynomial F ∈ A⊗P as a finitesum F =

∑k≥0 ak ⊗ zk, with ak ∈ A, then (IA ⊗ uS)(F ) =

∑k≥0 ak ⊗ Sk. If

(6.14) is valid, then IA ⊗ uS acts as an isometry on polynomials, and hence bydensity IA ⊗ uS is an isometry.

To prove (6.14), we consider two commuting completely contractive homo-morphisms π : A → B(H) and ρ : A(D) → B(H). We may assume (using 6.1.11)that ρ is unital. Thus ρ = uT for a contraction T commuting with the range ofπ (see 6.2.2). According to the result of McAsey–Muhly cited in 6.3.2, there is aunital w∗-continuous completely contractive homomorphism

wT : T ∞ −→ B(2(H)) = B(2)⊗B(H)

such that wT (Eij) = Eij ⊗ T j−i, for j ≥ i ≥ 1. Next we define a mappingπ : A → B(2(H)) by letting π(a) = I2 ⊗ π(a), for a ∈ A. Then π is completelycontractive and commutes with wT . We have∥∥∥∑

k≥0

Sk ⊗ π(ak)T k∥∥∥

B(2(H))= sup

∥∥∥∑k≥0

π(ak)T kzk∥∥∥

B(H): z ∈ C, |z| = 1

by Lemma 6.3.3 and the displayed formula in 2.2.7. However, wT (Sk) = (S⊗T )k,for k ≥ 0. Thus Sk ⊗ π(ak)T k = π(ak)wT (Sk), and so∥∥∥∑

k≥0

Sk ⊗ π(ak)T k∥∥∥

B(2(H))=

∥∥∥∑k≥0

π(ak)wT (Sk)∥∥∥ ≤

∥∥∥∑k≥0

ak ⊗ Sk∥∥∥

A⊗norT ∞.

The last two centered formulae imply that∥∥∥∑k≥0

π(ak)ρ(zk)∥∥∥

B(H)=∥∥∥∑

k≥0

π(ak)T k∥∥∥

B(H)≤∥∥∥∑

k≥0

ak ⊗ Sk∥∥∥

A⊗norT ∞.

Taking the supremum over all π and ρ = uT as above, we obtain (6.14).

Theorem 6.3.5 (Paulsen and Power) Let A be an approximately unital opera-tor algebra. Then the following assertions are equivalent:(i) A ⊗max A(D) = A ⊗min A(D) completely isometrically.(ii) For every integer n ≥ 1, A ⊗max T n = A ⊗min T n completely isometrically.(iii) A ⊗nor T ∞ = A ⊗min T ∞ completely isometrically.

Proof Since A⊗min A(D) ⊂ A⊗min T ∞ completely isometrically, the implica-tion ‘(iii) ⇒ (i)’ follows from Proposition 6.3.4. Note moreover that when proving‘(ii) ⇒ (iii)’ and ‘(i) ⇒ (ii)’, we will only need to prove that the desired identi-ties hold isometrically. Indeed, as in the proof of 6.3.4, the completely isometricstatements follow from the isometric ones and Corollary 6.1.15.

We assume (ii), and shall prove that A ⊗nor T ∞ = A ⊗min T ∞. We regardT n as a subalgebra of T ∞, and we let αn : T ∞ → T n denote the canonical


‘truncation’ operator. We clearly have b = w∗-limn αn(b), for b ∈ T ∞. Hence forany (π, w) ∈ Inor(A, T ∞), and any y =

∑k ak ⊗ bk ∈ A ⊗ T ∞, we have

π • w(y) =∑

k

π(ak)w(bk) = w∗ − limn

(∑k

π(ak)w(αn(bk)

)).

For n ≥ 1, let wn be the restriction of w to T n. Our hypothesis (ii) ensures thatπ • wn is completely contractive on A ⊗min T n. Hence∥∥∥∑

k

π(ak)w(αn(bk)

)∥∥∥ ≤∥∥∥∑

k

ak ⊗ αn(bk)∥∥∥

A⊗minT n

.

Moreover αn is a complete contraction, and so the right side of this inequalityis less than or equal to ‖y‖min. We deduce that ‖π • w(y)‖ ≤ ‖y‖min. Hence‖y‖max = ‖y‖min as expected.

Assume (i), and let n ≥ 2. To show the isometric version of (ii), we con-sider two commuting completely contractive homomorphisms π : A → B(H) andw : T n → B(H), with w unital, and we will show that

‖π • w : A ⊗min T n −→ B(H)‖ ≤ 1. (6.15)

Then taking the supremum over all possible (π, w) and applying 6.1.11 yieldsA ⊗min T n = A ⊗max T n isometrically. Recall from 6.3.2 that there exist aorthogonal decomposition H = H1 ⊕ · · · ⊕ Hn, such that w(Eii) = pi is theprojection onto Hi, and such that the contraction w(Eij) maps Hj into Hi andvanishes on H⊥

j , for 1 ≤ i ≤ j ≤ n. Define

T = w(E12 + E23 + · · · + E(n−1)n

).

Then T k = w(E1(k+1) + E2(k+2) + · · ·+ E(n−k)n

), for k ≥ 0. Applying this with

k = j − i, we deduce that for any ζ = (ζ1, . . . , ζn) and any η = (η1, . . . , ηn) inH = H1 ⊕ · · · ⊕ Hn, we have

〈T j−i(ζj), ηi〉 = 〈w(Eij)ζ, η〉, 1 ≤ i ≤ j ≤ n. (6.16)

Here of course we are identifying ζj with pj(ζ). Define V : H → 2n(H) by

V (ζ) =∑

1≤i≤n

ei ⊗ ζi, ζ = (ζ1, . . . , ζn) ∈ H.

Clearly V is an isometry. We claim that for any a ∈ A, we have

π(a)w(Eij) = V ∗(Eij ⊗ T j−iπ(a))V, 1 ≤ i ≤ j ≤ n. (6.17)

To see this, let ζ, η in H . Then⟨V ∗(Eij ⊗ T j−iπ(a)

)V (ζ), η

⟩= 〈T j−iπ(a)(ζj), ηi〉.


Since π(a) commutes with each projection pj , we have π(a)(ζj) = pjπ(a)(ζ).Therefore using (6.16) we have

〈T j−iπ(a)(ζj), ηi〉 = 〈T j−ipjπ(a)(ζ), ηi〉 = 〈w(Eij)π(a)ζ, η〉,

which shows (6.17). We now consider y ∈ A ⊗ T n. We may write

y =∑

1≤i≤j≤n

aij ⊗ Eij ,

for some aij ’s in A. Then by (6.17), we have∥∥π • w(y)∥∥ =

∥∥∥ ∑1≤i≤j≤n

π(aij)w(Eij)∥∥∥ ≤

∥∥∥ ∑1≤i≤j≤n

Eij ⊗ T j−iπ(aij)∥∥∥

Mn(B(H)).

It is clear that T is a contraction. Consider uT : A(D) → B(H) (see 6.2.2). Notethat T , and hence also uT , commutes with π. Then∑

1≤i≤j≤n

Eij ⊗ T j−iπ(aij) =(IMn ⊗ (π • uT )

)( ∑1≤i≤j≤n

Eij ⊗ aij ⊗ zj−i

).

We therefore deduce from our hypothesis (i) that∥∥π • w(y)∥∥

B(H)≤∥∥∥ ∑

1≤i≤j≤n

Eij ⊗ aij ⊗ zj−i∥∥∥

Mn(A⊗minA(D)).

Now recall that Mn(A⊗minA(D)) = A(D; Mn(A)) isometrically (see 2.2.7). Hencethe right side of the last inequality is equal to

sup∥∥∥ ∑

1≤i≤j≤n

Eij ⊗ zj−iaij

∥∥∥Mn(A)

: z ∈ C, |z| = 1

. (6.18)

For z ∈ T, consider the diagonal matrix Uz = Diag(z, z2, . . . , zn) ∈ Mn. ThenUz is a unitary and∑

1≤i≤j≤n

Eij ⊗ zj−iaij = U∗z

( ∑1≤i≤j≤n

Eij ⊗ aij

)Uz = U∗

z y Uz.

Since ‖U∗z y Uz‖min = ‖y‖min for z ∈ T, we see that the supremum in (6.18) is

equal to ‖y‖min. This shows that ‖π •w(y)‖B(H) is less than or equal to ‖y‖min,which completes the proof of (6.15).

The following ‘normal’ analogue of Theorem 6.3.5 has a similar proof.

Theorem 6.3.6 For a unital dual operator algebra M , the following assertionsare equivalent:


(i) A(D) ⊗nor M = A(D) ⊗min M completely isometrically.(ii) For every integer n ≥ 1, T n ⊗nor M = T n ⊗min M completely isometrically.

Corollary 6.3.7 The following identities hold completely isometrically:(1) If A is a C∗-algebra, and n ≥ 1 is any positive integer,

A ⊗max T n = A ⊗min T n and A ⊗nor T ∞ = A ⊗min T ∞.

(2) For any n ≥ 1,

A(D) ⊗max T n = A(D) ⊗min T n and A(D) ⊗nor T ∞ = A(D) ⊗min T ∞.

(3) For any n, m ≥ 1,

T m ⊗max T n = T m ⊗min T n and T m ⊗nor T ∞ = T m ⊗min T ∞.

Proof By Theorem 6.3.5, we obtain the first two assertions from the corre-sponding results in Section 6.2, that is, (6.9) and 6.2.5. In turn, (3) follows from(2) by applying 6.3.5 and 6.3.6.

Tensor products with triangular algebras are closely related to dilation theory,and in particular to the well-known commutant lifting theorem of Sz–Nagy–Foias(see [405]). This is well illustrated by the following consequence of 6.2.3:

Proposition 6.3.8 (Paulsen and Power) Let A be a unital-subalgebra of a uni-tal C∗-algebra C. The following are equivalent:(i) A ⊗max T 2 = A ⊗min T 2 completely isometrically.(ii) For any pair of unital completely contractive homomorphisms π : A → B(H)

and θ : A → B(H ′), and any contraction T : H ′ → H with the propertythat π(a)T = Tθ(a) for a ∈ A, there exist Hilbert spaces K, K ′, isometriesJ : H → K and J ′ : H ′ → K ′, unital ∗-representations π : C → B(K) andθ : C → B(K ′), as well as a unitary U : K ′ → K, such that π(c)U = Uθ(c)for all c ∈ C, and such that π(a)T = J∗π(a)UJ ′, π(a) = J∗π(a)J andθ(a) = J ′∗θ(a)J ′, for a ∈ A.

Proof Assume (i). Then A⊗maxT 2 ⊂ C⊗maxM2 completely isometrically (see6.3.1). Let H, H ′, π, θ, and T be as in (ii), and define σ : A → B(H ⊕ H ′) andτ : T 2 → B(H ⊕ H ′) by letting

σ(a) =[

π(a) 00 θ(a)

]and τ

([λ1 µ0 λ2

])=

[λ1IH µT

0 λ2IH′

]. (6.19)

Then σ and τ are unital completely contractive homomorphisms (see 2.2.11).Moreover our assumption that T intertwines π(a) and θ(a) for a ∈ A, ensuresthat σ and τ have commuting ranges. By the implication ‘(i)⇒ (ii)’ of Propo-sition 6.2.3, there exist a Hilbert space F , an isometry j : H ⊕ H ′ → F , andcommuting unital completely contractive homomorphisms σ : C → B(F ) and

248 Pisier’s delta norm

τ : M2 → B(F ), such that σ(a)τ(b) = j∗σ(a)τ(b)j, for a ∈ A, b ∈ T 2. Since τ isa ∗-representation of M2, there is a decomposition F = K ⊕ K ′ and a unitaryoperator U : K ′ → K such that

τ

([λ1 µν λ2

])=

[λ1IK µUνU∗ λ2IK′

],

for λ1, λ2, µ, ν in C. Since σ and τ commute, we see that K and K ′ are bothσ(C)-invariant. Thus we may write σ = π ⊕ θ for some unital ∗-representationsπ : C → B(K) and θ : C → B(K ′). Moreover U intertwines π(c) and θ(c), forc ∈ C. Next we check that j maps H into K and H ′ into K ′. Since j∗τ(b)j = τ(b),we have j∗PKj = PH . For ζ ∈ H , this implies that

‖ζ‖ ≤ ‖PKj(ζ)‖ ≤ ‖ζ‖,and j(ζ) = PKj(ζ). Hence j(H) ⊂ K. Similarly, j(H ′) ⊂ K ′. Let J : H → K andJ ′ : H ′ → K ′ be the corresponding restrictions. Then the last three assertions of(ii) follow from the relation σ(a)τ(b) = j∗σ(a)τ (b)j above, for b = E11, E12, E22

respectively.To prove conversely that (ii) implies (i), it suffices to reverse the argu-

ments and to appeal to the implication ‘(ii)⇒ (i)’ of 6.2.3. Given a Hilbertspace E, and unital completely contractive homomorphisms σ : A → B(E) andτ : T 2 → B(E), whose ranges commute, there is a decomposition E = H ⊕ H ′,a contraction T : H ′ → H , as well as π and θ as in (ii), such that τ and σ havethe form (6.19), and such that T intertwines π(a) and θ(a) for a ∈ A. Appealingto (ii), and using simple algebra, yields the desired maps in 6.2.3 (ii).

6.4 PISIER’S DELTA NORM

6.4.1 (Definitions) Let X be an operator space and let B be an approximatelyunital operator algebra. In this section we introduce a tensor norm on X ⊗ B,denoted by δ, and called the delta norm. Let Q0 : B ⊗ X ⊗ B → X ⊗ B be thelinear mapping defined by letting Q0(a⊗x⊗ b) = x⊗ab, for x ∈ X and a, b ∈ B.Recall that any c ∈ B can be written as c = ab, for some a and b in B (by A.6.2).Thus Q0 is onto. Hence for any z ∈ X ⊗ B, we may define δ(z) = inf‖y‖h,where the infimum runs over all y ∈ B ⊗X ⊗B such that Q0(y) = z. Here ‖y‖h

denotes the Haagerup tensor norm of y (see 1.5.4 and 1.5.5). A more explicitdefinition of δ is provided by the ‘3-variable’ version of (1.40). Namely,

δ(z) = inf∥∥[xij ]

∥∥Mn(X)

∥∥[a1 · · · an]∥∥

Rn(B)

∥∥[b1 · · · bn]t∥∥

Cn(B)

,

where the infimum runs over all decompositions of z of the form

z =∑

1≤i,j≤n

xij ⊗ aibj . (6.20)

It is clear that δ is a seminorm, and it is not hard to check that this is actually anorm on X ⊗B. Indeed we shall show in 6.4.2 below that ‖z‖min ≤ δ(z), for z in


X ⊗B. We denote the resulting completion by X ⊗δ B. By construction, and byan elementary Banach space argument, Q0 uniquely extends to a quotient map

Q : B ⊗h X ⊗h B −→ X ⊗δ B. (6.21)

6.4.2 (Comparisons with other tensor norms) Let H be a Hilbert space, letθ : X → B(H) be a complete contraction, and let π : B → B(H) be a completelycontractive homomorphism. If θ and π have commuting ranges, then we claim:∥∥θ • π(z)

∥∥B(H)

≤ δ(z), z ∈ X ⊗ B. (6.22)

Indeed we have θ(x)π(ab) = π(a)θ(x)π(b), for x ∈ X and a, b ∈ B. Hence if z isrepresented as in (6.20), we have∥∥θ • π(z)

∥∥ =∥∥∥∑ θ(xij)π(aibj)

∥∥∥ =∥∥∥∑π(ai)θ(xij)π(bj)

∥∥∥.The latter quantity is dominated by

∥∥∑ ai ⊗ xij ⊗ bj

∥∥h

(see the first paragraphof 1.5.8). Taking the infimum over all such representations of z, we obtain (6.22).

By a similar argument to that in 6.1.1, equation (6.22) implies that

‖y‖min ≤ δ(y), y ∈ X ⊗ B.

On the other hand, δ is dominated by the Haagerup norm on X⊗B. To see this,first observe that δ(x⊗ b) ≤ ‖x‖‖b‖, for x ∈ X and b ∈ B. Indeed if ‖x‖ < 1 and‖b‖ < 1, then by A.6.2 we can write b = b′b′′, with ‖b′‖ < 1 and ‖b′′‖ < 1. Hencex⊗ b = Q0(b′ ⊗ x ⊗ b′′), and we conclude that δ(x ⊗ b) ≤ ‖b′‖‖x‖‖b′′‖ < 1. Nowlet (et)t be a cai for B, and consider z =

∑xk ⊗ bk ∈ X ⊗ B, where (xk)k and

(bk)k are finite sequences in X and B respectively. We let zt =∑

xk ⊗ etbk forany t. Then the above point implies that δ(zt − z) → 0, and hence δ(zt) → δ(z).Moreover zt = Q0(

∑k et ⊗ xk ⊗ bk), and hence

δ(zt) ≤∥∥∥∑

k

et ⊗ xk ⊗ bk

∥∥∥h≤∥∥∥∑

k

xk ⊗ bk

∥∥∥h

= ‖z‖h.

We deduce that δ(z) ≤ ‖z‖h.It is also clear from (6.22) and 6.1.1 that if X = A is an approximately unital

operator algebra, then the maximal tensor norm ‖· ‖max on A⊗B is dominatedby δ. The first main result of this section is Corollary 6.4.4 below, which is a‘converse’ to (6.22). According to this result, δ can be regarded as an genuineanalogue of the maximal tensor norm. We will deduce 6.4.4 from the followingdescription of the dual space of X⊗δB:

Theorem 6.4.3 (Pisier) Let X be an operator space, and let B be an approxi-mately unital operator algebra. Given a functional ϕ : X ⊗B → C, the followingare equivalent:(i) ϕ extends to an element of (X⊗δB)∗, with ‖ϕ‖ ≤ 1.


(ii) There exist a Hilbert space H, vectors ζ1, ζ2 ∈ Ball(H), a completely con-tractive map θ : X → B(H), and a completely contractive homomorphismπ : B → B(H), such that θ and π have commuting ranges, and such that

ϕ(x ⊗ b) = 〈θ(x)π(b)ζ2 , ζ1〉, x ∈ X, b ∈ B.

Proof If (ii) holds, then ϕ(z) = 〈(θ • π)(z)ζ2, ζ1〉, for z ∈ X ⊗ B. This clearlyimplies (i) by (6.22).

Assume conversely that (i) holds. Composing ϕ with the contractive mappingQ from (6.21), we obtain

ϕ Q ∈(B ⊗h X ⊗h B

)∗, with ‖ϕ Q‖ ≤ 1.

By the CSPS theorem for trilinear maps (see the second paragraph of 1.5.8),there exist Hilbert spaces K1, K2, vectors e1 ∈ Ball(K1) and e2 ∈ Ball(K2),a completely contractive map v : X → B(K2, K1), and completely contractiverepresentations π1 : B → B(K1) and π2 : B → B(K2), such that

ϕ(x ⊗ ab) = 〈π1(a)v(x)π2(b)e2, e1〉, x ∈ X, a, b ∈ B. (6.23)

By 2.1.10, we may assume that π1 and π2 are nondegenerate. Consider a, b, c inB. Then writing acb = (ac)b or acb = a(cb), and computing ϕ(x⊗acb) by meansof (6.23), we find that

〈π1(a)π1(c)v(x)π2(b)e2, e1〉 = 〈π1(a)v(x)π2(c)π2(b)e2, e1〉. (6.24)

Since π1 and π2 are nondegenerate, the vectors e1 and e2 belong to [π1(B)∗e1]and [π2(B)e2] respectively (see the last line of 2.2.8). Replacing K1 and K2 by[π1(B)∗e1] and [π2(B)e2] if necessary, we may assume that the spaces π1(B)∗e1

and π2(B)e2 are dense in K1 and K2 respectively. Under these conditions, wededuce from (6.24) that

π1(c)v(x) = v(x)π2(c), x ∈ X, c ∈ B. (6.25)

Let H = K1⊕K2, and write any element of B(H) as a 2×2 matrix with respectto that decomposition. We define π : B → B(H) and θ : X → B(H) by letting

π(b) =(

π1(b) 00 π2(b)

)and θ(x) =

(0 v(x)0 0

), (6.26)

for x ∈ X and b ∈ B. Then (6.25) ensures that θ and π have commuting ranges.It is clear that θ and π are both completely contractive, and that π is a homo-morphism. Let ζ1 = (e1, 0) and ζ2 = (0, e2), elements in H . Then ‖ζ1‖ ≤ 1 and‖ζ2‖ ≤ 1. Moreover, 〈θ(x)π(b)ζ2, ζ1〉 = 〈v(x)π2(b)e2, e1〉, for x ∈ X and b ∈ B. If(et)t is a cai for B, then π1(et) → IK1 strongly. Hence

〈θ(x)π(b)ζ2 , ζ1〉 = limt〈π1(et)v(x)π2(b)e2, e1〉 = lim

tϕ(x ⊗ etb) = ϕ(x ⊗ b)

by (6.23). This proves (ii).


Corollary 6.4.4 (Pisier) Let X be an operator space and let B be an approxi-mately unital operator algebra. Then for any z ∈ X ⊗ B, we have

δ(z) = sup∥∥θ • π(z)

∥∥,

where the supremum runs over all triples (H, θ, π), in which H is a Hilbert space,θ : X → B(H) is a complete contraction, π : B → B(H) is a completely contrac-tive homomorphism, and θ and π have commuting ranges.

Proof This follows from (6.22), 6.4.3, and the Hahn–Banach theorem.

6.4.5 (Functoriality) Let u : X → Y be a completely contractive map betweentwo operator spaces, and let π : B → D be a completely contractive homo-morphism between approximately unital operator algebras. Then the mappingu ⊗ π : X ⊗ B → Y ⊗ D extends to a contraction from X⊗δB into Y ⊗δD. Thisfollows either directly from the definition of the delta norm, or from the abovecorollary, arguing as in 6.1.9.

In particular if X is a subspace of Y , and if B is a subalgebra of D, then wehave a canonical complete contraction X⊗δB → Y ⊗δD.

6.4.6 (The normal delta norm) For the purpose of later applications of thedelta norm, we will need the normal delta norm σδ. This is defined analoguouslyto the normal Haagerup tensor product of 1.6.8. Let X be a dual operator space,and let M be a unital dual operator algebra. We define

(X⊗δM

)∗σ⊂(X⊗δM

)∗to be the subspace of all functionals whose associated bilinear form X ×M → C

is separately w∗-continuous. Then we define the normal delta tensor productX⊗σδM to be the dual space of (X⊗δM

)∗σ. The norm of any z in X⊗σδM will

be denoted by σδ(z). We can regard X⊗M as a subspace of X⊗σδM in a obviousway. Clearly σδ(z) ≤ δ(z), for z ∈ X ⊗ M . In the notes for Section 6.6 we willshow that this inequality may be strict.

Lemma 6.4.7 If z ∈ X⊗σδM then σδ(z) ≤ 1 if and only if there exists a net(zt)t in X ⊗ M such that δ(zt) < 1 for any t, and 〈ϕ, z〉 = limt〈ϕ, zt〉, forϕ ∈

(X⊗δM

)∗σ.

Proof If σδ(z) ≤ 1, then the functional z :(X⊗δM

)∗σ→ C admits a contrac-

tive extension z ∈(X⊗δM

)∗∗ by the Hahn–Banach theorem. By Goldstine’sLemma (A.2.1), we can find a net (zt)t ⊂ X ⊗ M converging to z in the w∗-topology of

(X⊗δM

)∗∗, and such that δ(zt) < 1 for any t. This yields the ‘onlyif’ part. The converse is obvious.

Theorem 6.4.8 Let X be a dual operator space and let M be a unital dualoperator algebra. Given a functional ϕ : X⊗B → C, the following are equivalent:(i) ϕ extends to an element of

(X⊗δB

)∗σ, with ‖ϕ‖ ≤ 1.

(ii) There exist a Hilbert space H, vectors ζ1, ζ2 ∈ Ball(H), a w∗-continuouscompletely contractive map θ : X → B(H), and a w∗-continuous completely


contractive homomorphism π : M → B(H), such that θ and π have commut-ing ranges, and such that

ϕ(x ⊗ b) = 〈θ(x)π(b)ζ2 , ζ1〉, x ∈ X, b ∈ M.

Proof Clearly (ii) implies (i) by Theorem 6.4.3. For the other direction, weassume (i), and we will mimic the proof of the analoguous implication in Theorem6.4.3, with B = M . We will show that (6.23) can be achieved with the additionalproperty that π1 : M → B(K1), π2 : M → B(K2), and v : X → B(K2, K1), arew∗-continuous. Clearly, ϕ Q is a contractive functional on M ⊗h X ⊗h M .Since the product on M is separately w∗-continuous (see 2.7.4 (1)), and since ϕbelongs to

(X⊗δB

)∗σ, we see that the trilinear mapping associated to ϕ Q is

separately w∗-continuous as well. According to 1.6.10, there exist Hilbert spacesE1, E2, and w∗-continuous completely contractive maps w : X → B(E2, E1),u2 : M → B(C, E2) = Ec

2, and u1 : M → B(E1, C) = E∗r1 such that

ϕ(x ⊗ ab) = (ϕ Q)(a ⊗ x ⊗ b) = u1(a)w(x)u2(b),

for a, b ∈ M and x ∈ X . Applying a slight modification of Theorem 2.7.10,we find a Hilbert space K2, a vector e2 ∈ K2 with ‖e2‖ ≤ 1, a contractionW2 : K2 → E2, and a unital w∗-continuous completely contractive representationπ2 : M → B(K2) such that u2(b) = W2π2(b)e2, for b ∈ M . Likewise, u1 admits afactorization u1(a) = ξπ1(a)W1, where K1 is a Hilbert space, ξ is the functionalon K1 given by taking the inner product with a vector e1 ∈ Ball(K1), andπ1 : M → B(K1) is a unital w∗-continuous completely contractive representation,and W1 : E1 → K1 is a contraction. The linear map v = W1w(· )W2 is a w∗-continuous complete contraction, and it is clear from above that π1, π2, v, e1 ande2 satisfy (6.23) as expected. If we now follow the proof of 6.4.3, we see that πand θ defined by (6.26) are w∗-continuous. Hence that proof yields (ii).

Corollary 6.4.9 Let X be a dual operator space and let M be a unital dualoperator algebra. Then for any z ∈ X ⊗ M , we have

σδ(z) = sup∥∥θ • π(z)

∥∥,

where the supremum runs over all Hilbert spaces H, all w∗-continuous completelycontractive maps θ : X → B(H), and all w∗-continuous completely contractivehomomorphisms π : M → B(H), such that θ and π have commuting ranges.

Proof Pick z ∈ X ⊗M . By definition, σδ(z) = sup∣∣〈ϕ, z〉

∣∣, where the supre-mum is over all ϕ ∈

(X⊗δB

)∗σ

with ‖ϕ‖ ≤ 1. By Theorem 6.4.8 we deducethat σδ(z) is dominated by the supremum in the centered equation above. Tocheck the converse inequality, we assume that σδ(z) ≤ 1. It suffices to showthat ‖θ • π(z)‖ ≤ 1 for all θ and π as above. If ζ1, ζ2 ∈ Ball(H), define ϕ in(X⊗δM

)∗σ

by ϕ(x ⊗ b) = 〈θ(x)π(b)ζ2, ζ1〉, for x ∈ X and b ∈ M . Clearly ϕ iscontractive. Let (zt)t be the net in X ⊗M given by 6.4.7. Then

∣∣〈ϕ, zt〉∣∣ < 1 for

any t, and so∣∣〈ϕ, z〉

∣∣ ≤ 1. Taking the supremum over ζ1, ζ2 as above, we see that‖θ • π(z)‖ ≤ 1.


Proposition 6.4.10 Let X be an operator space and let B be an approximatelyunital operator algebra.(1) Given any ϕ ∈

(X⊗δB

)∗, the restriction of ϕ to X ⊗ B admits a uniqueextension to some Φ ∈ (X∗∗⊗δB

∗∗)∗σ. The resulting mapping ϕ → Φ is a

linear isometric isomorphism, that is,(X⊗δB

)∗= (X∗∗⊗δB

∗∗)∗σ

isometrically.

(2) We have (X⊗δB)∗∗ = X∗∗⊗σδB∗∗ isometrically.

(3) The following canonical embeddings are isometries:

X⊗δB ⊂ X∗∗⊗δB∗∗, X⊗δB ⊂ X∗∗⊗δB, X⊗δB ⊂ X⊗δB

∗∗.

Proof There is a canonical contraction j : X⊗δB → X∗∗⊗δB∗∗ given by 6.4.5.

The map Φ → Φ j from (X∗∗⊗δB∗∗)∗σ to (X⊗δB)∗ is a one-to-one linear

contraction. To prove that it is a surjective isometry, let ϕ ∈(X⊗δB

)∗. Recallfrom 6.4.2 that δ(z) ≤ ‖z‖h, for z ∈ X ⊗ B. It therefore follows from 1.6.7that ϕ : X ⊗ B → C extends to a (necessarily unique) separately w∗-continuousΦ: X∗∗ ⊗ B∗∗ → C. Our aim is to show that

Φ ∈ (X∗∗⊗δB∗∗)∗, with ‖Φ‖ ≤ ‖ϕ‖. (6.27)

We may assume that ‖ϕ‖ = 1. Fix z ∈ X∗∗ ⊗ B∗∗, with δ(z) < 1. By 6.4.1,there exists matrices a∗∗ = [a∗∗

1 · · · a∗∗n ] ∈ Rn(B∗∗), x∗∗ = [x∗∗

ij ] ∈ Mn(X∗∗),and b∗∗ = [b∗∗1 · · · b∗∗n ]t ∈ Cn(B∗∗), such that z =

∑x∗∗

ij ⊗a∗∗i b∗∗j with ‖b∗∗‖ < 1,

‖x∗∗‖ < 1 and ‖a∗∗‖ < 1. Since Mn(X∗∗) = Mn(X)∗∗ isometrically (see 1.4.11),Goldstine’s Lemma ensures that there is a net (xr)r in the unit ball of Mn(X)converging to x∗∗ in the w∗-topology of Mn(X)∗∗. Similarly, there exist nets(as)s in Ball(Rn(B)) and (bs′

)s′ in Ball(Cn(B)) converging to a∗∗ and b∗∗ in thew∗-topologies of Rn(B)∗∗ and Cn(B)∗∗ respectively. For any r, s, s′, we have

δ

( ∑1≤i,j≤n

xrij ⊗ as

i bs′j

)≤ 1,

and so ∣∣∣∣⟨ϕ,∑

1≤i,j≤n

xrij ⊗ as

i bs′j

⟩∣∣∣∣≤ 1 .

Taking the limit over r, we deduce from the w∗-continuity of Φ that∣∣∣∣⟨Φ,∑

1≤i,j≤n

x∗∗ij ⊗ as

i bs′j

⟩∣∣∣∣≤ 1

for any s, s′. Since the product on B∗∗ is separately w∗-continuous, we have

a∗∗i b∗∗j = w∗ − lim

slims′

asi b

s′j , 1 ≤ i, j ≤ n.

Then we finally deduce from the w∗-continuity of Φ in the second variable that|〈Φ, z〉| ≤ 1. This shows (6.27).

254 Factorization through matrix spaces

Item (2) follows from (1) at once. The first assertion in (3) follows from thefirst line of the proof of (1), together with (2) and the fact that σδ(z) ≤ δ(z).The proofs of the rest of (3) are similar.

6.5 FACTORIZATION THROUGH MATRIX SPACES

6.5.1 (Definition of ∆(u)) In this section we will describe the delta norm interms of factorizations through matrix spaces, by finite rank operators. Let Ybe an operator space and let B be an approximately unital operator algebra.We will use the canonical identification between Y ∗ ⊗ B and the space of finiterank operators from Y into B. If u : Y → B is such a map, and if z ∈ Y ∗ ⊗B isassociated to u, we define

∆(u) = δ(z).

If z =∑

xij ⊗ aibj , for [xij ] ∈ Mn(Y ∗), and a1, . . . , an, b1, . . . , bn in B, then

u(y) =∑

〈xij , y〉aibj, y ∈ Y. (6.28)

Let α : Y → Mn be associated to [xij ] via the relation Mn(Y ∗) = CB(Y, Mn).Then (6.28) asserts that u = βα, where β : Mn → B is the mapping defined by

β(Eij) = aibj , 1 ≤ i, j ≤ n. (6.29)

Here as usual (Eij) are the matrix units of Mn. Thus ∆(u) is given by the

expression inf‖α‖cb

∥∥[a1 · · · an]∥∥∥∥[b1 · · · bn]t

∥∥, where the infimum runs overall n ≥ 1, and all possible factorizations u = βα, with β defined by (6.29).

The following gives an alternative description of ∆(u) if Y = Mn:

Proposition 6.5.2 Let n ≥ 1 be an integer, let B be an approximately unitaloperator algebra, and let u : Mn → B be a linear mapping. Then ∆(u) < 1 if andonly if there exist an integer m ≥ 1, and two finite families (aik)1≤i≤n,1≤k≤m,(bkj)1≤j≤n,1≤k≤m in B such that∥∥∥∥∑

i,k

aika∗ik

∥∥∥∥< 1,

∥∥∥∥∑j,k

b∗kjbkj

∥∥∥∥< 1, (6.30)

and for s = [sij ] in Mn,u(s) =

∑i,j,k

aiksijbkj . (6.31)

Proof We shall use the dual operator space S1n = M∗

n = Rn ⊗h Cn. By theassociativity of the Haagerup tensor product and 1.5.14 (3), we can write

B ⊗h S1n ⊗h B ∼= Rn(B) ⊗h Cn(B).

Let u : Mn → B with ∆(u) < 1, and let z ∈ S1n ⊗ B be associated to u. Then

there exist w ∈ B ⊗ S1n ⊗ B such that ‖w‖h < 1 and Q(w) = z, where Q is the


quotient map considered in (6.21) for X = S1n. Suppose that w ∈ Rn(B)⊗Cn(B)

corresponds to w via the last centered equation. Thus ‖w‖h < 1, and so thereexist an integer m ≥ 1, and m-tuples c1, . . . , cm in Rn(B) and d1, . . . , dm inCn(B), such that

w =∑

1≤k≤m

ck ⊗ dk,∥∥(ck)k

∥∥Rm(Rn(B))

< 1, and∥∥(dk)k

∥∥Cm(Cn(B))

< 1.

However Rm(Rn(B)) = Rnm(B) and Cm(Cn(B)) = Cnm(B) isometrically. Writ-ing each ck ∈ Rn(B) as ck =

∑i aik ⊗ ei, with a1k, . . . , ank ∈ B, we have

∥∥(ck)k

∥∥Rm(Rn(B))

=∥∥∥∥∑

i,k

aika∗ik

∥∥∥∥ 12

.

Likewise, writing dk =∑

j ej ⊗ bkj , with bk1, . . . , bkn ∈ B, we have

∥∥(dk)k

∥∥Cm(Cn(B))

=∥∥∥∥∑

j,k

b∗kjbkj

∥∥∥∥ 12

.

It is now easy to check that u satisfies (6.31), which completes the proof of the‘only if’ part. The ‘if’ part is obtained by simply reversing the arguments.

6.5.3 (A factorization property) We may regard the above result as a specialfactorization property for maps u : Mn → B. If a = [aik] ∈ Mn,m(B), and ifb = [bkj ] ∈ Mm,n(B), then we will say that the mapping u : Mn → B defined by(6.31) is ‘implemented’ by the pair (a, b). Let W and V be a and b respectively,considered as ‘columns’ in Cnm(B). If C is any unital C∗-algebra containing B,consider the ∗-representation π : Mn → Mnm(C) taking s ∈ Mn to s⊗ Im ⊗ IC .It is easy to check that u = W ∗π(· )V . In particular, if B is a C∗-algebra and ifb = a∗, then V = W and u is completely positive. In this case, 1.3.4 gives

‖u‖ = ‖u‖cb ≤ ‖V ‖2 =∥∥∥∑

i,k

aika∗ik

∥∥∥.The following result reduces the computation of ∆(u) for general finite rank

operators, to the case considered in 6.5.2.

Lemma 6.5.4 For any finite rank operator u : Y → B, we have

∆(u) = inf‖α‖cb ∆(β)

,

where the infimum runs over all integers n ≥ 1, and all possible factorizationsu = βα, with Y

α−→ Mnβ−→ B.

Proof That ∆(u) dominates the infimum is clear from the last line in 6.5.1,together with 6.5.2. For the reverse inequality, we assume that u = βα is a


factorization of u through Mn, and we fix ε > 0. There exist an m ∈ N, a linearmap γ : Mn → Mm, and a1, . . . , am, b1, . . . , bm in B, with

‖γ‖cb

∥∥[a1 · · · an]∥∥ ∥∥[b1 · · · bn]t

∥∥ < ∆(β) + ε.

Also, β = β′γ, where the map β′ : Mm → B is defined by β′(Epq) = apbq, for1 ≤ p, q ≤ m. Write u = β′(γα). Since ‖γα‖cb ≤ ‖γ‖cb‖α‖cb, we have

∆(u) ≤ ‖α‖cb‖γ‖cb

∥∥[a1 · · · an]∥∥∥∥[b1 · · · bn]t

∥∥ .

Hence ∆(u) ≤ ‖α‖cb(∆(β) + ε). Letting ε → 0 yields the result.

6.5.5 (Factorization of w∗-to-norm continuous maps) Let u : Y → B be a finiterank operator from an operator space Y into an approximately unital operatoralgebra B. Assume that Y = Z∗ is a dual operator space and that u is w∗-to-norm continuous. It is easy to argue that this is equivalent to u being canonicallyassociated to some z ∈ Z ⊗ B. It follows from 6.4.10 (3) that ∆(u) = ‖z‖Z⊗δB.This implies that ∆(u) = inf

‖α‖cb ∆(β)

, where the infimum runs over possible

factorizations u = βα, where α : Y → Mn is w∗-continuous, and β : Mn → B.

6.5.6 (Decomposable maps) In the rest of this section, we focus on the casewhen B is a C∗-algebra. It turns out that in this case, the delta norm and itscompanion ∆, are closely related to Haagerup’s decomposable norm. We shallonly give a brief introduction to this topic and refer the reader to [180] for detailsand complements (see also [149, Section 5.4] and the Notes to this section).

Let B and C be two C∗-algebras. A linear mapping u : C → B is said tobe decomposable provided that u is a linear combination of completely positivemaps from C into B. It is plain that in that case, u may be written as

u = (u1 − u2) + i(u3 − u4), for completely positive uj : C → B. (6.32)

We define DEC(C, B) to be the vector space of all such maps. Note, for example,that any finite rank operator between C∗-algebras is decomposable.

Recall for any u : C → B, the mapping u : C → B defined by u(c) = u(c∗)∗

(see 1.2.25). A key observation is that u : C → B is decomposable if and only ifit is the ‘1-2-corner’ of a completely positive corner-preserving map

w =[

Φ1 uu Φ2

]: M2(C) −→ M2(B). (6.33)

Here Φ1, Φ2 are (necessarily completely positive) maps from C into B. To checkthis observation, observe that if (6.33) defines a completely positive map, thenv(λ) = Φ1 + λu + λu + Φ2 : C → B is completely positive, for any complexnumber λ such that |λ| = 1. Hence (6.32) holds, with 4u1 = v(1), 4u2 = v(−1),4u3 = v(i), and 4u4 = v(−i). To see the other direction, let λ be as above, andlet U(λ) be the diagonal unitary 2× 2 matrix with entries 1 and λ. If Φ: C → B


is a map which is completely positive, then so is U(λ)∗Φ2(·)U(λ). Hence if usatisfies (6.32), then the four operators[

u1 u1

u1 u1

],

[u2 −u2

−u2 u2

],

[u3 iu3

−iu3 u3

],

[u4 −iu4

iu4 u4

]are all completely positive. Their sum w : M2(C) → M2(B) is a completelypositive map of the required form (6.33), with Φ1 = Φ2 = u1 + u2 + u3 + u4.

For any u ∈ DEC(C, B), we define

‖u‖dec = inf‖w‖,

where the infimum runs over all completely positive corner-preserving mapsw : M2(C) → M2(B) with 1-2-corner equal to u. It is clear that ‖ · ‖dec is anorm on DEC(C, B). For example, to see that ‖λu‖dec = |λ|‖u‖dec, first notethat this is easy if λ ≥ 0. On the other hand, if |λ| = 1, and if w is as in(6.33) and if U(λ) is as above, then U(λ)∗w(·)U(λ) is completely positive too,with 1-2-corner λu. Thus ‖λu‖dec ≤ ‖u‖dec; and the other direction follows bysymmetry.

Let u : C → B be decomposable, and let Φ1, Φ2 : C → B be such that themapping w defined by (6.33) is completely positive. Let A be a C∗-algebra.Then IA ⊗ w extends to a completely positive map from A ⊗max M2(C) intoA ⊗max M2(B), whose norm is dominated by ‖w‖ (see 6.1.10 and its proof).Since A⊗max M2(C) = M2(A⊗max C) and A⊗max M2(B) = M2(A⊗max B) (see(6.7)), we deduce that IA ⊗ u is bounded from A ⊗max C into A ⊗max B, withnorm dominated by ‖w‖. Taking the infimum over all such w’s, we obtain:∥∥IA ⊗ u : A ⊗max C −→ A ⊗max B

∥∥cb

≤ ‖u‖dec. (6.34)

Arguing as in the proof of Lemma 6.1.10, it is easy to deduce that (6.34) holdsas well for a nonselfadjoint approximately unital operator algebra A.

Proposition 6.5.7 Let n ≥ 1 be a positive integer, let B be a C∗-algebra, andlet u : Mn → B be a linear mapping. Then ∆(u) = ‖u‖dec.

Proof We will prove that ∆(u) ≤ ‖u‖dec as a consequence of Corollary 6.4.4.Let z ∈ S1

n ⊗B be associated to u. Let H be a Hilbert space, let θ : S1n → B(H)

be a complete contraction, let π : B → B(H) be a ∗-representation, and assumethat θ and π have commuting ranges. We let M = [π(B)]′ ⊂ B(H) be thecommutant of the range of π, so that θ is valued in M . Thus θ : S1

n → M isassociated with some y ∈ Mn ⊗ M . We write

z =∑

k

ϕk ⊗ bk and y =∑

p

σp ⊗ cp,

where (ϕk)k ⊂ S1n, (bk)k ⊂ B, (σp)p ⊂ Mn, and (cp)p ⊂ M . Then


θ • π(z) =∑

k

θ(ϕk)π(bk) =∑k,p

〈σp, ϕk〉π(bk)cp.

On the other hand,(u ⊗ IM

)(y) =

∑p

u(σp) ⊗ cp =∑k,p

〈σp, ϕk〉bk ⊗ cp.

Let Ψ: B ⊗max M → B(H) be the completely contractive mapping taking anyb⊗c to π(b)c. The above calculation shows that θ•π(z) = Ψ

[(u⊗IM )(y)

], hence∥∥θ • π(z)

∥∥ ≤ ‖Ψ‖ ‖u‖dec ‖y‖max,

by property (6.34). Since Mn ⊗max M = Mn ⊗min M = CB(S1n, M), we have

‖y‖max = ‖y‖min = ‖θ‖cb ≤ 1.

Since ‖Ψ‖ ≤ 1, we have proved that∥∥θ • π(z)

∥∥ ≤ ‖u‖dec. We deduce by 6.4.4that δ(z) ≤ ‖u‖dec. That is, ∆(u) ≤ ‖u‖dec.

Conversely, if ∆(u) < 1, then by 6.5.2 there are matrices [aik] ∈ Mn,m ⊗ Band [bkj ] ∈ Mm,n ⊗ B satisfying (6.30) and (6.31). For any 1 ≤ i ≤ n and1 ≤ k ≤ m, consider the elements of M2(B) defined by

c′ik = E11 ⊗ aik and c′′ik = E21 ⊗ b∗ki.

Here E11 and E22 denote the diagonal matrix units of M2. Define

c =[

(c′ik)(c′′ik)

]∈ M2n,m ⊗ M2(B).

Let w : M2n → M2(B) be the linear mapping implemented by the pair (c, c∗), inthe notation of 6.5.3. Then the discussion in 6.5.3 shows that w is completelypositive. Furthermore, by the last line of 6.5.3,

‖w‖ ≤ max∥∥∥∑ aika∗

ik

∥∥∥ ,∥∥∥∑ b∗kjbkj

∥∥∥ < 1.

Under the identification M2n = M2(Mn), the mapping w is corner-preserving andits 1-2-corner is equal to u. For example, to see the last statement, note that fori, j ∈ 1, . . . , n, we have by careful inspection that w(E12 ⊗Eij) =

∑k c′ik c′′jk

∗.However this last quantity equals∑

k

(E11 ⊗ aik)(E12 ⊗ bkj) = E12 ⊗∑

k

aikbkj = E12 ⊗ u(Eij),

as expected. We deduce that ‖u‖dec < 1.


Corollary 6.5.8 Let Y be an operator space, let B be a C∗-algebra, and letu : Y → B be any finite rank operator. Then

∆(u) = inf‖α‖cb‖β‖dec,

where the infimum runs over all possible factorizations of u of the form u = βα,with Y

α−→ Mnβ−→ B.

Proof Combine Lemma 6.5.4 and Proposition 6.5.7.

6.6 NUCLEARITY AND SEMIDISCRETENESS FOR LINEAROPERATORS

6.6.1 (Nuclear C∗-algebras and semidiscrete W ∗-algebras) We briefly reviewsome fundamental results from C∗-theory, which will serve as a motivation anda guide for the results presented in this section. Let B be a C∗-algebra. ThenB is nuclear (i.e. A ⊗min B = A ⊗max B for every C∗-algebra A) if and only

if there exist nets of completely positive contractive maps Bαt−→ Mnt

βt−→ Bsuch that b = limt βtαt(b) for every b ∈ B. This result is due to Choi andEffros [89], and to Kirchberg [227]. The above property is usually referred toas the completely positive approximation property. Smith showed in [390] thatreplacing the words ‘completely positive contractive’ in the above, by ‘completelycontractive’, gives an equivalent characterization of nuclear C∗-algebras. A proofof Smith’s result may be found in 7.1.12 below. Likewise, if M is a W ∗-algebra,then A⊗min M = A⊗nor M for every C∗-algebra A if and only if there exist netsof completely positive contractions M

αt−→ Mnt

βt−→ M such that for b ∈ M wehave b = w∗-limt βtαt(b). This was proved by Effros and Lance [140], who calledsuch W ∗-algebras semidiscrete. Furthermore, it is known that a W ∗-algebra issemidiscrete if and only if it is injective [419]; and that a C∗-algebra B is nuclearif and only if its second dual B∗∗ is semidiscrete [89]. These two results rely onConnes’ famous work on injectivity [99].

Delta norms were introduced by Pisier (see [337]) in order to give a newproof of the Choi–Effros–Kirchberg characterization of nuclearity. We will seethat they can also be used to obtain approximation properties for certain classesof operator algebra valued linear maps.

6.6.2 (K-Nuclear operators) Let Y be an operator space, let B be an approxi-mately unital operator algebra, and let u : Y → B be a bounded operator. Givenany constant K ≥ 0, we say that u is K-nuclear if whenever A is an approxi-mately unital operator algebra, then IA ⊗u extends to a bounded operator fromA ⊗min Y into A ⊗max B, with∥∥IA ⊗ u : A ⊗min Y −→ A ⊗max B

∥∥ ≤ K. (6.35)

(We warn the reader that this notion of ‘nuclearity’ for maps is quite differentfrom the class of maps of the same name introduced by Grothendieck, which

260 Nuclearity and semidiscreteness for linear operators

there are also operator space variants of (e.g. see [149, Section 12.1]). See also7.1.1 below, for a different notion yet).

Some comments on our definition:(1) Replacing A by Mn(A) and applying Corollary 6.1.15 and (1.36), we see

that if u : Y → B is a K-nuclear map, then for any A as above, the tensor mapIA⊗u is completely bounded from A⊗minY into A⊗maxB, with ‖IA⊗u‖cb ≤ K.In particular, any K-nuclear map u is completely bounded, with ‖u‖cb ≤ K.

(2) We say that an operator algebra B is K-nuclear if the identity mapping IB

is K-nuclear. If B is a C∗-algebra, then 6.1.14 says that B is nuclear in the clas-sical sense if and only if it is 1-nuclear. Moreover if a C∗-algebra B is K-nuclearfor some K ≥ 1, then it is 1-nuclear. Indeed if A is a C∗-algebra, and if the iden-tity mapping I : A⊗min B → A⊗max B is bounded, then it is a ∗-representation,and hence is a contraction. Thus all of these notions of nuclearity coincide forC∗-algebras. The situation is rather different if B is nonselfadjoint. We will seein 7.1.8 that if an approximately unital operator algebra B is 1-nuclear, thenit has to be a C∗-algebra. We have met (for example, in 6.2.4) nonselfadjointoperator algebras B which satisfy A ⊗max B = A ⊗min B isometrically for anyC∗-algebra A; however such an algebra B need not be 1-nuclear. On the otherhand, any nonselfadjoint finite-dimensional operator algebra is clearly K-nuclearfor some K > 1, but not 1-nuclear (by 7.1.8).

(3) Let u : Y → B be a bounded operator, let K ≥ 0, and assume that B isa C∗-algebra. One can show by a variant of the proofs of 6.1.10 or 6.1.14, thatu is K-nuclear provided that (6.35) holds for any C∗-algebra A.

6.6.3 (K-Semidiscrete operators) Let Y be an operator space, let M be a uni-tal dual operator algebra, and let u : Y → M be a bounded operator. Given anyconstant K ≥ 0, we say that u is K-semidiscrete if whenever A is an approxi-mately unital operator algebra, then IA ⊗u extends to a bounded operator fromA ⊗min Y into A ⊗nor M , with∥∥IA ⊗ u : A ⊗min Y −→ A ⊗nor M

∥∥ ≤ K.

Similar comments to those in 6.6.2 apply to this definition. In particular, if Mis a W ∗-algebra, then M is semidiscrete in the classical sense if and only if it is1-semidiscrete, and if and only if it is K-semidiscrete for some K ≥ 1.

Theorem 6.6.4 (Pisier) Let Y be an operator space and let B be an approxi-mately unital operator algebra. We consider a bounded operator u : Y → B anda constant K ≥ 0. Then the following assertions are equivalent:(i) u is K-nuclear (in the sense of 6.6.2).(ii) For any operator space X, the mapping IX ⊗u extends to a bounded operator

from X ⊗min Y into X⊗δB, with∥∥IX ⊗ u : X ⊗min Y −→ X ⊗δ B∥∥ ≤ K.

(ii)’ Same as (ii) for finite-dimensional operator spaces X.


(iii) There exists a net ut : Y → B of finite rank operators converging to u in thepoint-norm topology, such that ∆(ut) < K for each t.

(iv) There exist nets of maps Yαt−→ Mnt

βt−→ B, such that βtαt converges to uin the point-norm topology, and such that ‖αt‖cb ∆(βt) < K for each t.

If further B is a C∗-algebra, these conditions are equivalent to:

(iv)’ There exist nets of maps Yαt−→ Mnt

βt−→ B, such that βtαt converges to uin the point-norm topology, and such that ‖αt‖cb‖βt‖dec < K for each t.

Proof The equivalence between (iii) and (iv) (and (iv)’ in the C∗-case) is anobvious consequence of Lemma 6.5.4 (and Corollary 6.5.8).

In order to prove ‘(iii) ⇒ (i)’, we shall first show that if u : Y → B is finiterank, then it is ∆(u)-nuclear. Going back to the definition of ∆(u), we assumethat u = βα, where β : Mn → B is defined by (6.29) for some [a1 · · · an] inRn(B) and [b1 · · · bn]t in Cn(B), and α : Y → Mn is a linear mapping. If A isan approximately unital operator algebra and [cij ] ∈ Mn(A), we have

(IA ⊗ β)([cij ]) =∑i,j

cij ⊗ aibj .

Hence δ((IA⊗β)([cij ])

)≤ ‖[cij ]‖‖[a1 · · · an]‖ ‖[b1 · · · bn]t‖. Since the delta norm

dominates the maximal tensor norm on A ⊗ B (see 6.4.2), this shows that∥∥IA ⊗ β : Mn(A) −→ A ⊗max B∥∥ ≤

∥∥[a1 · · · an]∥∥∥∥[b1 · · · bn]t

∥∥.Since Mn(A) = Mn ⊗min A, we have∥∥IA ⊗ α : A ⊗min Y −→ Mn(A)

∥∥ ≤ ‖α‖cb.

Hence we obtain by composition that∥∥IA ⊗ u : A ⊗min Y −→ A ⊗max B∥∥ ≤ ‖α‖cb

∥∥[a1 · · · an]∥∥∥∥[b1 · · · bn]t

∥∥.According to 6.5.1, this shows that u is ∆(u)-nuclear.

Assume (iii), let A be an approximately unital operator algebra, and letz ∈ A ⊗ Y with ‖y‖min ≤ 1. Applying the above to each ut, we obtain that∥∥(IA ⊗ut)(z)

∥∥max

< K. Since (IA ⊗ut)(z) converges to (IA ⊗u)(z) in A⊗max B,this implies that ∥∥(IA ⊗ u)(z)

∥∥max

≤ K.

This shows that u is K-nuclear.Assume (i), and let X be an operator space. To show (ii), it will suffice

by 6.4.4 to show that for any pair of commuting completely contractive mapsθ : X → B(H) and π : B → B(H) such that π is a homomorphism, we have∥∥(θ • π) (IX ⊗ u) : X ⊗min Y −→ B(H)

∥∥ ≤ K. (6.36)


For such θ and π, consider the commutant A = [π(B)]′ ⊂ B(H). Then∥∥θ ⊗ IY : X ⊗min Y −→ A ⊗min Y∥∥ ≤ 1. (6.37)

We let Φ: A⊗max B → B(H) be the map taking a⊗ b to aπ(b), which is clearlycontractive. Also, (θ • π) (IX ⊗ u) = Φ (IA ⊗ u) (θ ⊗ IY ) on X ⊗ Y . Henceapplying (6.37) and appealing to (6.35), we deduce the desired estimate (6.36).This yields (ii).

Finally, we assume (ii)’ and will show (iii). Let F be the set of all finite-dimensional subspaces E of Y , ordered by inclusion. Our net (ut)t will be indexedby F . We let (εt)t be a net of positive real numbers with limt εt = 0. Given anyt = E in F , we apply (ii)’ with X = E∗. If j ∈ X⊗Y is associated to the canonicalembedding of E into Y , then (IX ⊗u)(j) ∈ X⊗B is associated to the restrictionu|E : E → B of u to E. Since ‖j‖min = 1 (by 1.35), we have ∆(u|E) ≤ K. By6.5.4, there exist n ≥ 1, and linear maps α : E → Mn and β : Mn → B, suchthat u|E = βα, ‖α‖cb ≤ 1, and ∆(β) < K(1 + εt). By the injectivity of Mn (see1.2.10), α admits a completely contractive extension α : Y → Mn. Define

ut = (1 + 2εt)−1βα : Y −→ B.

By 6.5.4 we have ∆(ut) ≤ (1+2εt)−1‖α‖cb∆(β) < K. For y ∈ Y , the net (ut(y))t

is eventually equal to (1 + 2εt)−1u(y), and hence converges to u(y).

Theorem 6.6.5 Let Y be an operator space and let M be a unital dual operatoralgebra. We consider a bounded operator u : Y → M and a constant K ≥ 0.Then the following assertions are equivalent:

(i) u is K-semidiscrete (in the sense of 6.6.3).(ii) For any finite-dimensional operator space X, the mapping IX ⊗u extends to

a bounded operator from X ⊗min Y into X⊗σδB, with∥∥IX ⊗ u : X ⊗min Y −→ X⊗σδB∥∥ ≤ K.

(iii) There exists a net ut : Y → M of finite rank operators converging to u inthe point-w∗ topology, such that ∆(ut) < K for each t.

(iv) There exist nets of maps Yαt−→ Mnt

βt−→ M such that βtαt converges to uin the point-w∗ topology, such that ‖αt‖cb∆(βt) < K for each t.

If further M is a W ∗-algebra, these conditions are equivalent to:

(iv)’ There exist nets of maps Yαt−→ Mnt

βt−→ M so that βtαt converges to u inthe point-w∗ topology, and such that ‖αt‖cb‖βt‖dec < K for each t.

Proof As in 6.6.4, we only need to show the equivalence between (i), (ii),and (iii). We omit the proof that (i) implies (ii), since this is similar to thecorresponding implication in Theorem 6.6.4 (but uses 6.4.9 in place of 6.4.4).


To see that (iii) implies (i), let A be an approximately unital operator algebra.We consider an arbitrary element

∑k ak ⊗ yk in A ⊗ Y , (ak)k and (yk)k being

finite families of A and Y respectively. It follows from 6.6.4 that∥∥∥∑k

ak ⊗ ut(yk)∥∥∥

max≤ K

∥∥∥∑k

ak ⊗ yk

∥∥∥min

,

for any t. Let π : A → B(H) and ρ : M → B(H) be two commuting completelycontractive homomorphisms, such that ρ is w∗-continuous. Then∥∥∥∑

k

π(ak)ρ(ut(yk)

)∥∥∥ ≤ K∥∥∥∑

k

ak ⊗ yk

∥∥∥min

, (6.38)

for any t. The w∗-continuity of ρ ensures that ρ(u(yk)

)= w∗-limt ρ

(ut(yk)

), for

any k. We deduce that

π • ρ(∑

k

ak ⊗ u(yk))

=∑

k

π(ak)ρ(u(yk)

)= w∗ − lim

t

∑k

π(ak)ρ(ut(yk)

).

Taking (6.38) into account, this yields∥∥∥π • ρ(∑

k

ak ⊗ u(yk))∥∥∥ ≤ K

∥∥∥∑k

ak ⊗ yk

∥∥∥min

.

Taking the supremum over all such pairs (π, ρ), we deduce that∥∥∥∑k

ak ⊗ u(yk)∥∥∥

nor≤ K

∥∥∥∑k

ak ⊗ yk

∥∥∥min

.

This proves that u is K-semidiscrete.We now assume (ii), and will show (iii). Let F1 be the set of all finite subsets

I1 of Y , let F2 be the set of all finite subsets I2 of M∗, and set F = F1×F2×(0, 1).We endow this set with its canonical order, given by

(I1, I2, ε) ≤ (I ′1, I′2, ε

′) ⇐⇒ I1 ⊂ I ′1, I2 ⊂ I ′2, ε ≥ ε′.

We shall define a net (ut)t indexed by F which satisfies (iii). Fix t = (I1, I2, ε)in F , and let E ⊂ Y be the finite-dimensional subspace spanned by I1. Applying(ii) with X = E∗, and arguing as in the proof of Theorem 6.6.4, we find that thetensor z ∈ X⊗M associated to the restriction u|E of u to E satisfies σδ(z) ≤ K.By Lemma 6.4.7, there exists a net (zs)s of X ⊗ M with δ(zs) < K convergingto z in the w∗-topology of X⊗σδM . Let ws : E → M be associated to zs; thenwe have ∆(ws) < K. The convergence of zs to z is easily seen to imply that(ws) converges to u|E in the point w∗-topology. Thus we may find an operatorw : E → M such that ∆(w) < K, and∣∣〈u(y), ϕ〉 − 〈w(y), ϕ〉

∣∣ < ε, y ∈ I1, ϕ ∈ I2. (6.39)

As in the proof of 6.6.4, we may extend w to some ut : Y → M with ∆(ut) < K.It clearly follows from (6.39) that the net (ut) satisfies (iii).


6.6.6 (A characterization of nuclear and semidiscrete operator algebras) Let Bbe an approximately unital operator algebra and let K ≥ 1. Applying Theorem6.6.4 to IB, we see that B is K-nuclear (in the sense of 6.6.2 (2)) if and only

if there exist nets of maps Bαt−→ Mnt

βt−→ B such that ‖αt‖cb ∆(βt) < K, andb = limt βtαt(b), for b ∈ B. If B is a 1-nuclear C∗-algebra then with some morework (e.g. see [337, Section 12]) one can also ensure that αt, βt are completelypositive, recovering one direction of the Choi–Effros–Kirchberg result mentionedin 6.6.1. In any case, since ‖βt‖cb ≤ ∆(βt) (as is easily argued from 6.5.1 say, orfrom the comparison ‖ · ‖min ≤ δ(·) proved in 6.4.2), we can deduce as a specialcase of what we did above, that if B is 1-nuclear then there exists an ‘approximatefactorization’ of IB as above, where αt, βt are completely contractive.

Likewise, if M is a unital dual operator algebra, Theorem 6.6.5 implies thatM is K-semidiscrete if and only if there exist nets of maps M

αt−→ Mnt

βt−→ Msuch that ‖αt‖cb ∆(βt) < K, and b = w∗-limt βtαt(b), for b ∈ M .

6.6.7 (Local reflexivity) Let Y be an operator space. For a finite-dimensionaloperator space X , we may canonically identify (X ⊗min Y )∗∗ and X ⊗min Y ∗∗ astopological vector spaces. The corresponding ‘identity mapping’

JX : (X ⊗min Y )∗∗ −→ X ⊗min Y ∗∗ (6.40)

is a contraction. To see this, consider z ∈ X ⊗ Y ∗∗ and let u : X∗ → Y ∗∗ be theassociated linear mapping. If ‖z‖(X⊗minY )∗∗ ≤ 1, then there exists by Goldstine’sLemma (see A.2.1), a net (zt)t in X⊗Y which converges to z in the w∗-topology,such that ‖zt‖X⊗minY ≤ 1 for any t. If we let ut : X∗ → Y be the linear mappingassociated to zt, this is equivalent to saying that each ut is a complete contraction(by (1.32)), and that u(ϕ) is the w∗-limit of ut(ϕ), for ϕ ∈ X∗. These immediatelyimply that ‖u‖cb ≤ 1, that is, ‖z‖X⊗minY ∗∗ ≤ 1.

The inverse mapping J−1X is not a contraction in general, that is, the two

norms on X ⊗Y ∗∗ given by (X ⊗min Y )∗∗ and X ⊗min Y ∗∗ may be different (seethe Notes to this section for more). Given a constant K ≥ 1, we say that Y isK-locally reflexive if ‖J−1

X ‖ ≤ K for any finite-dimensional operator space X .

Corollary 6.6.8 Let Y be an operator space, let B be an approximately unitaloperator algebra, and let u : Y → B be a bounded operator.(1) Let K ≥ 0. If the second adjoint u∗∗ : Y ∗∗ → B∗∗ is K-semidiscrete, then u

is K-nuclear.(2) Let K ≥ 1 and K ′ ≥ 1, and assume that Y is K ′-locally reflexive. If u is

K-nuclear, then u∗∗ is KK ′-semidiscrete.

Proof Assume that u∗∗ is K-semidiscrete, and let X be a finite-dimensionaloperator space. Since the mapping JX in (6.40) is a contraction, it follows fromTheorem 6.6.5 that∥∥IX ⊗ u∗∗ : (X ⊗min Y )∗∗ −→ X⊗σδB

∗∗∥∥ ≤ K. (6.41)


Hence by 6.4.10 (2), we deduce that∥∥IX ⊗ u : X ⊗min Y −→ X⊗δB∥∥ ≤ K.

This shows that u is K-nuclear by Theorem 6.6.4.Assume conversely that u is K-nuclear. Reversing the arguments above, we

see that (6.41) holds true for any finite-dimensional operator space X . Composingwith J−1

X , we deduce that∥∥IX ⊗ u∗∗ : X ⊗min Y ∗∗ −→ X⊗σδB∗∗∥∥ ≤ K ‖J−1

X ‖.

Now if Y is K ′-locally reflexive, this is less than or equal to KK ′. Hence u∗∗ isKK ′-semidiscrete by Theorem 6.6.5.

We will continue our discussion of nuclearity and semidiscreteness in the firstsection of the next chapter.


6.1: If A and B are C∗-algebras, then their algebraic tensor product A⊗B is a∗-algebra: we set (a⊗b)∗ = a∗⊗b∗, for a ∈ A and b ∈ B. By definition a C∗-normis a norm ‖ · ‖α on A ⊗ B such that ‖xy‖α ≤ ‖x‖α‖y‖α, ‖y∗y‖α = ‖y‖2

α, and‖y‖α = ‖y∗‖α for any x, y ∈ A⊗B. Thus the resulting completion A⊗αB is a C∗-algebra. The study of C∗-norms goes back at least to the work of Turumaru [413],Takesaki [406], who proved that ‖ · ‖min is the smallest C∗-norm on A ⊗ B, andGuichardet [174], who proved that ‖·‖max is the greatest one. The term ‘nuclear’was introduced by Lance in the study [238]. The reader is referred, for example,to [407, Chapter IV], or to [420], for the basics on C∗-norms, and their history.We will not attempt below to reference carefully the huge number of papersdevoted to the selfadjoint versions of topics in this chapter.

The maximal tensor product for nonselfadjoint operator algebras was in-troduced by Paulsen and Power [316]. This is a generalization of the maximalC∗-norm for C∗-algebras (see 6.1.4). However there seems to be no good ana-logue of the theorem of Takesaki which asserts that ‖ · ‖min ≤ ‖ · ‖α if ‖ · ‖α is aC∗-tensor norm. Indeed in [316], an example is given of two unital operator alge-bras A and B and an algebra homomorphism Φ: A⊗B → B(H) such that eachof the maps a ∈ A → Φ(a⊗ 1B) and b ∈ B → Φ(1A ⊗ b) is a complete isometry,but for which ‖Φ(a ⊗ b)‖ < ‖a‖‖b‖ for some a ∈ A, b ∈ B. The normal tensorproduct for C∗-algebras was introduced by Effros and Lance in [140]. Its exten-sion to nonselfadjoint operator algebras appears in [316] under another name,see also [249]. There is also a binormal tensor product on pairs of W ∗-algebraswhich was introduced in [140]. It can be extended to the nonselfadjoint settingas follows. Let N and M be two unital dual operator algebras. For any n ≥ 1and any y ∈ Mn(N ⊗ M), define ‖y‖bin = sup‖(IMn ⊗ (π • ρ))(y)‖, where theinfimum runs over all pairs of commuting w∗-continuous completely contractivehomomorphisms π : N → B(H) and ρ : M → B(H). These are matrix norms on


N ⊗ M , and the resulting completion N ⊗bin M is a unital operator algebra. Itfollows from 6.1.7 that if A, B are approximately unital operator algebras, andif θ : A ⊗max B → M is a completely contractive homomorphism into a dualoperator algebra M , then θ extends to a separately w∗-continuous completelycontractive homomorphism θ : A∗∗ ⊗bin B∗∗ → M . Applying that property withM = (A⊗maxB)∗∗, one may deduce that A∗∗⊗binB∗∗ ⊂ (A⊗maxB)∗∗ completelyisometrically.

The two statements 6.1.2 and 6.1.14 are due to Paulsen and Power [316]. Thispaper only considers unital operator algebras, in which case Lemma 6.1.7, andhence Lemma 6.1.13 are quite transparent. We could not find any reference forthese statements in the nonunital case; or for the result in 6.1.11. Lemma 6.1.10is from [252].

Variants of the maximal tensor product, and nuclearity and semidiscreteness,for other classes of operator spaces such as TROs (see 4.4.1), have been intro-duced and studied by Kirchberg, and Kaur and Ruan, and others. See [226], forexample.

6.2: Besides Ando’s theorem, the main sources for joint dilations and theirrelationships with the maximal tensor product are the two papers [316] and [315]by Paulsen and Power. Related work also appears in [114] and [109]. Proposition6.2.3 is from [316], which also contains the identity (6.9). As noticed in the latterpaper, the examples by Crabb and Davie [103] and Varopoulos [416] showingthat von Neumann’s inequality fails for triples of commuting contractions, implythat A(D2) ⊗min A(D) and A(D2) ⊗max A(D) are not isometric.

Arveson’s dilation theorem discussed in 6.2.1 is from [21]. Proposition 6.2.4seems to be new, but the fact that A ⊗max A(D) = A ⊗min A(D) completelyisometrically for any C∗-algebra A is well-known. Indeed, Corollary 6.2.6 is an oldresult due to Arveson (unpublished). A remarkable generalization was obtainedby Parrott [299], as follows. Let A be a C∗-algebra, and assume for simplicity thatA is unital. Let π : A → B(H) be a unital ∗-representation, let ρ : A(D2) → B(H)be a unital completely contractive homomorphism, and assume that π and ρhave commuting ranges. If (z, w) denote the variables in D

2, let T = ρ(z) andS = ρ(w). These are commuting contractions on H , and since A is selfadjoint,the range of π commutes with S, S∗, T, T ∗. Hence by [299, Theorem 4], there exista Hilbert space K, an isometry J : H → K, commuting unitaries U, V ∈ B(K),and a unital ∗-representation π : A → B(K) whose range commutes with U andV , and such that

∑n,m π(anm)T nSm =

∑n,m J∗π(anm)UnV mJ for any finite

family (an,m)n,m≥0 in A. If we let ρ : C(T2) → B(K) denote the ∗-representationdefined by letting π(f) = f(U, V ), we obtain that π(a)ρ(f) = J∗π(a)ρ(f)J , fora ∈ A and f ∈ A(D2). Thus 6.2.3 (ii) is satisfied for C = A, B = A(D2), andD = C(T2). According to 6.2.5, this implies that A⊗max A(D2) = A⊗min A(D2)completely isometrically. It is not hard to deduce that A(D2; A1 ⊗max A2) andA(D; A1) ⊗max A(D; A2) coincide completely isometrically, for C∗-algebras A1

and A2. We leave this to the reader.6.3: The main result of this section, Theorem 6.3.5, is from [316]. Some of


its consequences, such as assertions (2) and (3) in 6.3.7 appeared earlier in [315].The McAsey–Muhly theorem mentioned in 6.3.2 admits a generalization to nestalgebras due to Paulsen, Power, and Ward [317]. See also [109] for related work.In fact, many results in Section 6.3 extend to the case when triangular algebrasare replaced by nest algebras. For example, if A is an approximately unitaloperator algebra and if A ⊗min A(D) = A ⊗max A(D) completely isometrically,then we have A ⊗min A = A ⊗max B for every finite-dimensional nest algebra B(see [316, Theorem 3.4]), and A ⊗min B = A ⊗nor B for every nest algebra A(by [315, Theorem 3.1]). It is also possible to prove a variant of 6.3.6 involvingthe binormal tensor product considered in the Notes to Section 6.1. Namely ifM is a unital dual operator algebra, then the conditions (i) and (ii) in 6.3.6 arealso equivalent to T ∞ ⊗min M being completely isometric to T ∞ ⊗bin M . Asa consequence, one obtains that T ∞ ⊗min T ∞ = T ∞ ⊗bin T ∞. More generally,if A and B are any two nest algebras, then A ⊗min B = A ⊗bin B completelyisometrically. This result is implicit in [315]. Proposition 6.3.8 is from [316].

6.4: Delta norms first appeared in 1996 in an early and preliminary versionof Pisier’s book [337]. The material from 6.4.1 to 6.4.5 dates back to that time.In fact, the δ norm was introduced for the purpose of giving an operator spaceproof of the fact that a C∗-algebra is nuclear if and only if it has the completelypositive approximation property (see 6.6.1). We refer the reader to [337, Chapter12] for that proof. There are other interesting proofs of 6.4.4, e.g. [265, p. 133].The normal delta norm was introduced by Le Merdy in [249], from which thesecond part of this section is taken.

Oikhberg and Pisier considered a variant of the δ norm for pairs of operatorspaces [294]. If X and Y are operator spaces, define µ(z) = sup‖θ • σ(z)‖ forz ∈ X ⊗ Y , where the supremum runs over all pairs of complete contractionsθ : X → B(H) and σ : Y → B(H) with commuting ranges. In analogy with 6.4.4,it is showed in [294] that µ(z) < 1 if and only if there are z1, z2 in X ⊗ Y suchthat z = z1 + z2, and ‖z1‖X⊗hY + ‖zf

2 ‖Y ⊗hX < 1. Here zf2 denotes the image of

z2 under the canonical ‘flip’ from X ⊗ Y to Y ⊗ X .6.5: The decomposable norm was introduced by Haagerup in [180]. This

norm has several remarkable properties that we briefly review. First, we have‖u‖cb ≤ ‖u‖dec for any decomposable map u : C → B between C∗-algebras. Next,we have ‖u‖dec = ‖u‖cb = ‖u‖ if u is positive, and we also have ‖u‖dec = ‖u‖cb

if B is injective. Indeed, this follows from Paulsen’s proof of 1.2.8. Furthermoreif A is a third C∗-algebra and if u : C → B and v : B → A are decomposable,then v u is decomposable and ‖v u‖dec ≤ ‖v‖dec‖u‖dec. The main resultof [180] asserts that if M is any W ∗-algebra, and if C is any infinite-dimensionalC∗-algebra, then M is injective if (and only if) every completely bounded mapu : C → M is decomposable. We have seen in (6.34) that decomposable maps‘tensorize’ for the maximal tensor product. This result turns out to be optimal.Indeed, Kirchberg showed that if M is a W ∗-algebra, C is a C∗-algebra, K ≥ 0is a constant, and u : C → M is a bounded map such that for any C∗-algebra A,‖IA ⊗u : A⊗max C → A⊗nor M‖ ≤ K, then u is decomposable and ‖u‖dec ≤ K.


We refer to [337, Chapter 14] for a proof. Thus if M is not injective and C isinfinite-dimensional, then there exist completely bounded maps u : C → M andC∗-algebras A such that IA⊗u is unbounded with respect to the maximal tensornorms.

The first part of Section 6.5 is composed of several simple observations. Re-sults 6.5.7 and 6.5.8 are due to Pisier (see [337]). In [208], Junge and Le Merdyproved a generalization of 6.5.7, namely, if u : C → B is a finite rank operatorbetween C∗-algebras, then ∆(u) = ‖u‖dec. Indeed the proof of 6.5.7 presentedhere is a simplification of the proof of this more general result. There is no goodanalogue of the above mentioned result with the completely bounded norm re-placing the decomposable one. To see this, if B, C are C∗-algebras and u : C → Bis finite rank, define γ(u) = inf‖α‖cb‖β‖cb, where the infimum runs over all in-

tegers n ≥ 1, and all possible factorizations u = βα, with Cα−→ Mn

β−→ B. If Bis a W ∗-algebra, it is shown in [208] that B is injective if and only if γ(u) = ‖u‖cb

for any C, and any finite rank operator u : C → B.6.6: Theorem 6.6.4 is from the preliminary version of [337] alluded to in the

Notes to 6.4. However the terms ‘K-nuclear’ and ‘K-semidiscrete’, and their usein a setting involving nonselfadjoint algebras, are from [249]. See 6.6.1, 6.6.2,and 6.6.3 for the connection with the usual notions of nuclear C∗-algebras andsemidiscrete W ∗-algebras. Theorem 6.6.5 and its corollary 6.6.8 are taken from[249]. In general, δ and σδ are not comparable norms. To see this, let M be asemidiscrete W ∗-algebra which is not a nuclear C∗-algebra (for instance takeM = B(2)), and let K ≥ 1. Then M cannot be K-nuclear (see 6.6.2 (2)),hence Theorem 6.6.4 ensures that there exist a finite-dimensional operator spaceX , and z ∈ X ⊗ M such that ‖z‖min ≤ 1 and δ(z) ≥ K. However we haveσδ(z) = ‖z‖min by 6.6.5. Thus δ(z) ≥ Kσδ(z).

Local reflexivity (see 6.6.7) is a fundamental notion for operator spaces, al-though we will barely touch this topic in this book. See [149, Part IV] for acomprehensive treatment. We merely recall that local reflexivity was defined forC∗-algebras by Effros and Haagerup in [136]. It is quite remarkable that thereare operator spaces which are not K-locally reflexive for any K. The full groupC∗-algebra over the free group F2, and B(H) (if H is infinite-dimensional) aresuch examples. On the other hand, it follows from a pioneering paper of Archboldand Batty [15] that any nuclear C∗-algebra is 1-locally reflexive. Other funda-mental papers on locally reflexive operator spaces include [137,141]. See also thecited papers of Kirchberg, and [296], and references therein.

7

Selfadjointness criteria

There is a well-known book by Burckel with the title Characterizations of C(X)among its subalgebras [80]. This chapter may be viewed as having a similar aim,namely to discuss several interesting criteria which force an operator algebrato be selfadjoint. Such results are in some sense ‘negative’ in nature, showingthat certain results or themes which are important for C∗-algebras, may notbe transferred to general operator algebras, or at least not in a literal way.Nonetheless, they are sometimes quite useful, for example when in the middle ofsome proof one needs to show that a certain algebra is a C∗-algebra.

Throughout this chapter, B is an operator algebra, usually approximatelyunital for specificity. We say that ‘B is a C∗-algebra’, or B is selfadjoint, if B iscompletely isometrically homomorphic to a C∗-algebra. Equivalently, B = ∆(B),in the notation of 2.1.2.

The sections of this chapter are mostly independent of each other. In fact,our first section is largely a continuation of the study of nuclearity and semidis-creteness begun in Chapter 6 (and particularly the last section, Section 6.6).

7.1 OS-NUCLEAR MAPS AND THE WEAK EXPECTATION PROPERTY

In Chapter 6, we studied a certain notion of nuclearity for linear mappings valuedin an operator algebra. We now consider a related but different notion. We alsodiscuss the weak expectation property for operator spaces.

7.1.1 (OS-nuclear maps) Let X and Y be two operator spaces, and supposethat u : X → Y is a bounded operator. We say that u is OS-nuclear if thereexist nets of completely contractive maps Y

αt−→ Mnt

βt−→ X such that βtαt

converges to u in the point norm topology. We say that the operator space Xis OS-nuclear if the identity mapping IX : X → X is OS-nuclear. Smith showedin [390] that for a C∗-algebra, this notion coincides with the usual notion ofnuclearity. A generalization of Smith’s result will be given in this section (see7.1.11 and 7.1.12).

It is clear that any OS-nuclear map u : Y → X is completely contractive.

7.1.2 (OS-nuclearity versus 1-nuclearity) Let Y be an operator space, let Bbe an approximately unital operator algebra and suppose that u : Y → B isa bounded operator. If u is 1-nuclear (in the sense of 6.6.2), then it is OS-nuclear, as we observed in 6.6.6. Although we will prove a partial converse later

270 OS-nuclear maps and the weak expectation property

in this section, OS-nuclearity does not imply 1-nuclearity in general. Indeedassume that B is a C∗-algebra, and let u : Mn → B be a linear mapping. Thenu : Mn → B is OS-nuclear if and only if ‖u‖cb ≤ 1. On the other hand, u is1-nuclear if and only if ∆(u) ≤ 1. Indeed assume that u is 1-nuclear and apply6.6.4 (ii) with Y = Mn and X = Y ∗ = S1

n. Let j ∈ S1n ⊗ Mn be the tensor

associated to the identity mapping on Mn. Then ‖j‖min = 1, by (1.32). Also,(IS1

n⊗ u)(j) ∈ S1

n ⊗ B is the tensor associated to u as in the first paragraphof 6.5.1, so that ∆(u) ≤ ‖j‖min = 1. The converse direction is clear, from 6.6.4(iii). Hence u is 1-nuclear if and only if ‖u‖dec ≤ 1, by 6.5.7. However if B is anoninjective W ∗-algebra, then by Haagerup’s work in [180], there exists a mapu as above with 1 = ‖u‖cb < ‖u‖dec.

7.1.3 (The weak expectation property) Let X be an operator space and letiX : X → X∗∗ be the canonical embedding. We say that X has the weak expecta-tion property (WEP in short) if there exist a Hilbert space H , and two completelycontractive mappings J : X → B(H) and P : B(H) → X∗∗, such that iX = PJ .In this case, since iX is completely isometric, the mapping J is necessarily com-pletely isometric. Conversely, if X has the WEP, and if J : X → B(H) is anarbitrary complete isometry for some Hilbert space H , then there exists a com-plete contraction P : B(H) → X∗∗ such that iX = PJ . Indeed this follows fromthe extension theorem 1.2.10.

If X is a dual operator space, then there is a completely contractive idempo-tent from X∗∗ onto X (namely the adjoint of iX∗ : X∗ → X∗). It is easy to seefrom this that X has the WEP if and only if X is injective.

Lemma 7.1.4 Let X, Y be operator spaces, with Y ⊂ B(H) completely isomet-rically, for some Hilbert space H, and let u : Y → X be an OS-nuclear map. TheniX u : Y → X∗∗ extends to a completely contractive map P : B(H) → X∗∗.

Proof By assumption, there exist nets of completely contractive mappingsY

αt−→ Mnt and Mnt

βt−→ X , such that βtαt converges to u in the point normtopology. By the extension theorem 1.2.10, there exists for any t a completelycontractive mapping αt : B(H) → Mnt extending αt. Since X∗∗ is a dual op-erator space, CB(B(H), X∗∗) =

(B(H)

⊗ X∗)∗ is a dual Banach space (see

(1.51)). Since the net (iX βtαt)t lies in the unit ball, it has a w∗-cluster pointP : B(H) → X∗∗, in that ball. For y ∈ Y and ϕ ∈ X∗, we have

limt〈iX βtαt(y), ϕ〉 = lim

t〈ϕ, βtαt(y)〉 = 〈ϕ, u(y)〉.

Hence 〈P (y), ϕ〉 = 〈ϕ, u(y)〉. Thus P|Y = iX u.

Corollary 7.1.5 Any OS-nuclear operator space has the WEP.

We proved in 4.2.8 (3) that if an approximately unital operator algebra isinjective, then it is selfadjoint (and unital). In Theorem 7.1.7 below we will extendthis result to operator algebras with the WEP. We will need the following result,

Selfadjointness criteria 271

which is of independent interest. As usual, ∆(· ) denotes the diagonal C∗-algebraof an operator algebra (see 2.1.2).

Lemma 7.1.6 Let B be an operator algebra, regarded as a subalgebra of B∗∗ inthe canonical way.(1) If B ⊂ ∆(B∗∗), then B is selfadjoint.(2) If B∗∗ is selfadjoint, then B is selfadjoint.(3) If the unitization B1 is selfadjoint (resp. completely isomorphic to a C∗-

algebra), then B is selfadjoint (resp. completely isomorphic to a C∗-algebra).

Proof For (3), simply use the fact that a two-sided ideal in a C∗-algebra isselfadjoint. Clearly (2) follows from (1). For (1), assume that B is a subalgebraof B(H). Then B∗∗ ⊂ B(H)∗∗. By A.2.3 (4), B∗∗ ∩ B(H) = B. Assume thatB ⊂ ∆(B∗∗), and let b ∈ B. Then b∗ both belongs to B(H) and B∗∗. Henceb∗ ∈ B by the above. Thus B = ∆(B).

Theorem 7.1.7 Let B be an approximately unital operator algebra with theWEP. Then B is selfadjoint.

Proof Let (et)t be a cai of B. The second dual B∗∗ is a unital operator algebra.Assume that B∗∗ is a unital subalgebra of B(H). Thus B ⊂ B∗∗ ⊂ B(H)completely isometrically, and et → IH strongly (as may be seen via 2.1.9, forexample). By 7.1.3 and the WEP assumption, there exists a complete contractionP : B(H) → B(H) taking values in B∗∗, such that P (b) = b for every b ∈ B.In particular P (et) = et for any t. Arguing as in the proof of 4.2.8, we deducethat P (IH) = IH . Thus P is completely positive, and selfadjoint (see 1.3.3).That is, P (T )∗ = P (T ∗), for every T ∈ B(H). Given any b ∈ B, we haveb∗ = P (b)∗ = P (b∗) ∈ B∗∗. This shows that B ⊂ ∆(B∗∗). Hence B is selfadjointby Lemma 7.1.6.

The following is a straightforward consequence of 7.1.5, 7.1.7, and 7.1.2:

Corollary 7.1.8 Any OS-nuclear (and hence any 1-nuclear) approximately uni-tal operator algebra is selfadjoint.

7.1.9 (OS-semidiscreteness) Let X be a dual operator space. We say that X

is OS-semidiscrete if there exist nets of completely contractive maps Xαt−→ Mnt

and Mnt

βt−→ X such that βtαt converges to IX in the point w∗-topology, that

is, βtαt(x) w∗−→ x, for every x ∈ X . It is also possible to define OS-semidiscrete

linear maps but we will not use these here. If a dual operator space X is OS-semidiscrete, then it is injective. To see this, suppose that X1 is a subspace ofan operator space X2, and that u : X1 → X is a complete contraction. Let αt, βt

be as above. Since Mn is injective, αt u has a completely contractive extensionαt : X2 → Mnt . By the argument in 7.1.4, (βt αt) has a w∗-cluster point u, say,in CB(X2, X), and u extends u. Thus X is injective. We shall see later in 8.6.2that the converse is true too, an injective dual operator space is OS-semidiscrete.

272 OS-nuclear maps and the weak expectation property

Let M be a unital dual operator algebra. Since injective unital operatoralgebras are selfadjoint (see 4.2.8), we deduce that if M is OS-semidiscrete,then M is a W ∗-algebra. On the other hand, the argument in 7.1.2, combinedwith Theorem 6.6.5, will show that M is OS-semidiscrete provided that it is1-semidiscrete. Hence M is a W ∗-algebra if M is 1-semidiscrete.

The next result, concerning uniform algebras, does not extend to generaloperator algebras. Indeed as we observed in 6.6.2 (2), any finite-dimensionaloperator algebra is K-nuclear for some K ≥ 1.

Proposition 7.1.10 A uniform algebra B which is K-nuclear for some K ≥ 1,is selfadjoint. More generally, assume that for some K ≥ 1, there exists a net ofpairs of linear mappings B

αt−→ Mnt

βt−→ B, such that βtαt converges strongly toIB, and such that ‖αt‖cb‖βt‖ < K, for all t. Then B is selfadjoint.

Proof In this proof we shall use (1.10) several times without announcement.Thus B(X, B) = CB(X, B) isometrically for any operator space X . By theargument in 6.6.6, we need only prove the ‘more generally’ part of the statement.Thus we assume the existence of nets (αt)t and (βt)t as above. We will use someBanach space theory which may be found in [324, Section 8.c], for example.We say that a Banach space B has a local unconditional structure if there is aconstant C > 0 with the following property: For any finite-dimensional subspaceE if B, there is a Banach space F with an unconditional basis, and two linearmappings w1 : E → F and w2 : F → B, such that w2w1 is equal to the canonicalembedding E → B, and ‖w1‖‖w2‖ ≤ C. A deep result of Kisliakov [232] assertsthat if B ⊂ C(Ω) is a uniform algebra, then B is actually equal to C(Ω) providedthat B has a local unconditional structure. It therefore suffices to prove that forany finite-dimensional subspace E of B, there exist an integer N ≥ 1, and twolinear mappings w1 : E → ∞N and w2 : ∞N → B, such that w2w1 = jE , thecanonical embedding of E into B, and ‖w1‖‖w2‖ ≤ K + 1.

We fix E ⊂ B as above, and let ε > 0. Since E is finite-dimensional, any linearmapping from E into B factors through a matrix space. More precisely, givenany u : E → B, there exist n ≥ 1, and two linear mappings α : E → Mn andβ : Mn → B, such that βα = u. Then we may define a factorization norm γ on thespace B(E, B), by letting γ(u) = inf‖α‖cb‖β‖, where the infimum runs overall integers n ≥ 1, and all possible factorizations as above. It is easy to see thatγ is indeed a norm on B(E, B). For example, to see the triangle inequality, letu1, u2 : E → B, and let c1, c2 be positive numbers with γ(uj) < cj , for j = 1, 2.Fix linear mappings αj : E → Mnj and βj : Mnj → B such that ‖αj‖cb ≤ 1,‖βj‖ < cj , and uj = βjαj . Let n = n1 + n2 and regard Mn1 ⊕ Mn2 ⊂ Mn

in the usual way. Then let P : Mn → Mn1 ⊕ Mn2 be the canonical conditionalexpectation. We define a map α : E → Mn by letting α(x) = (α1(x), α2(x)) forx ∈ E, and we define β : Mn → E by letting β(z) = β1(z1) + β2(z2) for z ∈ Mn,if P (z) = (z1, z2). Then βα = u1 + u2, ‖α‖cb ≤ 1, and ‖β‖ ≤ ‖β1‖ + ‖β2‖. Thisshows that γ(u1 + u2) < c1 + c2.


Since E is finite-dimensional, γ is clearly equivalent to the usual norm onB(E, B). Thus the topology induced by γ coincides with the point norm topologyon B(E, B). By assumption, βtαt|E converges to jE in this topology, and so

γ(jE) = limt

γ(βtαt|E) ≤ lim supt

‖αt|E‖cb‖βt‖ ≤ K.

Thus we may write iE = βα, for some α : E → Mn and β : Mn → B satisfying‖α‖cb‖β‖ ≤ K + ε. We now use the well-known elementary fact from Banachspace theory, that there exist an integer N ≥ 1, a subspace G of ∞N , and anisomorphism σ : G → E such that ‖σ−1‖‖σ‖ ≤ 1 + ε. By 1.2.10, the mappingασ : G → Mn extends to a map w : ∞N → Mn, with ‖w‖cb = ‖ασ‖cb ≤ ‖α‖cb‖σ‖.We let w2 = βw : ∞N → B, and we let w1 be the mapping σ−1 regarded as valuedin ∞N . Then w2w1 = jE , and moreover,

‖w1‖‖w2‖ ≤ ‖σ−1‖‖β‖‖w‖cb ≤ ‖σ−1‖‖σ‖‖α‖cb‖β‖ ≤ (K + ε)(1 + ε).

This yields the desired factorization property for jE .

Proposition 7.1.11 Let B be a unital operator algebra, and let Y be a unital-subspace of B. Then the canonical embedding Y → B is 1-nuclear if and only ifit is OS-nuclear.

Proof We said in 7.1.2 that 1-nuclearity implies OS-nuclearity. The proof ofthe other direction is a variant of that of Theorem 7.1.7. We let u : Y → Bbe the canonical embedding and assume that u is OS-nuclear. We let H bea Hilbert space such that B∗∗ ⊂ B(H) is a unital-subalgebra. We thus haveY ⊂ B ⊂ B∗∗ ⊂ B(H) completely isometrically, with IH ∈ Y . By Lemma 7.1.4,there is a completely contractive mapping P : B(H) → B(H) taking values inB∗∗, and such that P (y) = y, for every y ∈ Y . In particular, P is unital. As inthe proof of 7.1.7, P is completely positive, and selfadjoint. Hence P takes valuesin the diagonal C∗-algebra D = ∆(B∗∗).

Let A be an approximately unital operator algebra. Since P is completelypositive and contractive, we have∥∥IA ⊗ P : A ⊗max B(H) −→ A ⊗max D

∥∥ ≤ 1

by Lemma 6.1.10. Since D ⊂ B∗∗, the last equation together with 6.1.9 gives∥∥IA ⊗ P : A ⊗max B(H) −→ A ⊗max B∗∗∥∥ ≤ 1. (7.1)

Let v : Y → B(H) be the canonical embedding. Since u is OS-nuclear, v is OS-nuclear too. We claim that the map v is actually 1-nuclear. Indeed for any n ≥ 1,and any linear map β : Mn → B(H), we have ‖β‖cb = ‖β‖dec (see the Notes toSection 6.5). Hence our claim follows from 6.6.4. Thus∥∥IA ⊗ v : A ⊗min Y −→ A ⊗max B(H)

∥∥ ≤ 1. (7.2)

274 Hilbert module characterizations

The canonical map iB : B → B∗∗ induces a complete isometry from A ⊗max Bto A ⊗max B∗∗ (see 6.1.3). Since Pv = iBu, (7.1) and (7.2) gives∥∥IA ⊗ u : A ⊗min Y −→ A ⊗max B

∥∥ ≤ 1.

Thus u is 1-nuclear.

7.1.12 (A remark) It is not hard to extend Proposition 7.1.11 above to approx-imately unital operator algebras. In that case, the assumption that Y is unitalneeds to be replaced by the assumption that Y contains a cai of B. The mainadjustment one needs to make to the last proof, is that to see that P (IH) = IH ,one uses the argument in 7.1.7.

This extension of 7.1.11 applies in particular when B is a C∗-algebra andY = B. Thus we recover Smith’s theorem from [390], which states that B isnuclear if and only if it is OS-nuclear.

7.2 HILBERT MODULE CHARACTERIZATIONS

In this section we discuss characterizations of ‘selfadjointness’ in terms of Hilbertmodules, or equivalently in terms of the completely contractive representationsof B (see 3.1.6 and 3.1.7).

7.2.1 (Complemented submodules) Let π : B → B(H) be a completely contrac-tive representation of an operator algebra B on a Hilbert space H . If we regardH as a Hilbert B-module, then a submodule is simply a closed π(B)-invariantsubspace K of H (see 3.2.1). Let p ∈ B(H) be an idempotent map with rangeequal to K, and let K ′ = Ker(p). Then K ′ is π(B)-invariant if and only ifpπ(b) = π(b)p, for every b ∈ B. Indeed assume that K ′ is π(B)-invariant, and letζ ∈ H . Then π(b)pζ ∈ K, π(b)(1 − p)ζ ∈ K ′, and π(b)ζ = π(b)pζ + π(b)(1− p)ζ.Hence pπ(b)ζ = π(b)pζ. Conversely, suppose that p commutes with π(B). Ifζ ∈ K ′, then pζ = 0, hence pπ(b)ζ = π(b)pζ = 0. Thus π(b)ζ ∈ K ′.

In this chapter we will consider direct sum decompositions H = K ⊕ K ′, forclosed subspaces K and K ′, which are not necessarily mutually orthogonal. Wesay that a closed submodule K of H is a complemented submodule if H admitssuch a direct sum decomposition H = K ⊕K ′, with the topological complementK ′ also a closed π(B)-invariant subspace. Perhaps one should say topologicallycomplemented submodule here, but for brevity we use the shorter phrase. Re-ducing submodules (in the sense of 3.2.3) are those submodules for which thelatter property holds with K ′ = K⊥. By the last paragraph, a submodule Kis complemented (resp. reducing) if and only if there exists an idempotent mapp in the commutant [π(B)]′, whose range is equal to K (resp. the orthogonalprojection onto K belongs to [π(B)]′).

7.2.2 (Module complementation property) Let B be an operator algebra, whichwe assume to be approximately unital for specificity (the general case is discussedin the Notes). We say that B has the module complementation property if when-ever H is a Hilbert space and π : B → B(H) is a nondegenerate completely


contractive representation, then every π(B)-invariant subspace of H is a com-plemented submodule. Likewise we say that B has the reducing property if forevery nondegenerate completely contractive representation π : B → B(H), everyπ(B)-invariant subspace is reducing. Any C∗-algebra has the reducing property.

Assume that B has the module complementation property, and consider anondegenerate representation π : B → B(H), that we only assume to be com-pletely bounded. If K is a π(B)-invariant subspace of H , then we claim thatthere is a π(B)-invariant subspace K ′ of H such that H = K ⊕ K ′. Indeed byPaulsen’s theorem 5.1.2, there is an invertible operator S ∈ B(H) such thatπS = S−1π(· )S is completely contractive. Clearly S−1(K) is πS(B)-invariant.By the module complementation property, S−1(K) admits a topological com-plement V which also is πS(B)-invariant. Then S(V ) is π(B)-invariant, andH = K ⊕ S(V ), which proves the claim.

These properties may be viewed as a variant of the property of semisimplicitymet in pure algebra. Indeed one characterization of semisimple rings R is thatevery submodule of every left R-module X is an R-module summand of X (e.g.see [8, Propositions 9.6 and 13.9 (d)]).

7.2.3 (W ∗-module complementation property) There is a natural ‘w∗-topologyversion’ of the module complementation property. Namely, we will say that aunital dual operator algebra M has the w∗-module complementation property,provided that for every unital w∗-continuous completely contractive represen-tation π : M → B(H), every π(M)-invariant subspace of H is a complementedsubmodule. We define the w∗-reducing property similarly, by demanding that forany π as above, every π(M)-invariant subspace of H is reducing.

7.2.4 (Passing to the second dual) Let B be an approximately unital opera-tor algebra, and let π : B → B(H) be a completely contractive representation.Let π : B∗∗ → B(H) be the unique w∗-continuous extension of π (see 2.5.5).This is a completely contractive representation, and every unital w∗-continuouscompletely contractive representation of B∗∗ is of this form. Moreover if K isany π(B)-invariant subspace of H , then K is fairly evidently π(B∗∗)-invariant.Indeed, if η ∈ B∗∗, then by A.2.1 there exists a net (bt)t in B converging to η inthe w∗-topology. Thus π(bt) → π(η). For ζ1 ∈ K and ζ2 ∈ K⊥, we have

〈π(η)ζ1, ζ2〉 = limt〈π(bt)ζ1, ζ2〉 = 0,

as desired. It follows immediately from this that B has the module complemen-tation property if and only if B∗∗ has the w∗-module complementation property.Similarly for the reducing property, which we characterize next.

Theorem 7.2.5

(1) An approximately unital operator algebra has the reducing property (see7.2.2) if and only if it is selfadjoint.

(2) A unital dual operator algebra has the w∗-reducing property (see 7.2.3) ifand only if it is a W ∗-algebra.


Proof The ‘if’ parts of both assertions are clear. For the converses, we will infact prove more than is asserted. Suppose that M is a w∗-closed unital-subalgebraof B(H), such that every closed M -submodule of the countable multiple H (∞)

of H is reducing. Let L = H(∞), and view M ⊂ B(L). If ζ ∈ L, then [Mζ] isan M -invariant subspace. By hypothesis, [Mζ]⊥ is an M -invariant subspace too.Equivalently, [Mζ] is an M-invariant subspace. Thus M ζ ⊂ [Mζ]. By A.1.5,M ⊂ M , so that M is selfadjoint, and we have proved (2).

For (1), let H be any Hilbert A-module such that the canonically extendedrepresentation of A∗∗ on H is completely isometric (see 2.5.5 and 2.5.6). If everyclosed A-submodule of H(∞) is reducing, then by the argument in 7.2.4, everyclosed A∗∗-submodule of H(∞) is reducing. Hence by (1), A∗∗ is selfadjoint. By7.1.6 (2), A is selfadjoint too.

7.2.6 (Stability under isomorphism) Let A and B be approximately unitaloperator algebras, and assume that B has the module complementation property.It clearly follows from the second paragraph of 7.2.2 that if A is completelyisomorphic to B, then A also has the module complementation property. Moregenerally, if there is a completely bounded homomorphism θ : B → A with adense range, then A has the module complementation property. To see this, letπ : A → B(H) be a nondegenerate completely contractive representation, andconsider a π(A)-invariant subspace K of H . The composition mapping πθ is acompletely bounded representation of B on H , and K is πθ(B)-invariant. Since θis assumed to have a dense range, πθ is nondegenerate. Thus according to 7.2.2,there is a direct sum decomposition H = K ⊕ K ′, for some πθ(B)-invariantsubspace K ′. Since θ(B) is dense, K ′ is π(A)-invariant.

Since C∗-algebras have the reducing property, an approximately unital op-erator algebra which is completely isomorphic to a C∗-algebra, has the modulecomplementation property. It is apparently an open problem whether the con-verse is true. We next consider some sufficient conditions in this direction.

Proposition 7.2.7 Let B be a finite-dimensional unital operator algebra. ThenB has the module complementation property if and only if B is (completely)isomorphic to a C∗-algebra.

Proof We need only discuss the ‘only if’ part. Finite-dimensional spaces maybe viewed as Hilbert spaces, and maps between them (and representations of Bon them) are necessarily bounded. Hence if B has the module complementationproperty, then using the second paragraph of 7.2.2 we see that the criterion in thelast paragraph of 7.2.2 holds, but for B-modules X of finite linear dimension.Since B is finite-dimensional, it follows from basic algebra that B is actuallysemisimple (e.g. see the paragraph above 13.6 in [8]). Hence by the classicaltheory of algebras (e.g. see [8, 368] or [106, Theorem 1.5.9]), B is isomorphic toa C∗-algebra.


Lemma 7.2.8 Let C be an approximately unital operator algebra and let B bea subalgebra of C containing a cai of C. Suppose that B has the module com-plementation property, and that for any nondegenerate completely contractiverepresentation π : C → B(H), we have [π(B)]′ = [π(C)]′. Then B = C.

Proof Let B and C be as above, and fix ϕ ∈ C∗ with B ⊂ Ker(ϕ). We willshow that ϕ = 0, so that B = C by the Hahn–Banach theorem. By 1.2.8, forexample, there exist a Hilbert space H , a completely contractive homomorphismπ : C → B(H) and two vectors ζ, η ∈ H such that

ϕ(c) = 〈π(c)ζ, η〉, c ∈ C. (7.3)

Replacing H by [π(C)H ], and η by its projection onto [π(C)H ], we may clearlyassume that π is nondegenerate. Let K = [π(B)ζ]; this is a π(B)-invariant sub-space of H , and is therefore a complemented submodule. By 7.2.1, there is anidempotent p ∈ [π(B)]′ = [π(C)]′ with range K.

By assumption, B contains a net (bt)t which is a cai for C. Let c be anarbitrary element of C. For any t we have π(bt)ζ ∈ K hence π(bt)ζ = pπ(bt)ζ.Thus π(c)π(bt)ζ = π(c)pπ(bt)ζ. Since p commutes with π(c), we deduce that

π(c)π(bt)ζ = pπ(c)π(bt)ζ.

Since π(c) = limt π(c)π(bt), we obtain that π(c)ζ = pπ(c)ζ. This shows thatπ(c)ζ ∈ K. Now recall that ϕ|B = 0, so that η ∈ K⊥. Hence ϕ(c) = 0.

7.2.9 (Normal elements) Let B be an approximately unital operator algebra,and let C∗

e (B) denote its C∗-envelope (see 4.3.4). We say that b ∈ B is normalif b is a normal element of C∗

e (B). By Theorem 4.3.1, if D is another C∗-algebracontaining B as a subalgebra, and if b ∈ B is normal as an element of D, then bis normal in the above sense.

Theorem 7.2.10

(1) Let B be an approximately unital operator algebra and assume that B is gen-erated (as an operator algebra) by its normal elements. Then B is selfadjointif and only if it has the module complementation property.

(2) A uniform algebra is selfadjoint if and only if it has the module complemen-tation property.

Proof We first prove (1). We assume that B has the module complementationproperty, we let C = C∗

e (B), and we will show that B = C. Let N be the setof normal elements of B. By 2.1.7, B contains a cai for C. Hence it suffices tocheck that B and C satisfy the assumption of 7.2.8. Let π : C → B(H) be acompletely contractive representation. Then π is a ∗-representation (by 1.2.4).Hence for any N ∈ N , π(N) is a normal element of B(H). We let T ∈ [π(B)]′.


Then Tπ(N) = π(N)T , for every N ∈ N . By Fuglede’s theorem, this impliesthat Tπ(N)∗ = π(N)∗T . Equivalently,

Tπ(N∗) = π(N∗)T, N ∈ N .

Thus T commutes with π(N ) and with π(N ∗). Since C is generated by B andB as an operator algebra, our assumption implies that C is generated by Nand N . Hence T ∈ [π(C)]′, which shows that [π(B)]′ = [π(C)]′.

Clearly (2) follows from (1). Indeed if B is a uniform algebra then its C∗-envelope is commutative, and hence every element of B is normal.

In 3.6.3, we showed B(K, H) is ‘injective as an A-B-bimodule’, if A and Bare C∗-algebras acting on H and K respectively. In our next statement, we showthat such results are not generally true for nonselfadjoint operator algebras.

Proposition 7.2.11 Let B be an approximately unital operator algebra. ThenB is selfadjoint if (and only if) whenever K is a Hilbert B-module, Y is a leftoperator B-module, and X is a B-submodule of Y , then any completely contrac-tive B-module map u : X → B(K) admits a completely contractive B-moduleextension u : Y → B(K).

Proof Assume that B satisfies the above property. We will apply 7.2.5 to B.Let H be a Hilbert space, let π : B → B(H) be a nondegenerate completelycontractive representation, and let K be a π(B)-invariant subspace of H . ThusK is a Hilbert module over B. Let η ∈ K be a unit vector, and let u : Kc → B(K)be defined by u(ζ) = ζ⊗η, for ζ ∈ K. Then u is a complete isometry (see 1.2.23),and u is a B-module map. Let X = Kc and Y = Hc, equipped with its B-modulestructure induced by π. By the hypothesis, we obtain a completely contractivemap u : Hc → B(K), which extends u and satisfies

u(π(b)ξ) = π(b)u(ξ), b ∈ B, ξ ∈ H. (7.4)

Let T : H → K be defined by T (ξ) = u(ξ)η. Then ‖T ‖ ≤ 1 and moreover, wehave T (ζ) = u(ζ)η = ζ, for every ζ ∈ K. Hence T is the orthogonal projectiononto K. It clearly follows from (7.4) that this projection commutes with π(B).According to 7.2.5 (1), this shows that B is selfadjoint.

In the following statement we do not require the operator algebras to beapproximately unital.

Proposition 7.2.12

(1) An operator algebra B is selfadjoint if and only if for any Hilbert space H,and any completely contractive homomorphism π : B → B(H), the commu-tant algebra [π(B)]′ is selfadjoint.

(2) Let M be a dual operator algebra. Then M is a W ∗-algebra if and only iffor any Hilbert space H, and for any w∗-continuous completely contractivehomomorphism π : M → B(H), the commutant [π(M)]′ is selfadjoint.


Proof The ‘only if’ in (1) and (2) is clear from 1.2.4, so we need only prove the‘if’ parts. If B is nonunital, and if π : B1 → B(H) is a completely contractivehomomorphism on the unitization B1, then [π(B1)]′ = [π(B)]′ ∩ [π(1)]′. More-over π(1) is a selfadjoint projection, by A.1.1, hence [π(1)]′ is selfadjoint. Thus[π(B1)]′ is selfadjoint if [π(B)]′ is selfadjoint. Also, B is selfadjoint if B1 is self-adjoint (by 7.1.6 (3)). Thus, replacing B by B1 if necessary, we may henceforthsuppose that B is unital.

For (1), note that if [π(B)]′ is selfadjoint, then so is [π(B)]′′. Taking π to bea B-universal representation, we deduce by 3.2.14 that B∗∗ is selfadjoint. ThusB is selfadjoint, by 7.1.6 (2). A similar argument works for (2), using a dualalgebra variant of the double commutant property (see [72]).

For some alternative proofs of 7.2.12, see the Notes section.

7.3 TENSOR PRODUCT CHARACTERIZATIONS

Corollary 7.1.8 asserts that if B is an approximately unital operator algebra,then B is selfadjoint provided that A ⊗max B = A ⊗min B isometrically forevery operator algebra A. Our aim is to extend that result (as well as its dualcounterpart at the end of 7.1.9) to obtain a tensor product characterization ofselfadjointness not restricted to nuclear or injective objects. This will be achievedin Theorem 7.3.3 below.

Proposition 7.3.1 Let B be a unital operator algebra, let Y be an operatorspace and let u : Y → B be a linear mapping. If u is 1-nuclear and if there existse ∈ Y such that u(e) = 1 and ‖e‖ = 1, then u(Y ) ⊂ ∆(B).

Proof Suppose that B is a unital-subalgebra of B(H), and write 1 for IH .We apply Theorem 6.6.4 to the 1-nuclear mapping u. Using 6.5.1, we obtain anet ut : Y → B of finite rank operators converging to u point norm, with thefollowing property: For any t, there exist nt ≥ 1, a matrix [ϕt

ij ] ∈ Mnt(Y ∗), andat1, . . . , a

tnt

, bt1, . . . , b

tnt

in B, such that

ut(y) =∑

1≤i,j≤nt

ϕtij(y) at

ibtj , y ∈ Y ; (7.5)

and ∥∥[ϕtij ]∥∥ ≤ 1,

∥∥[ati]∥∥

Rnt (B)≤ 1,

∥∥(btj)∥∥

Cnt(B)≤ 1. (7.6)

By assumption, there is a norm one element e of Y such that u(e) = 1. We let

ctj =

∑i

ϕtij(e)a

ti and dt

i =∑

j

ϕtij(e)b

tj,

for any t, and for 1 ≤ i, j ≤ nt. Then∥∥[ct1 · · · ct

nt]∥∥ ≤

∥∥[at1 · · · at

nt]∥∥∥∥[ϕt

ij(e)]∥∥ ≤ 1, (7.7)

280 Tensor product characterizations

using (7.6). Likewise, we have an estimate∥∥(dti)∥∥

Cnt(B)≤ 1. (7.8)

Let θt = ut(e) for each t. It follows from (7.5) that we have

θt =∑

i

atid

ti and θt =

∑j

ctjb

tj . (7.9)

We let y ∈ Y , and we will show that u(y)∗ ∈ B. Set

zt = ut(y) and σt =∑i,j

ϕtij(y)ct

jdti,

for any t. We will show that

limt

‖σt − z∗t ‖ = 0. (7.10)

Since ut converges pointwise to u, we have limt ‖zt − u(y)‖ = 0. Thus (7.10)implies that limt ‖u(y)∗ − σt‖ = 0. However σt ∈ B for any t, and so u(y)∗ ∈ B.

It therefore remains to prove (7.10). For any t we have z∗t =

∑i,j ϕt

ij(y)bt∗j at∗

i

by (7.5). Hence σt − z∗t equals∑i,j

ϕtij(y)

(ctjd

ti − bt∗

j at∗i

)=∑i,j

ϕtij(y)

(ctj − bt∗

j

)dt

i +∑i,j

ϕtij(y) bt∗

j

(dt

i − at∗i

).

The first of these terms,∑

i,j ϕtij(y)

(ctj − bt∗

j

)dt

i, has norm dominated by∥∥[ϕtij(y)]

∥∥ ∥∥[(ct1 − bt∗

1 ) · · · (ctnt

− bt∗nt

)]∥∥ ∥∥(dt

i)∥∥

Cnt(B)

≤ ‖y‖∥∥[ct

j − bt∗j ]∥∥

Rnt(B(H)),

by (7.6) and (7.8). Similarly we have∥∥∥∑i,j

ϕtij(y) bt∗

j

(dt

i − at∗i

)∥∥∥ ≤ ‖y‖∥∥(dt

i − at∗i )

∥∥Cnt(B(H))

.

We deduce that ‖σt − z∗t ‖ is less than or equal to

‖y‖(∥∥[ct

j − bt∗j ]∥∥

Rnt (B(H))+∥∥(dt

i − at∗i )

∥∥Cnt(B(H))

). (7.11)

Let ξ be an arbitrary element of H . Then we have∑i

∥∥(dti − at∗

i )ξ∥∥2 =

∑i

(‖dt

iξ‖2 + ‖at∗i ξ‖2 − 〈dt

iξ, at∗i ξ〉 − 〈at∗

i ξ, dtiξ〉

)=∑

i

‖dtiξ‖2 +

∑i

‖at∗i ξ‖2 − 〈θtξ, ξ〉 − 〈θ∗t ξ, ξ〉


by (7.9). We deduce that∑i

∥∥(dti − at∗

i )ξ∥∥2 ≤

(∥∥(dti)∥∥2

Cnt(B)+∥∥[at

j ]∥∥2

Rnt(B)

)‖ξ‖2 − 〈θtξ, ξ〉 − 〈θ∗t ξ, ξ〉

≤⟨(2 − θt − θ∗t )ξ, ξ

⟩by (7.6) and (7.8). Taking the supremum over all ξ with ‖ξ‖ ≤ 1, we have∥∥(dt

i − at∗i )

∥∥Cnt(B(H))

≤∥∥2 − θt − θ∗t

∥∥ 12 .

The same estimate holds for the first term in (7.11). Thus

‖σt − z∗t ‖ ≤ 2‖y‖‖2− θt − θ∗t ‖12 .

Now recall that θt = ut(e) converges to u(e) = 1 in the norm topology. Thus thesum θt + θ∗t converges to 2, and (7.10) follows.

Proposition 7.3.2 Let M be unital dual operator algebra, let Y be an operatorspace and let u : Y → M be a linear mapping. If u is 1-semidiscrete and if thereexists e ∈ Y such that u(e) = 1 and ‖e‖ = 1, then Ran(u) ⊂ ∆(M).

Proof The proof is quite similar to the previous one; we just give an outline.Assume that M is a w∗-closed unital subalgebra of B(H). By assumption, thereexists a net ut : Y → M of finite rank operators converging in the point w∗-topology to u, and elements ϕt

ij , ati, b

tj, c

tj , d

ti, θt, y, zt and σt as above. Then θt

converges to 1 in the w∗-topology of B(H). Hence for ξ ∈ H , we have

limt

⟨(2 − θt − θ∗t )ξ, ξ

⟩= 0.

Arguing as in the proof of 7.3.1, we find that for any ξ, η ∈ H , we have∣∣⟨(σt − z∗t )ξ, η⟩∣∣ ≤ ‖y‖

(‖η‖

⟨(2 − θt − θ∗t )ξ, ξ

⟩ 12 + ‖ξ‖

⟨(2 − θt − θ∗t )η, η

⟩ 12).

This shows that limt〈(σt − z∗t )ξ, η〉 = 0, for ξ, η ∈ H. For any t, ‖σt‖ ≤ ‖y‖and ‖zt‖ ≤ ‖y‖. Hence (σt − z∗t )t is a bounded net. Thus σt − z∗t → 0 in thew∗-topology of B(H), by A.1.4. Moreover, zt = ut(y) converges to u(y), and soz∗t converges to u(y)∗ in the w∗-topology of B(H). Hence u(y)∗ = w∗ − limt σt.Since σt ∈ M , we may conclude as before that Ran(u) ⊂ ∆(M).

Theorem 7.3.3

(1) Let B be a unital operator algebra. Then B is selfadjoint if (and only if)there is a C∗-algebra D containing B as a subalgebra, such that for everyapproximately unital operator algebra A, we have

A ⊗max B ⊂ A ⊗max D isometrically. (7.12)

282 Amenability and virtual diagonals

(2) Let M be a unital dual operator algebra. Then M is a W ∗algebra if (and onlyif) there is a W ∗-algebra N containing M as a w∗-closed unital-subalgebra,such that for every approximately unital operator algebra A, we have

A ⊗nor M ⊂ A ⊗nor N isometrically.

Proof Let B ⊂ D as in (1), and assume that (7.12) holds for every such A. Thisproperty clearly remains true if D is replaced by a smaller C∗-algebra containingB, using 6.1.9. Hence we may assume that D is a C∗-cover of B (see 2.1.1). Wewill show that B is selfadjoint by showing that b∗ ∈ B, whenever b ∈ Ball(B).Fix such b ∈ B. As usual, T denotes the unit circle. For any n ∈ Z, let en ∈ C(T)be defined by en(z) = zn, for any z ∈ T. Then there is a (necessarily unique)completely positive map P : C(T) → D satisfying P (en) = bn for any n ≥ 0 andP (en) = b∗n for any n ≤ 0. Indeed assume that D acts nondegenerately on someHilbert space H . By Nagy’s dilation theorem (see 2.4.12), there is a Hilbert spaceK, an isometry J : H → K, and a unitary U ∈ B(K) such that bn = J∗UnJ ,for any n ≥ 0. Thus if we let π : C(T) → B(K) be the ∗-representation inducedby U , then the mapping P = J∗π(· )J has the required properties. Since C(T) isa nuclear C∗-algebra, it readily follows from Lemma 6.1.10 that P is 1-nuclear.Let Y be the two-dimensional subspace of C(T) spanned by e0 and e1, and letu : Y → D be the restriction of P to that subspace. Then u is 1-nuclear as well.To see this, note that for any approximately unital operator algebra A, we haveA ⊗min Y ⊂ A ⊗min C(T) isometrically, and hence∥∥IA ⊗ u : A ⊗min Y −→ A ⊗max D

∥∥ ≤ 1.

By (7.12), since u is valued in B, we have∥∥IA ⊗ u : A ⊗min Y −→ A ⊗max B∥∥ ≤ 1.

This shows that u : Y → B is 1-nuclear as a B-valued map. Since u(e0) = 1 and‖e0‖ = 1, Proposition 7.3.1 ensures that b∗ = u(e1)∗ belongs to B.

Assertion (2) can be proved similarly. Indeed, following the proof above, wefind P such that IA ⊗ P : A ⊗min C(T) → A ⊗max N is a contraction. Since themaximal norm dominates the normal tensor norm (see 6.1.3), we deduce thatIA ⊗ P : A ⊗min C(T) → A ⊗nor N is a contraction. We restrict to Y , as before,and see that IA ⊗ u : A ⊗min Y → A ⊗nor N is a contraction. Since u is valuedin M , and since A ⊗nor M ⊂ A ⊗nor N isometrically by hypothesis, we deducethat IA ⊗ u : A ⊗min Y → A ⊗nor M is a contraction. Thus u is 1-semidiscrete.Finally, we appeal to 7.3.2.

7.4 AMENABILITY AND VIRTUAL DIAGONALS

Amenability for Banach algebras was originally introduced by Barry Johnson(see [205]), and is by now a central research area which has impacted muchof modern mathematics. In this last section, we investigate amenable operator


algebras. We will mainly focus on some issues which are related to operatorspaces. In particular, we will emphasize the role of the Haagerup tensor productin the study of these objects. This will lead to selfadjointness results whichcomplement those in Section 7.2.

7.4.1 (Amenable Banach algebras) Let B be a Banach algebra and let X be aBanach B-bimodule. A derivation is a bounded linear mapping D : B → X suchthat D(ab) = aD(b) + D(a)b, for any a, b ∈ B. Given any x ∈ X , it is easy tocheck that the mapping D : B → X defined by letting D(b) = xb− bx, for b ∈ B,is a derivation. Such derivations are called inner.

If X is a Banach B-bimodule, then the dual space X∗ is a Banach B-bimodulefor the dual actions defined by

〈bϕ, x〉 = 〈ϕ, xb〉 and 〈ϕb, x〉 = 〈ϕ, bx〉

for every b ∈ B, x ∈ X, ϕ ∈ X∗. In this case, we say that X∗ is a dual Banachbimodule. Note that the left and right actions B×X∗ → X∗ and X∗×B → X∗

are w∗-continuous in the second and in the first variable respectively. By obviousiteration, X∗∗ also becomes a dual Banach B-bimodule.

By definition a Banach algebra B is amenable if whenever X∗ is a dual BanachB-bimodule, any derivation from B into X∗ is inner.

7.4.2 (Facts on amenability) We mention three simple results, for which werefer, for example, to [106] or [304, (1.30)]. First, let A and B be two Banachalgebras, and assume that B is amenable. If there is a bounded homomorphismθ : B → A with a dense range, then A is amenable as well. In particular, anyBanach algebra isomorphic to an amenable algebra is amenable. Second, anyamenable Banach algebra has a bounded approximate identity. Third, a unitalBanach algebra B is amenable provided that any derivation from B into X∗ isinner whenever X is a nondegenerate Banach B-bimodule (i.e. 1x = x = x1 forx ∈ X). For simplicity, we will often restrict our attention to unital algebras.

7.4.3 (Virtual diagonals) Let B be a unital Banach algebra. We may regardthe Banach space projective tensor product B⊗B (defined in A.3.3) as a BanachB-bimodule by letting

c(a ⊗ b) = ca ⊗ b and (a ⊗ b)d = a ⊗ bd (7.13)

for a, b, c, d ∈ B, and then extending by linearity and continuity. We let m bethe contractive linear mapping from B⊗B to B induced by the multiplicationon B. By definition, a virtual diagonal of B is an element u ∈ (B⊗B)∗∗ suchthat m∗∗(u) = 1 ∈ B, and such that cu = uc for any c ∈ B. In this definition,(B⊗B)∗∗ is equipped with its B-bimodule structure inherited from that on B⊗B(see 7.4.1). Not all Banach algebras admit a virtual diagonal:

Lemma 7.4.4 (Johnson) A unital Banach algebra is amenable if and only if itadmits a virtual diagonal.


Proof Assume that u ∈ (B⊗B)∗∗ is a virtual diagonal of B. Let X∗ be a dualBanach B-bimodule, and let D : B → X∗ be a derivation. By 7.4.2, we mayassume that X is nondegenerate. We define a bounded mapping

F : B⊗B −→ X∗

by letting F (a ⊗ b) = D(a)b, for any a, b ∈ B, and then extending by lin-earity and continuity. According to A.2.2, F admits a w∗-continuous extensionF : (B⊗B)∗∗ −→ X∗. We let ϕ = F (u) ∈ X∗, and we will show that

D(c) = ϕc − cϕ, c ∈ B. (7.14)

Let (ut)t be a net in B ⊗B converging to u in the w∗-topology of (B⊗B)∗∗. Forany t, we let (at

k)k and (btk)k be finite families of B such that

ut =∑

k

atk ⊗ bt

k.

We haveϕc − cϕ = w∗ − lim

t

(∑k

D(atk)bt

kc −∑

k

cD(atk)bt

k

). (7.15)

Indeed, ϕ is the w∗-limit of F (ut), and so (7.15) follows from the w∗-continuityproperties of the dual actions (see 7.4.1). On the other hand, cu and uc are thew∗-limits of cut and utc respectively, and so w∗-limt(cut − utc) = 0. Since F isw∗-continuous, this implies that w∗-limt

(F (cut) − F (utc)

)= 0, that is,

w∗ − limt

(∑k

D(catk)bt

k −∑

k

D(atk)bt

kc)

= 0. (7.16)

Now since D is a derivation, we have∑k

D(catk)bt

k = D(c)∑

k

atkbt

k + c∑

k

D(atk)bt

k

for any t. The first term in the right side above equals D(c)m(ut), and by as-sumption, m(ut) → 1 in the weak topology of B. Hence D(c)m(ut) → D(c) inthe weak topology of X∗, and therefore also in its w∗-topology. The identity(7.14) therefore follows by combining the above with (7.15) and (7.16). Thisshows that D is inner, and hence that B is amenable.

Assume conversely that B is amenable. The ‘product’ m : B⊗B → B is aB-bimodule map. Thus its kernel Ker(m) is a B-B-submodule of B⊗B. HenceKer(m)∗∗ is a dual Banach B-bimodule (see 7.4.1). Regard Ker(m)∗∗ as a sub-space of (B⊗B)∗∗, and define D : B → Ker(m)∗∗ by letting

D(b) = 1 ⊗ b − b ⊗ 1, b ∈ B.


Clearly, D is a derivation. By amenability, there exists v ∈ Ker(m)∗∗ such thatD(b) = vb− bv for any b ∈ B. Let u = 1⊗ 1− v ∈ (B⊗B)∗∗. Then u is a virtualdiagonal of B. Indeed, m∗∗(u) = m(1 ⊗ 1) = 1 since m∗∗(v) = 0. Also,

ub − bu = (1 ⊗ 1)b − vb − b(1 ⊗ 1) + bv = 1 ⊗ b − vb − b ⊗ 1 + bv

= D(b) − vb + bv = 0,

for any b ∈ B.

7.4.5 (Amenable C∗-algebras) The amenable C∗-algebras are exactly the nu-clear ones. This follows from the remarkable work of Connes (who showed thatamenability implies nuclearity [100]), and Haagerup [179] who showed directlythat any nuclear C∗-algebra admits a virtual diagonal. Thus any operator al-gebra which is isomorphic to a nuclear C∗-algebra, is amenable by Haagerup’stheorem. It is unknown whether the converse is true. We will present some partialanswers in the rest of this section.

The simplest nuclear (hence amenable) C∗-algebras are the finite-dimensionalC∗-algebras. For such algebras, it is easy to describe all (virtual) diagonals. Seethe Notes to this section. We merely mention here that a diagonal for a finite-dimensional C∗-algebra B may be constructed as follows. If U denotes the unitarygroup of B, and if we let dµ be the Haar measure on this compact group, then

u =∫U

z ⊗ z∗ dµ(z) ∈ B ⊗ B (7.17)

is a (virtual) diagonal of B.

7.4.6 (Virtual h-diagonals) We consider a variant of 7.4.3 adapted to operatoralgebras and the Haagerup tensor product of 1.5.4. Let B be a unital operatoralgebra. By 3.4.9, B⊗hB is a Banach B-bimodule for the actions given by (7.13).We let mh : B⊗hB → B denote the contraction induced by the multiplication onB. We define a virtual h-diagonal of B to be an element u ∈ (B⊗hB)∗∗ such thatm∗∗

h (u) = 1 ∈ B, and such that cu = uc for any c ∈ B. Since the Banach spaceprojective tensor product dominates the Haagerup tensor product (see 1.5.13),there is a canonical contraction κ : B⊗B → B ⊗h B, and we have m = mh κ.Moreover κ is a B-bimodule map. By the w∗-continuity properties of dual actions(see 7.4.1), we deduce that κ∗∗ : (B⊗B)∗∗ → (B ⊗h B)∗∗ is a B-bimodule map.Thus if v ∈ (B⊗B)∗∗ is a virtual diagonal of B, then u = κ∗∗(v) ∈ (B⊗h B)∗∗ isa virtual h-diagonal of B. The converse is not true (see 7.4.7 below). However ifB is a C∗-algebra, it turns out that B admits a virtual diagonal if and only if itadmits a virtual h-diagonal (and if and only if it is nuclear, by 7.4.4 and 7.4.5).See 7.4.9 and the Notes to this section.

7.4.7 (A virtual h-diagonal which is not a virtual diagonal) Let B be the uniti-zation of the C∗-algebra of compact operators on 2. That is, B = SpanS∞, I2,


the span in B(2). Let (Eij)i,j≥1 be the matrix units. For any n ≥ 1, we defineun =

∑ni=1 Ei1 ⊗ E1i, considered as an element of B ⊗ B. By 1.2.5,

∥∥[E11 · · · En1]∥∥

Rn(B)=∥∥∥ n∑

i=1

Ei1E∗1i

∥∥∥ 12

=∥∥∥ n∑

i=1

Eii

∥∥∥ 12 ≤ 1.

Likewise, the norm of [E11 · · · E1n]t in Cn(B) is less than or equal to 1. Hence‖un‖h ≤ 1. Being a bounded sequence in B ⊗h B, (un)n has a w∗-cluster pointu ∈ (B ⊗h B)∗∗. We have mh(un) =

∑ni=1 Eii, and the latter converges to 1 in

the weak topology of B. Hence m∗∗h (u) = 1. Moreover, we have

Ejkun = Ej1 ⊗ E1k = unEjk

whenever 1 ≤ j, k ≤ n. Hence Ejku = uEjk, for j, k ≥ 1. This readily impliesthat cu = uc for any c ∈ B. Thus u is a virtual h-diagonal of B.

Let κ : B⊗B → B ⊗h B be the canonical contraction (see 7.4.6). We claimthat u does not belong to the range of κ∗∗, so that, by 7.4.6, u ‘is not’ a virtualdiagonal of B. Assume to the contrary that u = κ∗∗(v) for some v. We will use the‘trace class’ S1 = S1(2), and the duality relations S∞∗ = S1 and S1∗ = B(2)(see A.1.2). Consider the natural action of S1 ⊗ S1 on B(2) ⊗ B(2) given by

〈y ⊗ z, a ⊗ b〉 = 〈y, a〉〈z, b〉 = tr(ya) tr(zb), y, z ∈ S1, a, b ∈ B(2). (7.18)

Since (B(2)⊗B(2))∗ coincided with bounded bilinear forms on B(2) (see A.3.3),it follows from the definition of the injective tensor product (in A.3.1) that (7.18)induces an isometric embedding S1⊗S1 → (B(2)⊗B(2))∗. Since S1 = S∞∗,the restriction of (7.18) to pairs (a, b) ∈ S∞ × S∞ also yields an isometry ofS1⊗S1 into (S∞⊗S∞)∗ ∼= B(S∞, S1), by a fact towards the end of A.3.3. SinceS∞ ⊂ B ⊂ B(2), we deduce that

S1⊗S1 → (B⊗B)∗ isometrically. (7.19)

There is thus a canonical scalar valued pairing between (B⊗B)∗∗ and S1 ⊗ S1.Similarly, we can regard S1⊗S1 as a subspace of (B⊗hB)∗, and we obtain a scalarpairing between (B ⊗h B)∗∗ and S1 ⊗ S1. The map κ∗ : (B ⊗h B)∗ → (B⊗B)∗,is the identity mapping on the copies of S1 ⊗ S1. Thus

〈w, u〉 = 〈w, v〉, w ∈ S1 ⊗ S1. (7.20)

For any m ≥ 1, let wm =∑m

i=1 E1i ⊗ Ei1, regarded as an element of S1 ⊗ S1.For any two integers n ≥ m ≥ 1, we have

〈un, wm〉 =m∑

i=1

〈Ei1, E1i〉〈E1i, Ei1〉 = m.

Since 〈wm, u〉 is a cluster point of the sequence (〈wm, un〉)n, we have 〈wm, u〉 = mfor every m ≥ 1. It therefore follows from (7.20) that 〈wm, v〉 = m. Applying(7.19), we deduce that


m ≤ ‖v‖(B⊗B)∗∗ ‖wm‖S1⊗S1 , m ≥ 1.

For any a = [aij ] and b = [bij ] in B(2), we have

|〈wm, a ⊗ b〉| =∣∣∣ m∑i=1

ai1b1i

∣∣∣ ≤(∑

i

|ai1|2) 1

2(∑

i

|b1i|2) 1

2 ≤ ‖a‖ ‖b‖.

According to A.3.1, this shows that ‖wm‖S1⊗S1 ≤ 1. Hence ‖v‖(B⊗B)∗∗ ≥ m forevery integer positive m, a contradiction.

7.4.8 (Normal virtual h-diagonals) We introduce an analogue of 7.4.6 in thedual setting, using the normal Haagerup tensor product (see 1.6.8). Let M be aunital dual operator algebra, and let mh : M ⊗h M → M be the multiplicationmapping. Since the product on M is separately w∗-continuous (see 2.7.4 (1)),the adjoint mapping m∗

h maps M∗ into the subspace (M ⊗h M)∗σ ⊂ (M ⊗h M)∗

of separately w∗-continuous completely bounded bilinear forms. We let

mσ : M ⊗σh M −→ M

denote the adjoint of the restriction m∗h|M∗ : M∗ → (M ⊗h M)∗σ. By definition,

mσ is a w∗-continuous complete contraction extending mh. By 7.4.6, M ⊗h Mis an M -bimodule. Taking dual actions (see 7.4.1), (M ⊗h M)∗ is therefore adual Banach M -bimodule. We observe that (M ⊗h M)∗σ is an M -M -submoduleof (M ⊗h M)∗. Indeed assume that ψ ∈ (M ⊗h M)∗, and let a, b ∈ M . Then forany c and d in M , we have

〈bψa, c ⊗ d〉 = ψ(ac, db).

Since the multiplication on M is separately w∗-continuous, bψa ∈ (M ⊗h M)∗σif ψ ∈ (M ⊗h M)∗σ. Using the dual bimodule actions from 7.4.1, we may equipM⊗σhM =

((M⊗hM)∗σ

)∗ with a Banach M -bimodule structure. One can easilycheck that the actions on M ⊗σh M extend those on M ⊗h M . By definition, anormal virtual h-diagonal of M is an element u ∈ M⊗σhM , such that mσ(u) = 1and cu = uc for every c ∈ M . It follows from the work of Connes [100], Effros[134], and Effros and Kishimoto [138], that if M is a W ∗-algebra, then M isinjective if and only if it admits a normal virtual h-diagonal.

7.4.9 (Second duals) Let B be a unital operator algebra. By (1.56), (B⊗hB)∗∗

and B∗∗ ⊗σh B∗∗ coincide as dual operator spaces. Applying 7.4.8, B∗∗ ⊗σh B∗∗

is equipped with a canonical B∗∗-bimodule structure, and it also has a naturalB-bimodule structure. It is easy to check that the latter structure coincides withthe bidual B-bimodule structure on (B ⊗h B)∗∗, under the above identification.Furthermore m∗∗

h and mσ are the same under that identification. This is becausethey are both w∗-continuous, and they coincide on B ⊗ B, which is a w∗-densesubspace of B∗∗ ⊗σh B∗∗ = (B ⊗h B)∗∗.

According to the above discussion, any normal virtual h-diagonal of B∗∗

is a virtual h-diagonal of B. One of the key results from [138] is that if B is


selfadjoint, then the converse holds true, and hence normal virtual h-diagonalsof B∗∗ coincide with virtual h-diagonals of B. Thus if a C∗-algebra B has avirtual h-diagonal, then B∗∗ is injective by the last assertion of 7.4.8. Thus B isnuclear, and hence admits a virtual diagonal by Haagerup’s theorem [179].

7.4.10 We shall relate virtual h-diagonals (and hence amenability) to themodule complementation property from 7.2.2. Let M be a unital dual opera-tor algebra. To any unital w∗-continuous completely contractive representationπ : M → B(H), we associate a linear mapping τπ : M ⊗M → CB

(B(H), B(H)

),

defined by letting

τπ(a ⊗ b)(T ) = π(a)Tπ(b), a, b ∈ M, T ∈ B(H).

Let a1, . . . , am, b1, . . . , bm ∈ M , and let u =∑

k ak ⊗ bk ∈ M ⊗ M . We letQ = τπ(u), so that Q(T ) =

∑k π(ak)Tπ(bk) for any T ∈ B(H). We define a

representation ρ : B(H) → Mm⊗B(H) B(2m(H)), by letting ρ(T ) = IMm ⊗T .

Define V, W : H → 2m(H) by

V (ζ) =(π(b1)ζ, . . . , π(bm)ζ

)and W (ζ) =

(π(a1)∗ζ, . . . , π(am)∗ζ

),

for any ζ ∈ H . Then Q = W ∗ρ(· )V , hence ‖Q‖cb ≤ ‖V ‖‖W‖. We deduce, usingthe fact that ‖π‖cb = 1, that

‖Q‖cb ≤∥∥[π(a1) · · · π(am)]

∥∥Rm(B(H))

∥∥[π(b1) · · · π(bm)]t∥∥

Cm(B(H))

≤∥∥[a1 · · · am]

∥∥Rm(M)

∥∥[b1 · · · bm]t∥∥

Cm(M).

By 1.5.4, τπ extends to a contraction from M ⊗h M to CB(B(H), B(H)

).

Lemma 7.4.11 Let π and τπ as above.

(1) The above mapping τπ extends uniquely to a contractive w∗-continuous linearmap from M ⊗σh M into CB

(B(H), B(H)

)(which we still denote by τπ).

(2) For any u ∈ M ⊗σh M , the mapping τπ(u) : B(H) → B(H) is a bimodulemap over [π(M)]′. That is,

τπ(u)(S2TS1) = S2

(τπ(u)(T )

)S1, S1, S2 ∈ [π(M)]′, T ∈ B(H). (7.21)

(3) Let u ∈ M ⊗σh M and assume that mσ(u) = 1. Then τπ(u) is unital.(4) Let u ∈ M⊗σhM and c ∈ M , and assume that cu = uc. Then π(c) commutes

with τπ(u)(T ), for any T ∈ B(H).

Proof For simplicity write τπ as τ below. The uniqueness in (1) is clear, sinceM ⊗ M is w∗-dense in M ⊗σh M , as we saw in 1.6.8. According to 7.4.10, weregard τ as a contraction on M ⊗h M . We recall that CB

(B(H), B(H)

)is the

dual Banach space of S1(H)⊗ B(H) (by (1.51)), and consider the restriction of


the adjoint map τ∗ to this predual. Let ϕ and T be arbitrary elements of S1(H)and B(H) respectively. Then for any a, b ∈ M , we have

〈τ∗(ϕ ⊗ T ), a⊗ b〉 = 〈ϕ ⊗ T, τ(a ⊗ b)〉 = 〈ϕ, π(a)Tπ(b)〉.

Since π is w∗-continuous and the multiplication mapping on B(H) is separatelyw∗-continuous, we deduce that τ∗(ϕ⊗T ) is separately w∗-continuous, and hencebelongs to

(M⊗hM

)∗σ. By linearity and continuity, τ∗ maps S1(H)

⊗ B(H) into(

M ⊗h M)∗σ. We let θ : S1(H)

⊗ B(H) →

(M ⊗h M

)∗σ

be the resulting mapping.The adjoint θ∗ : M ⊗σh M → CB

(B(H), B(H)

)is a w∗-continuous contraction.

By construction, θ∗(a ⊗ b) = τ(a ⊗ b) for every a, b ∈ M , which proves (1).To prove the last three assertions, we fix an element u ∈ M ⊗σh M , and let

Q = τ(u). Let (ut)t be a net in M ⊗ M converging to u in the w∗-topology ofM ⊗σh M . For any t, we let (at

k)k and (btk)k be finite families of M such that

ut =∑

k

atk ⊗ bt

k.

Since τ is w∗-continuous, Q is the w∗-limit of τ(ut). Hence

Q(T ) = w∗ − limt

∑k

π(atk)Tπ(bt

k). (7.22)

if T ∈ B(H). If S1, S2 ∈ [π(M)]′ and T ∈ B(H), then

∑k

π(atk)S2TS1π(bt

k) = S2

(∑k

π(atk)Tπ(bt

k))

S1,

for any t. Since the multiplication is separately w∗-continuous on B(H), (7.21)now follows from (7.22).

Assume that mσ(u) = 1. With the above notation, we have for any t∑k

π(atk)π(bt

k) = π(∑

k

atkbt

k

)= π

(m(ut)

).

Let us apply (7.22) with T = IH . Since 1 = mσ(u) is the w∗-limit of m(ut) andπ is w∗-continuous and unital, we deduce that Q(IH) = IH , which proves (3).

Suppose that c ∈ M , with cu = uc, and fix T ∈ B(H). Using (7.22) again,

π(c)Q(T ) = w∗ − limt

∑k

π(catk)Tπ(bt

k), (7.23)

whereasQ(T )π(c) = w∗ − lim

t

∑k

π(atk)Tπ(bt

kc). (7.24)


Since cu = uc, we have w∗-limt(cut−utc) = 0 in M⊗σh M . By the w∗-continuityof τ we deduce that w∗-limt

(τ(cut) − τ(utc)

)= 0. Therefore we have

w∗ − limt

(τ(cut)(T ) − τ(utc)(T )

)= 0,

and so

w∗ − limt

(∑k

π(catk)Tπ(bt

k) −∑

k

π(atk)Tπ(bt

kc))

= 0.

This, together with (7.23) and (7.24), gives Q(T )π(c) = π(c)Q(T ).

Proposition 7.4.12

(1) Let M be a unital dual operator algebra which has a normal virtual h-diagonalu ∈ M ⊗σh M . Then M has the w∗-module complementation property (see7.2.3). More precisely, let π : M → B(H) be a unital w∗-continuous com-pletely contractive representation. If K is a π(M)-invariant subspace ofH, then there exists a idempotent map p in [π(M)]′ ⊂ B(H), such thatRan(p) = K and ‖p‖ ≤ ‖u‖M⊗σhM .

(2) Let B be a unital operator algebra which possesses a virtual h-diagonal u in(B ⊗h B)∗∗. Then B has the module complementation property (see 7.2.2).Indeed, if π : B → B(H) is a unital completely contractive representation,and if K is a π(B)-invariant subspace of H, then there exists an idempotentmap p in [π(B)]′ ⊂ B(H), such that Ran(p) = K and ‖p‖ ≤ ‖u‖(B⊗hB)∗∗.

Proof (1) We consider a unital w∗-continuous completely contractive repre-sentation π : M → B(H), and we let τπ be the associated mapping consideredin 7.4.10 and 7.4.11. We let Q = τπ(u), u being a normal virtual h-diagonal. Ac-cording to 7.4.11 (3), Q is unital. It therefore follows from (7.21) that Q(S) = S,for any S ∈ [π(M)]′. On the other hand, since uc = cu for any c ∈ M , 7.4.11 (4)shows that Q is valued in [π(M)]′. Thus Q : B(H) → B(H) is an idempotent,with range equal to [π(M)]′.

As in the proof of Lemma 7.4.11, we let ut =∑

k atk ⊗ bt

k ∈ M ⊗ M besuch that ut → u in the w∗-topology of M ⊗σh M . Let K be a π(M)-invariantsubspace of H , and let PK ∈ B(H) be the orthogonal projection onto K. Thenπ(· )PK = PKπ(· )PK , and so∑

k

PKπ(atk)PKπ(bt

k) =∑

k

π(atk)PKπ(bt

k),

and ∑k

π(atk)PKπ(bt

k)PK =∑

k

π(atk)π(bt

k)PK = π(∑

k

atkbt

k

)PK ,

for any t. Applying (7.22), we deduce that

PKQ(PK) = Q(PK) and Q(PK)PK = PK . (7.25)

Applying (7.21) with S2 = IH , T = PK and S1 = Q(PK), and using the factthat Q is an idempotent, we deduce that Q(PK)2 = Q(PK). Now (7.25) shows


that the two idempotents PK and p = Q(PK) have the same range. Moreover,we have the desired estimate

‖p‖ ≤ ‖Q‖‖PK‖ = ‖Q‖ ≤ ‖τπ‖‖u‖σh = ‖u‖σh.

(2) Let B be a unital operator algebra with virtual h-diagonal u in (B⊗hB)∗∗,and let π : B → B(H) be a completely contractive representation. Let K be anyπ(B)-invariant subspace of H . Set M = B∗∗, and consider the w∗-continuousextension π : M → B(H) of π. According to 7.2.4, K also is a π(M)-invariantsubspace. As explained in 7.4.9, we identify (B ⊗h B)∗∗ with M ⊗σh M as dualB-bimodules. Under this identification, we have mσ(u) = 1 and cu = uc forany c ∈ B. Since [π(B)]′ = [π(M)]′, (1) above shows that if we let Q = τπ(u),and if we let PK be the orthogonal projection onto K, then p = Q(PK) is anidempotent whose range equals K, with ‖p‖ ≤ ‖u‖(B⊗hB)∗∗ .

Corollary 7.4.13 (Gifford) Let B be a unital operator algebra. If B is amenablethen B has the module complementation property.

Proof We assume that B is amenable. By Lemma 7.4.4, B has a virtual diag-onal, and hence a virtual h-diagonal (see 7.4.6). Now apply 7.4.12 (2).

Corollary 7.4.14 Let B be a finite-dimensional unital operator algebra. ThenB is amenable if and only if B is (completely) isomorphic to a C∗-algebra.

Proof The ‘only if’ part follows from Proposition 7.2.7 and Corollary 7.4.13.The ‘if’ part was discussed in 7.4.5.

Corollary 7.4.15 (Curtis and Loy) Let B be a unital operator algebra andassume that B is amenable. If B is generated (as an operator algebra) by itsnormal elements (see 7.2.9), then B is selfadjoint.

Proof Combine Theorem 7.2.10 (1), and Corollary 7.4.13.

Corollary 7.4.16 (Sheinberg) A uniform algebra is amenable if and only if itis selfadjoint.

Proof The ‘only if’ part follows from Theorem 7.2.10, (2), and Corollary 7.4.13.The ‘if’ part is a special case of 7.4.5.

7.4.17 (Norms of virtual diagonals) Let M be a unital dual operator algebra,and assume that u ∈ M ⊗σh M is a normal virtual h-diagonal. Since mσ is acontraction from M ⊗σh M into M taking u to 1, we have ‖u‖σh ≥ 1. Like-wise, if B is a unital operator algebra with a virtual h-diagonal u ∈ (B ⊗h B)∗∗,then ‖u‖(B⊗hB)∗∗ ≥ 1. A fortiori, if u ∈ (B⊗B)∗∗ is a virtual diagonal, then‖u‖(B⊗B)∗∗ ≥ 1. It turns out these rough estimates are optimal for selfadjoint al-gebras. Namely, it follows from Haagerup’s work [179] on amenable C∗-algebras,that if B is a nuclear C∗-algebra, then B admits a virtual diagonal u such that‖u‖(B⊗B)∗∗ = ‖u‖(B⊗hB)∗∗ = 1. Likewise, it follows from [134] that any injectiveW ∗-algebra M admits a normal virtual h-diagonal u such that ‖u‖σh = 1. Itturns out that these properties characterize the amenable C∗-algebras amongamenable operator algebras:


Theorem 7.4.18

(1) Let M be a unital dual operator algebra. If M possesses a normal virtualh-diagonal of norm 1, then M is selfadjoint (and hence M is an injectiveW ∗-algebra, by 7.4.8).

(2) A unital operator algebra which admits a virtual h-diagonal of norm 1, isselfadjoint (and hence is a nuclear C∗-algebra, by 7.4.6).

Proof We only prove (1), the proof of (2) being similar. We apply Theorem7.2.5 (2). Let π : M → B(H) be a unital w∗-continuous completely contractiverepresentation, and let K be a π(M)-invariant subspace of H . According to 7.4.12(1), there is an idempotent p in [π(M)]′ such that Ran(p) = K and ‖p‖ ≤ 1.This estimate forces p to be the orthogonal projection onto K. This shows thatK is reducing, and hence that M has the w∗-reducing property.


Characterizations of selfadjointness for operator algebras are scattered through-out the literature (e.g. see [354]). We have made no attempt to survey all suchresults here; no doubt every researcher in this field has proved a theorem ofthis type. We have focused on some results related to operator space theory,although admittedly some of our results hold at the Banach algebra level too,with essentially the same proof.

7.1: The weak expectation property was originally introduced for C∗-algebrasby Lance [238]. Its definition for operator spaces appears in [141]; Corollary 7.1.5is from that paper. Results in 7.1.6 and 7.1.7 may be new. The second assertionof 7.1.8 appeared in Pisier’s book [337]. The WEP is currently attracting muchinterest from C∗-algebraists. For remarkable developments going far beyond thescope of this book we refer the reader to, for example, [136, 181, 228, 296], andthe references therein.

Proposition 7.1.10 is a new result based on Kisliakov’s remarkable paper [232].The main result in the latter paper asserts that if B ⊂ C(Ω) is a uniformalgebra, and if there is an idempotent P : C(Ω) → C(Ω) with Ran(P ) = B, thenB = C(Ω). The facts and arguments in 7.1.11 and 7.1.12 are new, we believe.

7.2: The material here may be viewed as having birth in the famous (andstill open) reductive algebra problem of Radjavi and Rosenthal: If M is a weaklyclosed unital-subalgebra of B(H), and if every M -invariant subspace of H isreducing, then is M selfadjoint? See [354] for details and progress. The assertionis false if ‘weakly closed’ is replaced by ‘w∗-closed’, as Loebl and Muhly showedin [256]. The module complementation property defined in 7.2.2 was introducedin a slightly different form by Gifford in [169]. This remarkable paper was un-fortunately never published. The fact that any unital amenable operator algebrahas the module complementation property (Corollary 7.4.13) was proved there.Gifford studied whether the module complementation property implies selfad-jointness for approximately unital operator algebras. He showed that the answeris positive provided that A can be represented as a subalgebra of the compact


operators S∞(H) for some Hilbert space H . A special case appears in [430].The first part of Theorem 7.2.5 is from [281, Theorem 3.1]. This theorem mayalso be deduced from the methods of 7.2.8. In connection with 7.2.2, some othernotions of semisimplicity for operator algebras are studied in [200, 223, 281], forexample. Note that every finite-dimensional uniform algebra A is selfadjoint. In-deed, the one-dimensional representations of A separate points, and since A isfinite-dimensional it is easily argued that such an algebra is semisimple in eitherthe algebraic or the Banach algebraic sense. An application of Wedderburn’stheorem as in the proof of 7.2.7, shows that A is isomorphic (and therefore alsoisometric, by A.5.4) to a commutative C∗-algebra.

The proof of 7.2.8 was inspired by that of [281, Theorem 3.1]. The result in7.2.10 (1) is an adaptation of 7.4.15. Proposition 7.2.11 was proved in [50], asa simple consequence of [281, Theorem 3.1]. Those papers contain other relatedcharacterizations of C∗-algebras. There is probably an ‘isomorphic’ version of7.2.11. Indeed, we leave it as an exercise using 5.1.2 and 3.6.2, and some diagramchasing, that if B is completely isomorphic to a C∗-algebra, and if K, X, Y, andu are as in 7.2.11, then u extends to a completely bounded A-module map fromY to B(K). Conversely, the idea of the proof of 7.2.11 shows that if B(K) hassuch a ‘completely bounded module map extension property’, for every HilbertB-module K, then B has the module complementation property. One may alsoshow using the ideas of 3.2.14, that if B has the module complementation prop-erty, then every representation of B has the double commutant property 3.2.13.This was first noticed by Gifford [169]. Also in connection with module map ex-tensions, in [391] Smith exhibits a unital subalgebra B of M6, a very simple andconcrete left B-submodule X of M6, and a B-module map u : X → M6 whichadmits no B-module extension from M6 to M6.

Le Merdy proved Proposition 7.2.12 in [252]. An alternative proof using 7.2.5was given shortly thereafter by Blecher (see [50]). We sketch Le Merdy’s argu-ment, which uses 7.3.3. Let D be the maximal C∗-algebra of B (see 2.4.3). LetA be an approximately unital operator algebra, and consider z ∈ A ⊗ B. Letπ : A → B(H) and ρ : B → B(H) be two commuting completely contractive rep-resentations on some Hilbert space H . By assumption, [ρ(B)]′ is selfadjoint, andso C = [ρ(B)]′′ is a C∗-algebra. Since ρ(B) ⊂ C, there exists a ∗-representationρ : D → C extending ρ (see 2.4.2). Now ρ commutes with π, since the latter isvalued in [ρ(B)]′ = C′. Hence

∥∥π • ρ(z)∥∥ =

∥∥π • ρ(z)∥∥ ≤ ‖z‖A⊗maxD. We de-

duce that ‖z‖A⊗maxB ≤ ‖z‖A⊗maxD, by taking the supremum over all possiblepairs (π, ρ). Hence A⊗max B ⊂ A⊗max D isometrically. By Theorem 7.3.3, B isselfadjoint. A similar argument proves 7.2.12 (2).

It is easy to extend the module complementation property and reducing prop-erty to nonunital operator algebras A; one simply drops the requirement thatthe representations π considered be nondegenerate. Using 2.1.13, it is clear thatthese properties are equivalent to the unitization A1 having the correspondingproperty in 7.2.2. We deduce from 7.2.10 (1) and 7.1.6 (3) that A has the reduc-ing property if and only if A is selfadjoint. Similarly, one may drop the unital


hypothesis in 7.2.7.7.3: Propositions 7.3.1 and 7.3.2 are due to Le Merdy [252], but their proofs

were inspired by Pisier’s proof of 7.1.8 contained in [337]. The first part of 7.3.3is also taken from [252] and its second part is an easy adaptation. A classicalrelated result of Lance [238] asserts that if B is a C∗-algebra, then B has theWEP if and only if it satisfies the following ‘inclusion’ property: for any C∗-algebra D containing B as a C∗-subalgebra, and for any approximately unitaloperator algebra A, we have A ⊗max B ⊂ A ⊗max D isometrically. In Lance’soriginal statement A is a C∗-algebra, but it is easy to see that one may allow Ato be nonselfadjoint.

7.4: Amenable Banach algebras were introduced by Barry Johnson in [205]. Inthat memoir, he proved that for a locally compact group G, L1(G) is an amenableBanach algebra if and only if G is an amenable group. The fact that commutativeC∗-algebras are amenable (a special case of 7.4.5), as well as Lemma 7.4.4, werealso proved in that memoir. For abundant information on amenability, and thehistory of that subject, we refer the reader to [106,304,377], for example.

Virtual h-diagonals and normal virtual h-diagonals were implicitly introducedin the selfadjoint setting by Haagerup [179], Effros [134], and Effros and Kishi-moto [138]. These fundamental papers are a primary inspiration for the materialin this section. The construction in 7.4.10 and Lemma 7.4.11 essentially go backto [138]. If u ∈ M ⊗σh M is a normal virtual h-diagonal, then the mappingQ = τπ(u) constructed in the proof of 7.4.12 is a completely bounded ‘quasi-expectation’ from B(H) onto [π(M)]′, in the sense of Bunce and Paschke [79]. Itis shown in that paper that if a von Neumann algebra M ⊂ B(H) is the rangeof a quasi-expectation, then it is injective.

In 7.4.9, we mentioned that if a C∗-algebra B admits a virtual h-diagonal,then it is nuclear. We sketch a proof of this using quasi-expectations. If u is a vir-tual h-diagonal in (B ⊗h B)∗∗, and if π : B → B(H) is a unital ∗-representation,then [π(B)]′ is the range of a quasi-expectation by the proof of 7.4.12 (2).Thus [π(B)]′ is injective (by the last paragraph). By the well-known equiva-lence of semidiscreteness and injectivity for W ∗-algebras, [π(B)]′ is semidiscrete.By [66, Proposition 3.7], [π(B)]′′ is semidiscrete. Taking π to be the universalrepresentation shows that B∗∗ is semidiscrete. By 6.6.8 (1), B is nuclear. A vari-ant of this proof is given in [311]. This paper contains remarkable related results,such as the fact that if a C∗-algebra is the range of a quasi-expectation on B(H),then it is injective. See [163] for a beautiful short proof of this.

The statements in 7.4.12 and 7.4.18 are new, and 7.4.13 is from [169]. Corol-lary 7.4.15 is from [105], whereas Sheinberg’s theorem 7.4.16 was proved in [382].

Let B be a unital Banach algebra. A diagonal of B is a virtual diagonalbelonging to B⊗B. Likewise if B is an operator algebra, then an h-diagonalof B is a virtual h-diagonal belonging to B ⊗h B. Paulsen and Smith provedin [319] that if a unital operator algebra possesses an h-diagonal, then it isfinite-dimensional, and hence isomorphic to a C∗-algebra (see 7.4.14). This is afortiori true if B has a diagonal. However it is unknown whether there may exist


an infinite-dimensional Banach algebra with a diagonal. See [377, Section 4.1] formore on this topic. Dual algebras with a virtual diagonal in the weak* Haageruptensor product are discussed in [70].

Ruan introduced operator amenability for matrix normed algebras in [374]. Bydefinition, a matrix normed algebra B is operator amenable if for every matrixnormed B-bimodule X , every completely bounded derivation D : B → X∗ isinner. For a matrix normed algebra B, let

m : B

⊗ B → B be the canonical map

induced by the multiplication on B. By definition, a virtual operator diagonal isan element u ∈ (B

⊗ B)∗∗ such that

m

∗∗(u) = 1, and cu = uc for any c ∈ B. The

following analogue of 7.4.4 holds true (with essentially the same proof): A matrixnormed algebra B is operator amenable if and only it admits a virtual operatordiagonal. Ruan proved in [374] that a locally compact group G is amenable ifand only if its Fourier algebra A(G) is operator amenable, if we regard A(G) asthe operator space predual of the group von Neumann algebra of G. Moreover,operator amenability does not imply amenability for a matrix normed algebra.Related results appear in [375].

The arguments in the proof of 7.4.4 can also be easily adapted to show thata unital operator algebra B admits a virtual h-diagonal if and only if wheneverX is an operator space and a Banach B-bimodule such that X∗ is an operatorB-bimodule, then every completely bounded derivation D : B → X∗ is inner. Itis tempting to call such operator algebras h-amenable. Corollaries 7.4.13–7.4.16hold true with ‘amenable’ replaced by ‘h-amenable’. Unfortunately, we do notknow any example of an h-amenable operator algebra which is not amenable.

It is elementary to describe all diagonals of a finite-dimensional C∗-algebra,and it is instructive to look at their norms. Consider for simplicity the case whenB = Mn for some integer n ≥ 1. Let (Eij)1≤i,j≤n be the canonical basis of Mn.Then if [tij ] is an element of Mn such that

∑nj=1 tjj = 1,

u =∑

1≤i,j,k≤n

tkjEij ⊗ Eki, (7.26)

is a diagonal of Mn. Moreover all diagonals of Mn ⊗ Mn have this form. Thespecial diagonal defined by (7.17) when B = Mn, corresponds to the case whentij = 0 if j = i and tjj = 1

n if 1 ≤ j ≤ n. Let u be given by (7.26), with∑nj=1 tjj = 1. Then it is not hard to check that ‖u‖h =

∥∥[tij ]∥∥

S1 . In particular,‖u‖h = 1 if and only if [tij ] is a positive matrix. On the other hand, ‖u‖∧ = 1 ifand only if (7.17) holds. We leave this as an exercise for the reader.

8

C∗-modules and operator spaces

This chapter only depends on (parts of) Chapters 1–4 of our text. It has sev-eral goals. The first is to study Hilbert C∗-modules (and their W ∗-algebra vari-ant, W ∗-modules) as operator modules. We aim to show that the theory ofC∗-modules fits comfortably into the operator module framework. Indeed the op-erator space viewpoint will lead us in a streamlined way through several aspectsof the theory of C∗-modules. Although we will not say much about this here, ourmethods also permit the generalization of these modules to the nonselfadjointoperator algebra case (see the Notes section for references). In contrast, Banachmodule methods are not generally compatible with C∗-module constructions,and indeed completely break down when one attempts the aforementioned non-selfadjoint generalization. The second goal of our chapter is to consider someimportant TROs or C∗-modules which are associated to every operator spaceX . In particular, we will discuss further here the noncommutative Shilov bound-ary T (X) of X . TRO methods, and this Shilov boundary, provide importantinsights into the structure of X . Third, we will illustrate how C∗-module andTRO methods can lead to interesting results about operator spaces.

Thus there is a profound two-way interaction between C∗-modules and op-erator spaces, which has attracted much interest in recent years. Because oflimitations of space, we cannot reproduce here many of the operator space ap-plications of C∗-module theory which appear in the literature. Instead, our moremodest goal, and this is the fourth purpose of our chapter, is to lay out, in asystematic way, most of the basic concepts, theory, and connections, which areneeded for such applications.

There are many different ‘pictures’, or ways of looking at, C∗-modules, as wellas many routes through this theory. We will start from scratch, moving quickly,and later we will begin to add the operator space and TRO perspectives. Ourpresentation is essentially selfcontained. However, we will concentrate on materialnot in the standard sources on C∗-modules (for example, see [35, 173, 239, 269,356,421,423]). We refer the reader to those texts for topics not covered here.

We will mostly state our results for right modules. However, as we shall see,there is a striking ‘left-right symmetry’ to the theory; in particular every rightC∗-module is also, canonically, a left C∗-module over another C∗-algebra. Whenwe need to apply to a module which is being viewed as a left module, a resultestablished earlier for right modules, we often refer to the ‘other-handed version’

C∗-modules and operator spaces 297

of the earlier result.

8.1 HILBERT C∗-MODULES—THE BASIC THEORY

Throughout this section and the next, A and B are C∗-algebras.

8.1.1 (The definition) A (right) C∗-module over A is a right A-module Y ,together with a map 〈·|·〉 : Y × Y → A, which is linear in the second variable,and which also satisfies the following conditions:(1) 〈y|y〉 ≥ 0 for all y ∈ Y ,(2) 〈y|y〉 = 0 if and only if y = 0,(3) 〈y|za〉 = 〈y|z〉a for all y, z ∈ Y, a ∈ A,(4) 〈y|z〉∗ = 〈z|y〉 for all y, z ∈ Y ,(5) Y is complete in the norm ‖y‖ = ‖〈y|y〉‖ 1

2 .We call 〈·|·〉 the A-valued inner product on Y . It follows from (3) and (4) that〈ya|z〉 = a∗〈y|z〉 for all y, z ∈ Y, a ∈ A.

In (5), the fact that ‖ · ‖ is a norm follows just as for Hilbert spaces fromthe following Cauchy–Schwarz inequality: ‖〈y|z〉‖ ≤ ‖y‖‖z‖ for y, z ∈ Y . Thisfollows from the relation

〈y|z〉〈z|y〉 ≤ ‖z‖2〈y|y〉,which in turn may be proved by using the fact that 0 ≤ 〈y + zb|y + zb〉, andtaking b = −〈z|y〉/‖z‖2. The last calculation is an easy exercise using the factthat c∗ac ≤ ‖a‖c∗c for any c ∈ A, a ∈ A+.

If the linear span of the range of the inner product 〈·|·〉 is dense in A, thenY is called full.

Left C∗-modules are defined analoguously. Here Y is a left module over a C∗-algebra A, the A-valued inner product is linear in the first variable, and condition(3) in the above is replaced by 〈ay|z〉 = a〈y|z〉, for y, z ∈ Y, a ∈ A. Note thatif A = C, then these are exactly the Hilbert spaces. On the other hand, a rightC∗-module over C is also a Hilbert space in the usual (mathematical) sense, withthe ‘reversed inner product’ ζ × η → 〈η|ζ〉.

If Y is a right C∗-module over A, then there is a canonical left C∗-module Yover A, which is simply the conjugate vector space of Y with left action ay = ya∗,and inner product 〈y|z〉 = 〈y|z〉. This is usually called the conjugate C∗-modulein the literature. We will favour the term adjoint module instead.

8.1.2 (Equivalence bimodules) If Y is an A-B-bimodule, then we say thatY is an equivalence bimodule, if Y is a full right C∗-module over B, and afull left C∗-module over A, and the two inner products are compatible in thesense that x〈y|z〉 = [x|y]z, for all x, y, z ∈ Y . Here we have written [·|·] forthe A-valued inner product. Equivalence bimodules are also sometimes calledimprimitivity bimodules, or strong Morita equivalence A-B-bimodules, or equiva-lence A-B-bimodules. If there exists such an equivalence bimodule, we say thatA and B are strongly Morita equivalent.

298 Hilbert C∗-modules—the basic theory

Good examples of equivalence bimodules are furnished by the ternary ringsof operators, or TROs, encountered in 4.4.1, and in Example 3.1.2 (6). If Z is aTRO, then we recall that D = ZZ and C = ZZ are C∗-algebras. Clearly Zis both a full right C∗-module over D, and a full left C∗-module over C. Indeedsuch Z is clearly an equivalence C-D-bimodule.

Thus we have one direction of the correspondence between TROs and C∗-modules. We will return to the other direction later, in 8.1.19 and 8.2.8.

8.1.3 (C∗-modules are Banach modules) If Y is a right C∗-module over A,then Y is a nondegenerate Banach A-module (see A.6.1). Indeed,

‖ya‖2 = ‖〈ya|ya〉‖ = ‖a∗〈y|y〉a‖ ≤ ‖a‖2‖y‖2,

for y ∈ Y and a ∈ A. If (et)t is a positive cai for A, then

〈y − yet|y − yet〉 = 〈y|y〉 − et〈y|y〉 − 〈y|y〉et + et〈y|y〉et −→ 0.

Thus yet → y for all y ∈ Y . Hence Y is a nondegenerate Banach A-module.

8.1.4 (The ideal I) Throughout this chapter we reserve the symbol I for theclosure of the linear span of the C∗-algebra valued inner product on a rightC∗-module. If Y is a right C∗-module over A then it is clear that I is a closedtwo-sided ideal in A, and that Y is a full right C∗-module over I. We make somesimple but important remarks:

(1) The canonical map I → B(Y ), taking a ∈ I to the operator y → ya, isa linear isometry. Indeed it is clearly contractive. If Y a = 0 then we have that〈Y |Y a〉 = 〈Y |Y 〉a = 0. This implies that Ia = 0, so that a∗a = 0. Hence a = 0.Thus the map is one-to-one. It follows from A.5.9 (or from the other-handedversion of the later result 8.1.15, for example) that this map is isometric.

(2) Since Y is a C∗-module, and hence a nondegenerate Banach module,over I, we have Y I = Y by Cohen’s factorization theorem A.6.2. In particular,the linear span of terms of the form x〈y|z〉, for x, y, z ∈ Y , is dense in Y .

(3) If Z is a Banach A-module then BA(Y, Z) = BI(Y, Z). Indeed, supposethat u ∈ BI(Y, Z), x ∈ Y and a ∈ A. By (2) and A.6.2, we may write x = ya′,with a′ ∈ I, y ∈ Y . Then u(xa) = u(ya′a) = u(y)a′a = u(ya′)a = u(x)a.

(4) Conversely to the first paragraph of 8.1.4, if Y is a right C∗-module overA, and if C is a C∗-algebra containing A as an ideal, then Y is also a right C∗-module over C. The module action here is the canonical one: (ya)c = y(ac), fory ∈ Y, a ∈ A, and c ∈ C. The fact that this action is well defined, and that 8.1.1(3) holds, follows from the relation 〈z|y(ac)〉 = 〈z|y〉ac = 〈z|ya〉c, for z ∈ Y .

Lemma 8.1.5 Let u : Y → Z be a bounded A-module map between right C∗-modules over A. Then 〈u(y)|u(y)〉 ≤ ‖u‖2〈y|y〉, for all y ∈ Y .

Proof We may suppose that ‖u‖ ≤ 1, and (by 8.1.4 (4)) that A is unital. Thenthe result follows by a trivial modification of the argument for (4.10).


Proposition 8.1.6 Suppose that A is a C∗-subalgebra of B(H). The norm andinner product on a right C∗-module Y over A are related by the formula:

〈〈y|y〉ζ, ζ〉 = sup‖f(y)ζ‖2 : f ∈ Ball(BA(Y, A)), y ∈ Y, ζ ∈ H.

Proof For f ∈ Ball(BA(Y, A)), y ∈ Y, and ζ ∈ H , we have

‖f(y)ζ‖2 = 〈f(y)∗f(y)ζ, ζ〉 ≤ 〈〈y|y〉ζ, ζ〉,

by 8.1.5. Next, set f = 〈zn| ·〉, where zn = y(〈y|y〉+ 1n )−

12 . Here n ∈ N. We have

〈zn|zn〉 = (〈y|y〉+ 1n )−

12 〈y|y〉(〈y|y〉+ 1

n )−12 ≤ 1. Hence f ∈ Ball(BA(Y, A)). Note

that f(y)∗f(y) = 〈y|y〉(〈y|y〉 + 1n )−1〈y|y〉. By spectral theory,

limn

〈y|y〉(〈y|y〉 +1n

)−1〈y|y〉 = 〈y|y〉.

Putting these facts together gives the desired equality.

Proposition 8.1.6, together with the polarization identity (1.1), shows thatthe inner product on a C∗-module Y is completely determined by, and may berecovered from, the Banach module structure of Y .

8.1.7 (Adjointable maps) If Y, Z are right C∗-modules over A, then we writeBA(Y, Z) (or simply B(Y, Z)) for the set of adjointable maps from Y to Z, thatis, the set of maps T : Y → Z such that there exists a map S : Z → Y with

〈T (y) | z 〉 = 〈 y |S(z) 〉, y ∈ Y, z ∈ Z.

It is easy to see that such an S is unique; it is denoted by T ∗. It follows imme-diately from the centered equation that (i) T is a right A-module map, (ii) T isbounded (use the closed graph theorem), (iii) T ∗ is a bounded right A-modulemap, and (iv) T ∗∗ = T . It is also easy to verify relations such as (T1T2)∗ = T ∗

2 T ∗1 .

Also, BA(Y, Z) is a norm closed subspace of BA(Y, Z). Indeed, if (Tn)n is Cauchyin BA(Y, Z), then (T ∗

n)n is Cauchy too. The limits of these two sequences evi-dently are the adjoints of each other. Writing BA(Y ) = BA(Y, Y ), it is easy toverify that BA(Y ) is a C∗-algebra with respect to the usual norm of a bounded op-erator. For example, the more difficult direction of the C∗-identity ‖T ∗T ‖ = ‖T ‖2

follows immediately from the fact that if y ∈ Ball(Y ), then

‖Ty‖2 = ‖〈T (y)|T (y)〉‖ = ‖〈T ∗T (y)|y〉‖ ≤ ‖T ∗T ‖.

We define KA(Y, Z) (or simply K(Y, Z)) to be the closure, in B(Y, Z), of thelinear span of the ‘rank-one’ operators |z〉〈y|, for y ∈ Y, z ∈ Z. We are using ‘bra-ket’ notation here; thus |z〉〈y| is the operator which takes an x ∈ Y to z〈y|x〉.It is easy to verify familiar relations such as (|z〉〈y|)∗ = |y〉〈z|, and that theproduct of ‘rank one’ operators is ‘rank one’: (|y〉〈z|)(|y′〉〈z′|) = |y(〈z|y′〉)〉〈z′|.Also |ya∗〉〈z| = |y〉〈za| if a ∈ A. From these relations, it is easy to see that


KA(Y ) = KA(Y, Y ) is a C∗-subalgebra of BA(Y ). Indeed KA(Y ) is a closed idealof BA(Y ), as may be seen from the simple relations T (|y〉〈z|) = |T (y)〉〈z| and(|y〉〈z|)S = |y〉〈S∗z| for T ∈ BA(Y ) and S ∈ BA(Y ). Similarly,

KA(Y, Z) is canonically a BA(Z)-BA(Y )-bimodule. (8.1)

We say that a map u : Y → Z between right C∗-modules over A is unitary if uis a surjective A-module map such that 〈uy|uy〉 = 〈y|y〉, for all y ∈ Y . By thepolarization identity (1.1) this is equivalent to: 〈uy|uz〉 = 〈y|z〉 for all y, z ∈ Y .It is clear that any unitary map is adjointable. If there exists such a unitary thenwe say Y ∼= Z unitarily. If u is unitary then u∗ = u−1. If Y = Z then u is unitaryif and only if it is a unitary in the C∗-algebra BA(Y ).

Corollary 8.1.8 An A-module map u : Y → Z between right C∗-modules overA is unitary if and only if u is isometric and surjective.

Proof The one direction is obvious. The other direction follows from 8.1.5applied to both u and u−1.

8.1.9 (The direct sum) If Yi : i ∈ I is a collection of right C∗-modules overA, then we define the direct sum C∗-module by

⊕ci Yi =

(yi) ∈

∏i∈I

Yi :∑

i

〈yi|yi〉 converges in norm in A

.

With the canonical inner product 〈(yi)|(zi)〉 =∑

i〈yi|zi〉 (which converges bythe polarization identity (1.1)), and obvious A-module action, ⊕c

i Yi is a rightC∗-module over A. We omit the routine proof. In passing, we remark that thereare alternative ways to define the direct sum. See, for example, 8.2.14.

For a cardinal I, CI(Y ) denotes the C∗-module direct sum of I copies of Y . Wewill see in 8.2.3 (4) that there is no conflict with earlier operator space notation:viewed as an operator space, CI(Y ) means exactly what it meant before.

It is important, and easily seen, that the canonical inclusion and projectionmaps between ⊕c

i Yi and its summands Yi are adjointable.We say that a right C∗-module Z is the internal orthogonal direct sum of

closed submodules Y and W , if Z = Y + W and Y ⊥ W (i.e. 〈y|w〉 = 0 for ally ∈ Y, w ∈ W ). In this case, Z ∼= Y ⊕c W unitarily, quite clearly. We say thatY is orthogonally complemented in Z if there exists such a W . It is clear that Yis orthogonally complemented in Z if and only if Y is the range of a projection(i.e. a selfadjoint idempotent) P in the C∗-algebra BA(Z).

8.1.10 (A characterization of module maps) Result 8.1.5 may be improved.In fact, a linear map u : Y → Z between C∗-modules over B is a contractiveB-module map, if and only if 〈u(y)|u(y)〉 ≤ 〈y|y〉, for all y ∈ Y . This is alsoequivalent to ‖(u(y), z)‖ ≤ ‖(y, z)‖, for all y ∈ Y and z ∈ Z, where the normshere are those of Z ⊕c Z and Y ⊕c Z respectively. We omit the proofs of theseassertions since we shall not need them (see [302,56]). We remark that the lattercondition is the analogue for C∗-modules of condition 4.5.2 (ii).


Proposition 8.1.11 Suppose that Y is a right C∗-module over A. Then:(1) Y ∼= KA(A, Y ) isometrically.(2) Y ∼= KA(Y, A) isometrically.(3) KA(Cn(Y )) ∼= Mn(KA(Y )) and BA(Cn(Y )) ∼= Mn(BA(Y )) as C∗-algebras.

Proof (1) That the canonical map L : Y → BA(A, Y ) is isometric, with rangecontained inside KA(A, Y ), is a simple modification of the proof of 3.5.4 (1).Since that range is a closed vector subspace which contains every ‘rank one’operator, it must equal KA(A, Y ).

(2) Define Φ: Y → BA(Y, A) by Φ(y)(z) = 〈y|z〉, for y, z ∈ Y . Clearly Φ isisometric and linear. We show that Φ maps onto KA(Y, A). Indeed, by A.6.2 wemay write any y ∈ Y as y′a, for y′ ∈ Y, a ∈ A. Then Φ(y) = a∗〈y′|·〉, the generic‘rank one’ operator in KA(Y, A). We conclude as in (1).

(3) There is a canonical homomorphism θ : Mn(KA(Y )) → BA(Cn(Y )). It iseasy to check that θ(a)∗ = θ(a∗), so that θ is a ∗-homomorphism into BA(Cn(Y )).Clearly θ is one-to-one. We leave it as an exercise that Ran(θ) = KA(Cn(Y ))(using the (adjointable) canonical inclusion and projection maps between Cn(Y )and its summands, and (8.1)). The other assertion is similar.

Lemma 8.1.12 Let Y be a right C∗-module over A, and suppose that T is alinear map on Y . Then T ∈ BA(Y )sa (resp. T ∈ BA(Y )+) if and only if 〈Ty|y〉is selfadjoint (resp. 〈Ty|y〉 ≥ 0) for all y ∈ Y .

Proof We sketch a proof of the difficult implication. If 〈Ty|y〉 is selfadjoint forall y ∈ Y , then 〈Ty|y〉 = (〈Ty|y〉)∗ = 〈y|Ty〉. By the polarization identity (1.1),T is adjointable on Y , with T ∗ = T . If, further, 〈Ty|y〉 ≥ 0, for y ∈ Y , then T isselfadjoint, by the above. To see that T ≥ 0, suppose that A ⊂ B(H), and applythe following simple general fact about C∗-algebras to the collection of positivefunctionals 〈〈y| · y〉ζ, ζ〉 on BA(Y ), for y ∈ Ball(Y ), ζ ∈ Ball(H). Namely, let Bbe a C∗-algebra, and let S be a set of positive contractive functionals on B with

‖b‖ = supϕ(b) : ϕ ∈ S, for all b ∈ B+.

Then if b ∈ Bsa and if ϕ(b) ≥ 0 for all ϕ ∈ S, then b ∈ B+. To prove this generalfact, suppose that the endpoints of the spectrum of b are α and β, with α ≤ β.The centered equation applied to β1 − b, gives

β − α = ‖β1 − b‖ = supβϕ(1) − ϕ(b) : ϕ ∈ S ≤ β.

From this it is clear that α ≥ 0. Hence b ≥ 0.

Corollary 8.1.13 If a = [aij ] ∈ Mn(A), then ‖a‖n equals

sup‖ab‖Cn(A) : b ∈ Ball(Cn(A)) = sup

∥∥∥∑i,j

c∗i aijbj

∥∥∥ : b, c ∈ Ball(Cn(A))

.

Moreover, the following are equivalent:


(i) a is positive in Mn(A),(ii)

∑i,j b∗i aijbj ≥ 0 in A, for all b1, . . . , bn ∈ A,

(iii) a is a finite sum of matrices of the form [a∗i aj], with a1, . . . , an ∈ A.

Proof Define θ : Mn(A) → BA(Cn(A)) by the canonical left action of Mn(A) onCn(A). Clearly θ is a one-to-one ∗-homomorphism into the C∗-algebra BA(Cn(A))of 8.1.7. Then the first statement is clear from A.5.8. The equivalence of (i) and(ii) follows from 8.1.12. The equivalence with (iii) is a simple exercise.

8.1.14 (C∗-modules are equivalence bimodules) Any right C∗-module Y overB is canonically also a full left C∗-module over KB(Y ), using |·〉〈·| for the innerproduct. When checking this, the only nontrivial point is that |y〉〈y| ≥ 0, whichfollows quite easily from 8.1.12. It is now clear that if Y is a right C∗-moduleover B, then Y is also an equivalence KB(Y )-I-bimodule. Here I is as in 8.1.4.

Note also that the norm on Y induced by this new inner product correspond-ing to this left C∗-module action, is the same as the old norm. In fact it is evidentthat ‖|y〉〈y|‖ ≤ ‖y‖2. The reverse inequality follows from the fact that

‖|y〉〈y‖2 ≥ ‖|y〉〈y|(z)‖2 = ‖〈z|y〉〈y|y〉〈y|z〉‖, z ∈ Ball(Y ).

Setting z = y/‖y‖ yields the desired inequality.The proof of the following result shows, conversely, that given an equivalence

A-B-bimodule Y , then A ∼= KB(Y ) ∗-isomorphically, and via this isomorphismthe action of A on Y corresponds exactly to the canonical KB(Y ) action, and theA-valued inner product corresponds to the KB(Y )-valued inner product. Thuswe see that full right C∗-modules are essentially the same things as strong Moritaequivalence bimodules. This gives another ‘picture’ of C∗-modules, as the strongMorita equivalence bimodules.

For the next result, we write λ for the canonical map from A into B(Y ), fora left Banach A-module Y .

Lemma 8.1.15 If Y is an equivalence A-B-bimodule, then λ(A) = KB(Y ).Indeed, A ∼= KB(Y ) ∗-isomorphically, via the map λ above. Also, the two normsdefined on Y via each of the two inner products coincide.

Proof Writing [·|·] for the A-valued inner product, we have

x〈ay|z〉 = [x|ay]z = [x|y]a∗z = x〈y|a∗z〉, x, y, z ∈ Y, a ∈ A.

Thus by 8.1.4 (1), we have 〈ay|z〉 = 〈y|a∗z〉. Hence λ(A) ⊂ BB(Y ), and λ isa ∗-homomorphism into BB(Y ). Clearly λ([y|z]) = |y〉〈z|, so that it follows bycontinuity and density that the range of λ is KB(Y ). By the left module versionof 8.1.4 (1), λ is one-to-one. Thus λ is an isometry, by A.5.8.

For the last assertion, since λ is isometric, and by the second paragraph of8.1.14, we have ‖[y|y]‖ = ‖|y〉〈y|‖ = ‖〈y|y〉‖.

Proposition 8.1.16


(1) If Y is an equivalence A-B-bimodule then the canonical map from the mul-tiplier algebra M(A) to B(Y ) (see 3.1.11) is isometric, and indeed is a∗-isomorphism onto BB(Y ). This ∗-isomorphism extends the canonical iso-morphism between A and KB(Y ) (see 8.1.15).

(2) Further, the ∗-isomorphism in (1) extends to an isometric isomorphism be-tween LM(A) and BB(Y ).

(3) If Y is a right C∗-module over B, then BB(Y ) ∼= LM(KB(Y )) isometrically(as Banach algebras), and BB(Y ) ∼= M(KB(Y )) ∗-isomorphically.

(4) BB(Y ) is the span of the Hermitian elements (see A.4.2) of BB(Y ).

Proof Suppose that Y is an equivalence A-B-bimodule, and that (et)t is apositive cai for A. By 3.1.11, Y is a Banach LM(A)-module, and we have acorresponding contractive homomorphism θ : LM(A) → BB(Y ) given by

θ(η)(y) = limt

η(et)y = η(a)y′, y, y′ ∈ Y, a ∈ A, such that y = ay′. (8.2)

It follows from this, and 8.1.4 (1), that θ is a one-to-one homomorphism extendingλ. Next fix η ∈ M(A). By (8.2) and 8.1.15,

〈θ(η)y|z〉 = limt

〈η(et)y|z〉 = limt

〈y|η(et)∗z〉, y, z ∈ Y.

Writing z = az′ for a ∈ A, z′ ∈ Y , the latter quantity equals

limt

〈y|etη∗(a)z′〉 = 〈y|η∗(a)z′〉 = 〈y|θ(η∗)z〉,

using (8.2) again. Thus θ restricts to a ∗-homomorphism from M(A) into BB(Y ).Suppose that Y is simply a right C∗-module. We may assume, by 8.1.4 (3),

that Y is full, and we apply the above with A = KB(Y ). By (8.1), we may definea contractive homomorphism ρ : BB(Y ) → LM(KB(Y )) by ρ(S)(T ) = ST , forS ∈ BB(Y ), T ∈ KB(Y ). Here we are viewing LM(A) as the right A-modulemaps on A. It is easy to see that θ ρ is the identity map. Hence θ is surjective,and ρ is isometric. Also, by 2.6.8, it is easy to check that ρ takes BB(Y ) intoM(KB(Y )). Thus we have proved (3).

In the situation of (2), by (3) and 8.1.15 we have isometric isomorphisms

LM(A) −→ LM(KB(Y )) −→ BB(Y ).

It is easy to check that the composition of these maps is the map θ in (8.2). Thisproves (2), and (1) is similar. Assertion (4) follows from (3) and 2.6.9.

8.1.17 (The linking C∗-algebra) Suppose that Y is a right C∗-module over B,and write A = KB(Y ). We define the linking C∗-algebra L(Y ) to be the set of2 × 2 matrices:

L(Y ) =[

A YY B

].

We turn this set into an algebra, using the usual product of 2× 2 matrices, andusing the inner products and module actions. For example, the product yz, of a


term y from the 1-2-corner, and a term z from the 2-1-corner, is taken to mean|y〉〈z| ∈ A = KB(Y ). Or the product za, for a ∈ A, is given by a∗z. We definethe involution of one of these 2 × 2 matrices in the obvious way.

Define a map π : L(Y ) → B(Y ⊕c B) by the obvious action (i.e. viewing anelement of Y ⊕c B as a column with two entries, and formally multiplying a 2×2matrix and such a column. It is easy to check that π(m) ∈ BB(Y ⊕c B) for eachmatrix m ∈ L(Y ), and moreover that π is a ∗-homomorphism into BB(Y ⊕c B).Also, one can quickly check that π is one-to-one, and that π is an isometry whenrestricted to each of the four corners of L(Y ). Hence the range of π is closed. Wegive L(Y ) a norm by pulling back the norm from BB(Y ⊕c B) via π, thus L(Y ) isa C∗-algebra ∗-isomorphic to the range of π. Indeed we may regard BB(Y ⊕c B)as a 2 × 2 matrix C∗-algebra consisting of matrices t = [tij ] whose four entriesare adjointable maps. To see this, note that tij are defined in terms of t andthe (adjointable) projection and inclusion maps between Y ⊕c B and its twosummands. With this in mind, it is clear from 8.1.11 and (8.1), that KB(Y ⊕c B)is exactly the range of π. Hence the linking C∗-algebra may simply be thoughtof as KB(Y ⊕c B).

By the last fact, and 8.1.16 (3), we see that the multiplier algebra M(L(Y ))is BB(Y ⊕c B). We define the unitized linking C∗-algebra L1(Y ) of Y to be thelinear span within M(L(Y )) of KB(Y ⊕c B) and the two diagonal idempotentmatrices p = 1⊕0 and q = 0⊕1. The last two 1’s may be viewed as the identitiesof the unitizations of A and B respectively (where we take the unitization of aunital algebra to be itself). Then L1(Y ) is a unital C∗-algebra with identity1 = p + q. Clearly Y is the 1-2-corner of both L1(Y ) and L(Y ). In particular,

Y ∼= pL(Y )(1 − p). (8.3)

If we take a general C∗-algebra A, and if Y is an equivalence A-B-bimodule,then we may form L(Y ) as above, but using A instead of KB(Y ). Of course by8.1.15 this is essentially the same thing; that is, the resulting linking algebraswill be ∗-isomorphic. In this case we say that L(Y ) is the Morita linking algebraof Y . We will also use this terminology even when A is not specified, takingA = KB(Y ), however we insist that Y be full over B in this case.

Corollary 8.1.18 If Y is a right C∗-module over B, then L(Y ) is stronglyMorita equivalent to B (via the equivalence bimodule Y ⊕c B).

Proof Clearly Y ⊕c B is a full B-module. Since KB(Y ⊕c B) ∼= L(Y ), we haveby 8.1.14 that L(Y ) is strongly Morita equivalent to B, via Y ⊕c B.

8.1.19 (C∗-modules and corners) One great advantage of the linking C∗-algebra of a C∗-module Y , is that the inner products and module actions havebeen replaced by concrete multiplication of elements in a C∗-algebra. To see this,we employ the completely isometry in (8.3). This is simply the ‘corner map’ c,taking y ∈ Y to the matrix in L(Y ) with y in the 1-2-corner and zeroes elsewhere.If we identify B with the 2-2-corner in a similar way, then 〈y|z〉 is simply the


product c(y)∗c(z) in the C∗-algebra L(Y ). Indeed it is convenient, and usuallyleads to no difficulties, to suppress the ‘c’ map and simply write y∗z for the lastexpression above. Similarly, if we write xy∗z the reader will have no difficulty inseeing that what is meant is the element x〈y|z〉, or equivalently |x〉〈y|(z). Similarconventions apply to longer such products.

Thus, by (8.3), any right C∗-module Y is a corner in a C∗-algebra, in thesense of 2.6.14. Conversely, any corner pAq in a C∗-algebra A, is clearly a rightC∗-module over qAq. This gives another ‘picture’ of C∗-modules, as the cornersof C∗-algebras. In the language of 4.4.1, any C∗-module may be viewed as aTRO, namely, as a subtriple of L(Y ). This, together with the second paragraphof 8.1.2, gives another ‘picture’ of C∗-modules, as the TROs. We will tighten upthis observation further in 8.2.8.

If Y is an equivalence A-B-bimodule, then one can also view LM(A), and itsaction on Y , in terms of the linking algebra. Indeed, LM(A) ⊂ A∗∗ ⊂ L(Y )∗∗.The composition of these canonical inclusions is easily seen (since Y = AY B) tohave range within LM(L(Y )). That is, we may regard as subalgebras:

LM(A) ⊂ LM(L(Y )) ⊂ L(Y )∗∗.

Similar assertions hold for RM(B).We turn to some more corollaries of 8.1.16:

Corollary 8.1.20 If A and B are strongly Morita equivalent, then the centersof their multiplier algebras are ∗-isomorphic, via a ∗-isomorphism θ satisfying

θ(η)y = yη, for all y ∈ Y, η ∈ Z(M(B)).

Here Y is the associated equivalence A-B-bimodule.

Proof By 3.1.11, Y is a right Z(M(B))-module, with action yη = limt yη(et),for y ∈ Y, η ∈ Z(M(B)). Here (et)t is a cai for B. As in A.6.1, this defines acontractive unital homomorphism π : Z(M(B)) → B(Y ). Clearly π maps intoBB(Y ). By 8.1.16 (4), together with A.4.2, π maps Z(M(B))sa, and hence alsoZ(M(B)), into BB(Y ). Thus by 8.1.16 (1), there exists a unique ν ∈ M(A) suchthat νy = yη for all y ∈ Y . Clearly this implies that νay = ayν = aνy, if a ∈ A.This, and 8.1.4 (1), implies that ν ∈ Z(M(A)). Moreover, if we define θ(η) = νthen θ is a homomorphism from Z(M(B)) to Z(M(A)). Now it is easy to see,by symmetry, that θ must be an isomorphism. By the last part of A.5.4, forexample, θ is a ∗-isomorphism.

Corollary 8.1.21

(1) Let P be a contractive idempotent A-module map on a right C∗-moduleY over A. Then P is adjointable. Indeed P is an orthogonal projection inthe C∗-algebra B(Y ), and the range of P is an orthogonally complementedsubmodule of Y .

(2) Suppose that Y and Z are right C∗-modules over A, and that α : Y → Z andβ : Z → Y are contractive module maps with βα = IY . Then these maps


are adjointable, with β = α∗. Moreover, α is a unitary module map onto anorthogonally complemented submodule of Z.

Proof (1) By 8.1.16 (3), P corresponds to a contractive idempotent in theoperator algebra LM(KA(Y )). By 2.1.3, the last idempotent is Hermitian. ThusP is Hermitian, and is adjointable by 8.1.16 (4). The rest follows from 8.1.9.

(2) Note that α is an isometry onto the closed submodule W = Ran(αβ) ofZ. Since αβ satisfies the conditions of (1), it is adjointable. By 8.1.9 we see thatW is orthogonally complemented. By 8.1.8 we have that α is unitary, and henceadjointable as a map into W . It is then easy to see that α is adjointable as amap into Z. Indeed,

〈βz|y〉 = 〈(αβ)z|αy〉 = 〈z|(αβ)αy〉 = 〈z|αy〉,

for y ∈ Y, z ∈ Z.

From 8.1.21, one may deduce a universal property of the direct sum:

Proposition 8.1.22 Suppose that Yi : i ∈ I is a collection of right C∗-modules over A, and that Y is a right C∗-module over A, such that there existcontractive module maps εi : Yi → Y and Pi : Y → Yi such that Pi εj = δi,jIYi ,and such that

∑i εi Pi converges strongly on Y . Then there exists an orthogo-

nally complemented submodule W of Y such that Y ∼= (⊕ci Yi)⊕c W unitarily. If∑

i εi Pi converges strongly to IY , then W = 0.

Proof By 8.1.21, each εi and Pi are adjointable, with Pi = ε∗i , and Qi = εi Pi

is an adjointable projection on Y . Moreover, the Qi are mutually orthogonal,and of course positive, elements of the C∗-algebra B(Y ). Set R(y) =

∑i Qi(y),

for y ∈ Y . Then R is a module map on Y , and

0 ≤ 〈R(y)|y〉 = limJ

⟨∑j∈J

Qjy|y⟩

≤ 〈y|y〉, y ∈ Y,

the limit over finite subsets J of I. Thus R is contractive and positive (by 8.1.12).Clearly RQi = Qi, which implies that R is idempotent. Hence, by 8.1.21, R andIY −R are adjointable projections on Y . Clearly I −R is orthogonal to each Qi.If W = Ran(I − R) and W⊥ = Ran(R), then Y ∼= W⊥ ⊕c W unitarily. Defineu : W⊥ → ⊕c

i Yi by u(y) = (Qiy). It is easy to check that 〈uy|uy〉 = 〈y|y〉, sothat u is an isometry. Since Ran(u) is norm dense, u is a unitary.

8.1.23 (Finite rank approximation) If Y is a right C∗-module over B, then weclaim that the C∗-algebra KB(Y ) has a cai (et)t of the form

et =n(t)∑k=1

|xtk〉〈xt

k|, (8.4)

for elements xtk in Y . To obtain this, first pick a cai (ft)t from the dense ideal of

‘finite rank’ operators in KB(Y ). Let et = f∗t ft, then (et)t is also a cai for KB(Y ).


If ft =∑n

i=1 |yi〉〈zi|, then by relations in the second paragraph of 8.1.7 we havef∗

t ft =∑n

i,j=1 |ziaij〉〈zj |, where [aij ] = [〈yi|yj〉]. The latter is a positive matrixP with entries in B (this may be seen by the criterion 8.1.13 (ii), for example).Factoring P as the square of its square root, and regrouping, we obtain (8.4).

Since KB(Y ) acts nondegenerately on Y (by 8.1.3 and 8.1.14), for any y ∈ Y

we have that∑n(t)

k=1 xtk〈xt

k|y〉 → y. From this we deduce the following:

Corollary 8.1.24 Let Y be a right C∗-module over B. Then there exists a net(n(t))t of positive integers, and contractive B-module maps αt : Y → Cn(t)(B)and βt : Cn(t)(B) → Y , such that βt(αt(y)) → y for every y ∈ Y . Indeed this canbe done with α∗

t = βt.

Proof We use the notation above. For y ∈ Y , define αt(y) ∈ Cn(t)(B) to havekth entry 〈xt

k|y〉. Also, define βt(b) =∑n(t)

k=1 xtkbk. Here b has kth entry bk ∈ B.

We have that βt(αt(y)) =∑n(t)

k=1 xtk〈xt

k|y〉 = ety → y, as we saw immediatelyabove the corollary. It is easily checked that αt, βt are adjointable, with α∗

t = βt,so that they have the same norm. We have:

‖αt(y)‖2 =∥∥∥∑

k

〈y|xtk〉〈xt

k|y〉∥∥∥ =

∥∥∥⟨y∣∣∣∑

k

xtk〈xt

k|y〉⟩∥∥∥ = ‖〈y|ety〉‖ ≤ ‖y‖2 .

Thus αt is contractive, and hence so is βt.

8.1.25 (Asymptotic factorization) Thus any right C∗-module Y over B ‘factorsasymptotically’ through spaces of columns over B. In passing, we remark that asimple modification of the last proof, using 8.1.14 and the left-handed version of8.1.23, shows that if Y is a full C∗-module over B, then B factors asymptotically(in the sense of the last result) through spaces of the form Cn(t)(Y ). Similarly,one can show that KB(Y ) factors asymptotically (via completely contractivelinear maps) through spaces of the form Mn(t)(B). Since we shall not use these,we omit the details (see [46, p. 391]).

The converse of 8.1.24 is also true: if Y is a right Banach B-module for whichthere exist nets of contractive B-module maps as in the lemma, then Y is aright C∗-module over B. This gives a characterization of C∗-modules among theBanach B-modules. Indeed we have:

Theorem 8.1.26 Suppose that B is a C∗-algebra and that Y is a right BanachB-module. Suppose further that there is a net (Yt)t of right C∗-modules overB, and contractive B-module maps αt : Y → Yt and βt : Yt → Y , such thatβt(αt(y)) → y for every y ∈ Y . Then Y is a right C∗-module over B, and thenorm on Y coincides with the norm induced by the inner product (see 8.1.1 (5)).The inner product on Y is given by the formula

〈y|z〉 = limt

〈αt(y)|αt(z)〉, y, z ∈ Y.

The limit here is in the norm topology of B.

308 C∗-modules as operator spaces.

A page long proof of this result, which uses only the triangle and Cauchy–Schwarz inequalities, and 8.1.5, may be found in [65, p. 41].

For finitely generated or countably generated C∗-modules one may improveon the last result considerably. By an algebraically finitely generated B-module,we mean a module Y for which there exists y1, . . . , yn ∈ Y such that the mapf : Cn(B) → Y given by f((ak)) =

∑k ykak, is surjective.

Theorem 8.1.27(1) If Y is an algebraically finitely generated right C∗-module over B, then Y is

unitarily isomorphic to an orthogonally complemented submodule of Cn(B),for some n ∈ N.

(2) A right C∗-module Y over B is algebraically finitely generated if and only ifKB(Y ) is unital. In this case, KB(Y, Z) contains all B-module maps fromY to Z, for any C∗-module Z over B.

Proof (1) Let f be the map above 8.1.27, which is easily seen to be adjointable.By [423] 15.3.8 (or rather the obvious variant of that result to maps between twopossibly different C∗-modules over B), there is a polar decomposition f = u|f |;W = Ran(|f |) is a closed orthogonally complemented submodule of Cn(B); andY is unitarily isomorphic to W via the partial isometry u.

(2) For the first part, it is easy to see by A.6.2 that Y is algebraically finitelygenerated over B if and only if it is also finitely generated over B1. Thus we mayassume that B is unital. If KB(Y ) is unital, then since Y is a nondegenerateKB(Y )-module (by 8.1.3 and 8.1.14), this identity is IY . Given ε > 0, we canfind et =

∑k |yk〉〈yk| as in 8.1.23, with ‖et − IY ‖ < ε. Hence et is invertible

with inverse S say, so that IY =∑

k |S(yk)〉〈yk|, from which it is immediate that(S(yk))k generates Y .

Conversely, if Y is algebraically finitely generated, let u : Cn(B) → Y be thesurjective partial isometry in the proof of (1). We have uu∗ = IY . Put yk = u(ek),where ek has 1B in the kth entry, and is zero elsewhere. Then∑

k

|yk〉〈yk| = u(∑

k

|ek〉〈ek|)

u∗ = uu∗ = IY .

Hence IY ∈ KB(Y ).Finally, any B-module map v : Y → Z into a Banach B-module Z satisfies

v(y) = v( n∑

k=1

yk〈yk|y〉)

=n∑

k=1

v(yk)〈yk|y〉, y ∈ Y.

Thus v is bounded, and the stated assertion is clear.

8.2 C∗-MODULES AS OPERATOR SPACES.

8.2.1 (C∗-modules are operator spaces) If Y is a right C∗-module over B, andif n ∈ N, then Mn(Y ) is a right C∗-module over Mn(B), with inner product


〈[yij ]|[zij ]〉 =[ n∑

k=1

〈yki|zkj〉], [yij ], [zij ] ∈ Mn(Y ). (8.5)

One way to see this is to identify Y with the 1-2-corner of the linking C∗-algebraL(Y ), and B with the 2-2-corner, as in 8.1.19, so that 〈y|z〉 = y∗z for y, z ∈ Y . Ifwe do this then, first, Mn(Y ) may be identified with a corner of the C∗-algebraMn(L(Y )), and the canonical inner product inherited from the latter C∗-algebrais y∗z = [

∑nk=1 y∗

kizkj ], for y = [yij ], z = [zij ] ∈ Mn(Y ). This gives (8.5). Second,Y inherits a canonical operator space from the C∗-algebra L(Y ). We call thisthe canonical operator space structure on Y . It is given explicitly by

‖[yij ]‖n =∥∥∥[ n∑

k=1

〈yki|ykj〉]∥∥∥ 1

2, [yij ] ∈ Mn(Y ), (8.6)

as may be seen by using the C∗-identity in Mn(L(Y )). When we consider aC∗-module as an operator space, it will always be with respect to this structure.

The formula (8.6) is also valid for nonsquare matrices. For instance, for acolumn y = [y1 · · · yn]t ∈ Cn(Y ) = Mn,1(Y ), we have ‖y‖ = ‖∑n

k=1〈yk|yk〉‖12 .

Viewing Y as a subspace of the linking C∗-algebra, we also have

‖y‖ = ‖yy∗‖ 12 = ‖[yiy

∗j ]‖ 1

2 = ‖[|yi〉〈yj |]‖12 .

Note that this shows that Cn(Y ) is isometric to the C∗-module direct sum of ncopies of Y . Similarly, if [x1 · · ·xn] ∈ Rn(Y ), then

‖[x1 · · ·xn]‖Rn(Y ) = ‖[〈xi|xj〉]‖12 =

∥∥∥ n∑k=1

|xk〉〈xk|∥∥∥ 1

2. (8.7)

Proposition 8.2.2 For C∗-modules Y and Z over B, every bounded B-modulemap u : Y → Z is completely bounded, with ‖u‖ = ‖u‖cb. If u is unitary then itis a complete isometry.

Proof This may be seen in many ways. For example, assume that ‖u‖ ≤ 1, andthat x1, . . . , xn ∈ Y, b1, . . . , bn ∈ B. Set z =

∑i xibi. Then∑

i,j

b∗i (〈xi|xj〉 − 〈uxi|uxj〉)bj = 〈z|z〉 − 〈uz|uz〉 ≥ 0,

using 8.1.5. By 8.1.13 (ii) it follows that [〈xi|xj〉]− [〈uxi|uxj〉] ≥ 0. Then the firstresult follows from an easier variant of the proof of the implication ‘(v) implies(ii)’ of Theorem 4.5.2. We leave the second as an exercise.

Henceforth, we give BB(Y, Z) = CBB(Y, Z) the operator space structurefrom 3.5.1. We assign to BB(Y, Z) and KB(Y, Z) the operator space structureswhich they inherit as subspaces of CBB(Y, Z). We shall see in 8.2.3 (7) belowthat if Y = Z, the latter operator space structures coincide with their canonicalC∗-algebra operator space structure.


8.2.3 (Operator space variants of C∗-module facts) Most of the results inSection 8.1 have operator space variants. We list the key points below; some ofthese will be used often in the rest of the chapter.

(1) Any right C∗-module Y over B is a right operator B-module. Indeedthis follows immediately from 3.1.2 (5) and (8.3). By 3.1.11, Y is also a rightoperator module over M(B), or over RM(B). By symmetry, Y is also a leftoperator module over KB(Y ), and (using also 8.1.16 (3)) over BB(Y ). Thus Y isan operator BB(Y )-B-bimodule. Similarly, if Y is an equivalence A-B-bimodule,then Y is an operator A-B-bimodule, and an operator M(A)-M(B)-bimodule.

(2) Let Y be an equivalence A-B-bimodule. Viewing Y as the 1-2-corner ofthe linking C∗-algebra, and the ‘adjoint module’ Y as the 2-1-corner, one sees thecanonical operator space structure on Y , as exactly the adjoint operator spacestructure Y from 1.2.25. Note that Y is an operator B-A-bimodule (see 3.1.16).

(3) If ⊕ci Yi is a direct sum of right C∗-modules over B, equipped with its

canonical operator space structure, then Mn(⊕ci Yi) ∼= ⊕c

i Mn(Yi) unitarily asMn(B)-modules. We leave this to the reader.

(4) For any cardinal I, the right C∗-module direct sum of I copies of Y ,is completely isometrically isomorphic to the operator space CI(Y ) defined in1.2.26. To see this, recall the canonical complete isometry c : Y → L(Y ) from8.1.19. The amplification cI,1 of c is a completely isometric embedding fromCI(Y ) into CI(L(Y )) (see 1.2.26). It is clear that

‖cI,1((yi))‖2 =∥∥∥∑

i

c(yi)∗c(yi)∥∥∥ =

∥∥∥∑i

〈yi|yi〉∥∥∥, (yi) ∈ CI(Y ).

This proves the isometric case of our result. The complete isometry may bededuced from the isometric case together with (3) above.

(5) There is an ‘operator space version’ of 8.1.26, which we may state asfollows. Let Y be an operator space and a right B-module, and suppose thatthere exist maps αt and βt satisfying all the conditions in 8.1.26. If αt and βt

are completely contractive, then in addition to the conclusions of 8.1.26, thegiven matrix norms on Mn(Y ) coincide with the norm (8.6) induced by theinner product. To prove this, notice that the amplifications (αt)n and (βt)n arecontractive Mn(B)-module maps. Applying 8.1.26 to Mn(Y ), and to these maps,yields the desired assertion.

(6) We consider the operator space version of 8.1.16. By (1) above, any rightC∗-module Y over B, is a left operator A-module, where A = KB(Y ). By 3.1.11it is also a left operator LM(A)-module. Thus by 3.1.5 (1), the isomorphismθ : LM(A) → BB(Y ) given by (8.2), is a completely contractive homomorphisminto CBB(Y ). Also, the map ρ : CBB(Y ) → LM(A) in the proof of 8.1.16, isclearly completely contractive (for example, because CB(Y ) is a matrix normedalgebra, as we observed in 2.3.9). Thus LM(A) ∼= CBB(Y ) completely isometri-cally isomorphically. Variants of several of the facts in 2.6.6 hold for C∗-modules.For example, there is a canonical isometric isomorphism from Mn(CBB(Y )) ontoBB(Cn(Y )). Indeed Mn(CBB(Y )) ∼= Mn(LM(KB(Y ))) by the above, and


Mn(LM(KB(Y ))) ∼= LM(Mn(KB(Y ))) ∼= LM(KB(Cn(Y ))) ∼= BB(Cn(Y )),

using 2.6.6 (3), 8.1.11 (3), and 8.1.16 (3).(7) The canonical operator space structure on the C∗-algebra BB(Y ) coin-

cides with the inherited operator space structure from CBB(Y ). Indeed, by (6),we have Mn(CBB(Y )) ∼= BB(Cn(Y )). On the other hand, by 8.1.11 (3), we haveMn(BB(Y )) ∼= BB(Cn(Y )) ⊂ BB(Cn(Y )).

By the discussion in (6) and (7) above, together with 8.1.16 (4), we have:

Corollary 8.2.4 If Y is a right C∗-module over B, then CBB(Y ) is a unitaloperator algebra completely isometrically isomorphic to LM(KB(Y )). Moreover,∆(CBB(Y )) = BB(Y ) (see 2.1.2 for this notation).

8.2.5 (Countably generated modules) We will use operator space column androw notation (see 1.2.26) to lead us through the important ‘stabilization the-orems’. We say that a Banach B-module X is countably generated if there isa sequence (xn) in X such that Spanbxn : b ∈ B, n ∈ N is dense in X . Bya (countable) right quasibasis of a right C∗-module Y over B, we mean a row[yk] ∈ Rw(Y ) (see 1.2.26 for this notation), such that

y =∞∑

k=1

yk〈yk|y〉, y ∈ Y, (8.8)

the sum converging in norm. Clearly if there exists a right quasibasis, then Yis countably generated over B. Indeed, clearly if Y has a right quasibasis, thenKB(Y ) has a countable approximate identity. By 8.1.15, this is equivalent to Ahaving a countable approximate identity, if Y is an equivalence A-B-bimodule.It is also easy to see that if KB(Y ) has a countable approximate identity, thenY is countably generated over B. We shall not use this fact, but conversely, if Yis a countably generated right C∗-module, then KB(Y ) has a ‘strictly positiveelement’ (see [65, Proof of 7.13]), and hence a countable approximate identity[320, Proposition 3.10.5]; following the proof of [75] Lemmas 2.1–2.3, one seesthat Y has a right quasibasis (see e.g. [46, Theorem 8.2]).

We claim that if Y has a right quasibasis, then Y is unitarily isomorphic to anorthogonally complemented submodule of C(B). Here of course C(B) = CI(B)(see 8.2.3 (4)) when I = N. This is a variant of Theorem 8.1.27 (1). To prove this,note that by (8.8) we have 〈y|y〉 =

∑∞k=1〈y|yk〉〈yk|y〉, for all y ∈ Y . This permits

us to define an isometric B-module map α : Y → C(B), by α(y) = (〈yk|y〉)k.By a simple calculation analoguous to that in 1.2.27, there is a well definedcontractive B-module map β : C(B) → Y given by β((bk)) =

∑k ykbk. Clearly

β α = IY . Our claim then follows from 8.1.21 (2).A left quasibasis for Y is a column (zk) ∈ Cw(Y ) with y =

∑∞k=1 y 〈zk|zk〉

for all y ∈ Y . If Y is full, then the latter condition is equivalent to the samecondition, but for y ∈ B, since in that case B = Y Y and Y = Y B. Takingadjoints, we see the latter is also equivalent to

312 C∗-modules as operator spaces.∞∑

k=1

〈zk|zk〉b = b, b ∈ B. (8.9)

By the other-handed version of an assertion made in the second last paragraph,(8.9) is also equivalent to B having a countable approximate identity. We shallavoid using this though, since we have not proved it.

In the following, to avoid a notational conflict, we write K∞(Y ) for the spacewe wrote as K(Y ) in 1.2.26. That is, K∞(Y ) ∼= K ⊗min Y .

Corollary 8.2.6 (Brown–Kasparov stabilization) Suppose that Y is a rightC∗-module over B. Then (using the notation above):(1) C(B) ⊕c Y ∼= C(B) unitarily, if Y has a right quasibasis.(2) C(B) ⊕c C(Y ) ∼= C(B) unitarily, if Y has a right quasibasis.(3) C(B) ⊕c C(Y ) ∼= C(Y ), if Y is full, and has a left quasibasis.(4) C(B) ∼= C(Y ), under the hypotheses of both (2) and (3).(5) If Y is an equivalence A-B-bimodule satisfying the hypotheses of both (2)

and (3), then K∞(B) ∼= K∞(Y ) ∼= K∞(A) linearly completely isometrically.

Proof For (1), by the ‘claim’ proved in 8.2.5, we may write C(B) ∼= Y ⊕c Wfor a submodule W of C(B). By the ‘associativity’ of the C∗-module sum, wemay employ the ‘Eilenberg Swindle’:

C(B) ∼= C(B) ⊕c C(B) ⊕c · · ·∼= (Y ⊕c W ) ⊕c (Y ⊕c W ) ⊕c · · ·∼= Y ⊕c (W ⊕c Y ) ⊕c (W ⊕c Y ) ⊕c · · ·∼= Y ⊕c C(B).

‘Associativity’ of the sum also gives (2). For example, we have using (1),

C(B) ∼= C(C(B)) ∼= C(C(B) ⊕c Y ) ∼= C(C(B)) ⊕c C(Y ) ∼= C(B) ⊕c C(Y ).

For (3), suppose that (zk) is a left quasibasis. Define a map α : B → C(Y ) bythe prescription α(b) = (zjb). That (zjb) ∈ C(Y ) is easily seen from (8.9). Defineβ : C(Y ) → B by β((yj)) =

∑j〈zj |yj〉. The latter sum converges by the argument

in 1.2.27, and indeed this argument shows that ‖β‖ ≤ 1. Then α and β arecontractive B-module maps which compose to the identity mapping on B, againby (8.9). Consequently, by 8.1.21, B is unitarily isomorphic to an orthogonallycomplemented B-submodule of C(Y ). Then (3) follows by an argument similarto that of (1) and (2).

Item (4) is clear from (2) and (3). For (5) note that K∞(Y ) ∼= R(C(Y )) (thisis easily deduced from (1.37), for example). By (4) we deduce that

K∞(Y ) ∼= R(C(Y )) ∼= R(C(B)) ∼= K∞(B).

The assertion about A follows by symmetrical arguments (replacing C(B) aboveby R(A), and so on).


8.2.7 (The Brown–Green–Rieffel stable isomorphism theorem) Since this is inmost of the cited C∗-module texts, we will be quick here. Suppose that Y is anequivalence A-B-bimodule which has both a left and a right quasibasis (which oc-curs, as we mentioned in passing in 8.2.5, exactly when A and B both have count-able approximate identities). We saw in 8.2.6 (5) that K∞(A) ∼= K∞(B) linearlycompletely isometrically. Thus by 4.5.13, K∞(A) ∼= K∞(B) ∗-isomorphically.Conversely, if K∞(A) ∼= K∞(B) ∗-isomorphically, then it is easy to see that Aand B are strongly Morita equivalent (see the hints in the Notes section).

8.2.8 (Representations of C∗-modules) Suppose that Y is an equivalence bi-module over A and B, and that we are given a nondegenerate ∗-representationπ : L(Y ) → B(H) of the Morita linking algebra of Y (see 8.1.17). If p is theprojection introduced above (8.3), then, using the notation and facts in 2.6.15,q = π(p) is a projection in B(H), and we may decompose B(H) as a 2 × 2 ma-trix operator algebra. Indeed the i-j-corner of B(H) is simply B(Hj , Hi), whereRan(q) = H1 and Ker(q) = H2. By 2.6.15, π is corner-preserving, and we may de-compose π as [πij ]. The maps πij are complete contractions, which are completeisometries if π is faithful. Also π11 and π22 are ∗-representations of A and B on H1

and H2 respectively. In fact π11 and π22 are also nondegenerate. Indeed, if (bβ) isa cai for L(Y ), then qπ(bβ)(qζ) → qζ, for all ζ ∈ H . However qπ(bβ)q = π(pbβp),in the language of 2.6.12 and 2.6.15. Since pbβp is a cai for A, π11 is nondegener-ate. A similar argument applies to π22. Note also that [π12(Y )H2] = H1, since wehave H1 = [π11(A)H1] = [π12(Y )π21(Y )H1] ⊂ [π12(Y )H2]. A similar argumentshows that [π12(Y )H1] = H2.

If c is the corner map mentioned in 8.1.19, then we have

q⊥π(c(y)∗)qπ(c(z))q⊥ = q⊥π(c(y)∗)π(c(z))q⊥ = q⊥π(c(y)∗c(z))q⊥.

From this, it follows immediately that

π12(y)∗π12(z) = π22(〈y|z〉), y, z ∈ Y. (8.10)

Similarly,

π12(xy∗z) = qπ(c(x)c(y)∗c(z))q⊥ = π12(x)π12(y)∗π12(z), x, y, z ∈ Y.

Thus π12 is a triple morphism (see 4.4.1). Hence the range of π12 is a TRO insideB(H2, H1). If π is faithful, then we have represented Y completely isometricallyas a TRO in B(H2, H1). In fact this must hold even if Y is only a C∗-module,since we saw in 8.1.14 that every C∗-module Y is an equivalence bimodule.

8.2.9 (Rigged C∗-modules) In C∗-module theory, the most important ten-sor product is the so-called interior tensor product. We will discuss this tensorproduct momentarily; for now we will just say that it is formed from a rightC∗-module Y over A, and a so-called B-rigged A-module Z. By the latter term,we will mean a right C∗-module Z over B together with a ∗-homomorphismθ : A → BB(Z), such that θ is nondegenerate in the sense that Z, considered as a


left Banach A-module in the canonical way (see A.6.1), is nondegenerate in thesense of A.6.1.

Lemma 8.2.10 Suppose that A and B are C∗-algebras, and that Z is a rightC∗-module over B, which is also a left A-module. Then Z is a B-rigged A-moduleif and only if Z with its canonical operator space structure as a right C∗-module(see (8.6)), is also a nondegenerate left operator A-module.

Proof If Z is a B-rigged A-module, then by the observations above, Z is cer-tainly a nondegenerate left B-module. By 8.2.3 (1), Z is a left operator moduleover BB(Z). By 3.1.12, Z is an operator A-module.

If Z is a left operator A-module, then since Z is also a right operator moduleover B, we see by the last assertion in 4.6.7 that Z is an A-B-bimodule. Defineθ : A → CBB(Y ) by θ(a)(y) = ay. Then θ is a contractive homomorphism.Indeed θ is a ∗-homomorphism into ∆(CBB(Y )) = BB(Y ), by the last assertionin 2.1.2 and 8.2.4. The rest is clear.

We emphasize that this lemma shows that the bimodules met with in thetheory of C∗-modules, are operator bimodules. By virtue of the lemma, it makessense to define the module Haagerup tensor product (see Section 3.4) of a rightC∗-module over A and a B-rigged A-module.

Theorem 8.2.11 Suppose that Y is a right C∗-module over A, and that Z isa B-rigged A-module. Then the module Haagerup tensor product Y ⊗hA Z is aright C∗-module over B, with B-valued inner product determined by the formula

〈y ⊗ z|y′ ⊗ z′〉 = 〈z|〈y|y′〉z′〉, y, y′ ∈ Y, z, z′ ∈ Z. (8.11)

Moreover, the usual operator space structure on Y ⊗hA Z coincides with thecanonical operator space structure induced by the inner product as in (8.6).

Proof By 8.1.24 and 8.2.2, there exist completely contractive A-module mapsαt : Y → Cnt(A) and βt : Cnt(A) → Y such that βt αt → IY strongly onY . By the functoriality of the module Haagerup tensor product (see 3.4.5), weobtain contractive B-module maps αt ⊗ IZ : Y ⊗hA Z → Cnt(A) ⊗hA Z andβt ⊗ IZ : Cnt(A) ⊗hA Z → Y ⊗hA Z. By density of the elementary tensors, thenet of maps (βt ⊗ IZ) (αt ⊗ IZ) converges strongly to the identity map onY ⊗hA Z. By 3.4.11, we have Cnt(A) ⊗hA Z ∼= Cnt(Z), and the latter is a rightC∗-module over B. Via this isomorphism, it is easily checked that the inducedinner product on Cn(A) ⊗hA Z is given by the formula

〈a ⊗ z|a′ ⊗ z′〉 = 〈z|θ(a∗a′)z′〉, a, a′ ∈ Cn(A), z, z′ ∈ Z.

By 8.1.26, we conclude that Y ⊗hA Z is a right C∗-module over B. By 8.1.26used twice more, the inner product on Y ⊗hA Z is specified by its value on a pairof rank one tensors y ⊗ z and y′ ⊗ z′, as follows:

limt

〈αt(y) ⊗ z|αt(y′) ⊗ z′〉 = limt

〈z|θ(αt(y)∗αt(y′))z′〉 = 〈z|〈y, y′〉z′〉.

This, together with 8.2.3 (5), proves the result.


In the remainder of this section, we will simply write Y ⊗A Z for Y ⊗hA Z.

8.2.12 (Properties of the tensor product) One may view the last result as theassertion that the well-known interior tensor product of Y and Z of C∗-modules(see any of the texts cited at the start of this chapter), coincides with the mod-ule Haagerup tensor product. This is helpful in many ways, partly because theHaagerup tensor product has many useful properties. For example, one advan-tage of 8.2.11 is that it gives most of the important properties of the interiortensor product ‘for free’. For example:

(1) (Functoriality) If u : Y → Y ′ is a bounded right A-module map betweenright C∗-modules over A, and if v : Z → Z ′ is a bounded A-B-bimodule map be-tween B-rigged A-modules, then u⊗v extends to a bounded right B-module mapbetween the interior tensor products: Y ⊗A Z → Y ′ ⊗A Z ′ (of norm ≤ ‖u‖‖v‖).This follows immediately from the functoriality property of the Haagerup tensorproduct (see 3.4.5), and 8.2.2.

(2) (Associativity) We have (Y ⊗A Z) ⊗B W ∼= Y ⊗A (Z ⊗B W ) unitarily,if Y is a right C∗-module over A, if Z is a B-rigged A-module, and if W isa C-rigged B-module. This follows immediately from the associativity of theHaagerup tensor product (see 3.4.10).

(3) (Commutation with the direct sum) We have

⊕ci (Yi ⊗A Z) ∼= (⊕c

i Yi) ⊗A Z unitarily, (8.12)

for right C∗-modules Yi over A, and a B-rigged A-module Z. Also,

⊕ci (Y ⊗A Zi) ∼= Y ⊗A (⊕c

i Zi)

for a right C∗-module Y over A, and B-rigged A-modules Zi. These relationsboth follow from the universal property of the direct sum in 8.1.22. To see (8.12),let εi and Pi be the canonical inclusion and projection maps between ⊕c

i Yi andits summands. Define maps ε′i = εi ⊗ IZ and P ′

i = Pi ⊗ IZ between (⊕ci Yi)⊗A Z

and Yi⊗AZ. These are contractive by (1) above, and are easily checked to satisfythe hypotheses of 8.1.22.

The second centered relation above is almost identical, however one firstshould check that ⊕c

i Zi is indeed a B-rigged A-module. Observe that the canon-ical left action of A on ⊕c

i Zi is well defined since by 8.1.5,∑j

〈θj(a)zj |θj(a)zj〉 ≤ ‖a‖2∑

j

〈zj |zj〉, a ∈ A, zj ∈ Zj.

Here θj : A → BB(Zj) is the homomorphism associated with the left A-moduleactions on Zj . Furthermore, it is easy to check that this action of A on ⊕c

i Zi isas ‘adjointable operators’, and that the action is nondegenerate. Hence ⊕c

i Zi isa B-rigged A-module.

(4) (The adjoint module) By the definition of the module Haagerup tensorproduct in Section 3.4, it is easy to see using 8.2.3 (2) that the ‘adjoint C∗-module’ of Y ⊗A Z is Z ⊗A Y , completely isometrically.


Corollary 8.2.13 Suppose that Y is a right C∗-module over B, and that θ is anondegenerate ∗-representation of B on a Hilbert space H. Then:(1) Y ⊗B Hc is a Hilbert space.(2) If Y is a B-rigged A-module, then Y ⊗B Hc is a Hilbert A-module. If θ and

the canonical map from A into B(Y ) are both one-to-one, then so is thecanonical map from A into B(Y ⊗B Hc).

Proof (1) This follows from 8.2.11, which shows that Y ⊗B Hc is a right C∗-module over C. That is, it is a Hilbert space with inner product ζ × η → 〈η|ζ〉.

(2) In this case, Y and Y ⊗B Hc are left operator A-modules, by 8.2.10 and3.4.9. Thus the first assertion follows from (1) and 3.1.7. If a(Y ⊗B Hc) = 0then aY = 0, by the relation 〈a(y ⊗ ζ), y′ ⊗ η〉 = 〈ζ|〈ay|y′〉η〉 from (8.11). Herea ∈ A, y, y′ ∈ Y , and ζ, η ∈ H .

8.2.14 (Avoiding the inner product) As we have already seen, many C∗-moduleconstructions can be done, if need be, without explicit reference to the innerproduct. See 8.1.6, 8.2.12, and 8.4.2, for example. Here we use 8.2.13 to take thisthought a little further, omitting full proofs. If Y is a right C∗-module over B,and if B is a nondegenerate ∗-subalgebra of B(K), say, then define a B-modulemap Φ: Y → B(K, Y ⊗B Kc) by Φ(y)(ζ) = y ⊗ ζ, for y ∈ Y, ζ ∈ K. It is easyto see that Φ(y)∗Φ(z) = 〈y|z〉, for y, z ∈ Y (using (8.11)). Also, Φ(Y ) is a C∗-module over B with the inner product (y, z) → Φ(y)∗Φ(z), and Φ is a unitaryB-module map. Thus the inner product on Y is completely determined by thenorm on the space Y ⊗B Kc. The latter norm has reformulations avoiding useof the inner product, mentioned at the end of the Notes to Section 8.2.

We use the above to give an alternative description of the C∗-module directsum of 8.1.9. For specificity we discuss the direct sum of two right C∗-modules,Y1 and Y2, over B. Let K be as above, and let Hi = Yi ⊗B Kc. We will suppressmention of the map Φ in the last paragraph, and simply write yiζ for yi ⊗ ζ.Then Y1 ⊕c Y2 may be identified with the B-submodule W of B(K, H1 ⊕ H2)consisting of the maps ζ → (y1ζ, y2ζ), for ζ ∈ K, y1 ∈ Y1, y2 ∈ Y2. Indeed, thecanonical inner product on W , namely S ×T → S∗T , takes values in B, makingW into a C∗-module over B which is unitarily B-isomorphic to Y1 ⊕c Y2.

For the next result, we will need to extend the definition of KB(Y, Z) from8.1.7, to allow Z to be any right operator B-module. Namely, we define KB(Y, Z)to be the closure in CB(Y, Z) of the span of the ‘rank one’ operators y → z〈y′|y〉(here y, y′ ∈ Y, z ∈ Z).

Henceforth, we shall assume that all operator modules are nondegenerate.

Corollary 8.2.15 Let Y be a right C∗-module over B, and let W be a right C∗-module over B (or more generally let W be a right operator B-module). Then:(1) W ⊗B Y ∼= KB(Y, W ) completely isometrically.(2) If Y is an equivalence A-B-bimodule, then KB(Y, W ) = CBess

B (Y, W ), inthe notation of the second paragraph of 3.5.2.


Proof (1) Define ρ : W ×Y → KB(Y, W ) by ρ(w, y)(x) = w〈y|x〉, for x, y ∈ Y ,and w ∈ W . As in 8.1.19, we write 〈y|x〉 as y∗x, interpreted as a product in L(Y ).Let [wij ], [z∗ij ] and [xrs] be matrices with entries in W, Y , and Y respectively.Since W is a right h-module (see 3.1.3), it is not hard to see that∥∥∥[∑

k

wik〈zkj |xrs〉]∥∥∥ ≤ ‖[wij ]‖‖[z∗ijxrs]‖ ≤ ‖[wij ]‖‖[z∗ij ]‖‖[xrs]‖,

using also the fact that L(Y ) is an operator algebra, and hence a matrix normedalgebra (see 2.3.9). Thus ρ is completely contractive in the sense of 1.5.4. By 3.4.2,ρ induces a complete contraction from W ⊗B Y to KB(Y, W ), which we will stillwrite as ρ. Next, take a cai for KB(Y ) of the form in (8.4). Given T ∈ KB(Y, W ),define θt(T ) ∈ W ⊗B Y to be the element

∑k T (xt

k)⊗ xtk. From (1.40) and (8.6),

one may easily check that θt : KB(Y, W ) → W ⊗B Y is completely contractive.However, for w ∈ W, y ∈ Y ,

θt(ρ(w ⊗ y)) =∑

k

w〈y|xtk〉 ⊗ xt

k = w ⊗∑

k

〈y|xtk〉xt

k −→ w ⊗ y.

By density, θt(ρ(u)) → u, for all u ∈ W ⊗B Y . This implies, by a simple mod-ification of the principle in 1.2.7, that ρ is a complete isometry. Now it is clearthat ρ maps onto KB(Y, W ).

(2) If Y is an equivalence A-B-bimodule, then A ∼= KB(Y ) by 8.1.15. It thenfollows from the argument for (8.1), that CBess

B (Y, W ) ⊂ KB(Y, W ). Conversely,every ‘rank one’ operator |w〉〈y| in KB(Y, W ) is the limit of |w〉〈ety| = |w〉〈y|et,which lie in CBess

B (Y, W ). Here (et)t is as in 8.1.23. It follows that |w〉〈y| is inCBess

B (Y, W ), and so KB(Y, W ) ⊂ CBessB (Y, W ).

8.2.16 (Hom–tensor relations) The tensor product identity in 8.2.15 (1) is aC∗-module variant of what is called a Hom–tensor relation in algebra. There arebimodule versions of this particular identity too, namely W ⊗B Y ∼= KB(Y, W )as bimodules, if in addition Y and W are operator bimodules. Also, there is amatching left-handed result: if Z is a right C∗-module over A, and W is a leftoperator A-module, then Z⊗A W ∼= AK(Z, W ). We leave these assertions to theinterested reader.

As a sample application of the relation in 8.2.15, we show that for a rightC∗-module Y over B, KB(C(Y )) is ∗-isomorphic to the minimal tensor productKB(Y ) ⊗min K∞, where K∞ = K(2) again. To see this, note that the adjointC∗-module of C(Y ) is R(Y ). Hence by 8.2.15 used twice, and by basic propertiesof the Haagerup tensor product from Sections 1.5 and 3.4, we have

KB(C(Y )) ∼= C(Y ) ⊗B R(Y ) ∼= C ⊗h (Y ⊗B Y ) ⊗h R ∼= KB(Y ) ⊗min K∞.

Setting Y = B, we may deduce from this and 8.1.14, that B is strongly Moritaequivalent to B ⊗min K∞.


From 8.2.15 one may deduce further Hom–tensor relations such as:

KB(Y ⊗A Z, W ) ∼= KA(Y, KB(Z, W )) completely isometrically.

Here Y is a right C∗-module over A, Z is a B-rigged A-module, and W is a rightoperator module over B. To see the last centered relation, note that by 8.2.15,KB(Y ⊗A Z, W ) ∼= W ⊗B (Y ⊗A Z). Thus

KB(Y ⊗A Z, W ) ∼= W ⊗B Z ⊗A Y ∼= KB(Z, W ) ⊗A Y ∼= KA(Y, KB(Z, W )),

using 8.2.15 twice, and 8.2.12 (2) and (4). Indeed, it is also true that for Y, Z, Was above, we have

CBB(Y ⊗A Z, W ) ∼= CBA(Y, CBB(Z, W ))

completely isometrically. The proof of this, and other such Hom-tensor relations,may be found in [47].

One may also define the well-known ‘exterior tensor product’ of right C∗-modules in operator space terms:

Theorem 8.2.17 Let Y and Z be right C∗-modules over A and B respectively.Then the minimal tensor product Y ⊗min Z (see 1.5.1) is a right C∗-module overA ⊗min B; with inner product determined by

〈y ⊗ z|y′ ⊗ z′〉 = 〈y|y′〉 ⊗ 〈z|z′〉, y, y′ ∈ Y, z, z′ ∈ Z.

We omit the proof of this, which is extremely similar to that of 8.2.11. The-orem 8.2.17 also easily implies results analoguous to those in 8.2.12.

Strong Morita equivalence may be restated concisely in the language of op-erator modules as follows:

Theorem 8.2.18 If Y is an equivalence A-B-bimodule, then Y ⊗A Y ∼= B com-pletely A-A-isometrically, and Y ⊗B Y ∼= A completely B-B-isometrically. Con-versely, if Y and X are respectively operator A-B- and B-A-bimodules, suchthat X ⊗hA Y ∼= B completely B-B-isometrically, and also Y ⊗hB X ∼= A com-pletely A-A-isometrically, then A and B are strongly Morita equivalent, Y is anequivalence A-B-bimodule, and X ∼= Y completely B-A-isometrically.

Proof We will only need the first statement later, and will only prove thisone here. We refer the reader to [52] Proposition 1.3 in conjunction with [65]Theorem 6.2, for a proof of the second statement.

If Y is an equivalence A-B-bimodule. then by 8.2.15 and 8.1.15, we haveY ⊗B Y ∼= KB(Y ) ∼= A. Tracing through these identifications, one sees that theisomorphism holds as A-A-bimodules too. Similarly, Y ⊗A Y ∼= B.

8.2.19 (Induced representations and Morita equivalence) Let Y be an equiv-alence A-B-bimodule. If Z is a left operator B-module, then G(Y ) = Y ⊗B Z


is an operator A-module (see 3.4.9). If u : Z1 → Z2 is a completely contractiveB-module map between left operator B-modules, then IY ⊗ u : G(Z1) → G(Z2)is a completely contractive A-module map (see 3.4.5). That is, G(—) = Y ⊗B —is a functor from the category BOMOD to the category AOMOD (see 3.5.1). In-deed the map u → G(u) is linear and contractive on BCB(Z1, Z2). It is easy, buttedious, to check that this map is actually completely contractive. Consequently,we call G a completely contractive functor. By 8.2.13, G maps the subcategoryBHMOD to the category AHMOD. By 8.2.10 and 8.2.11, G also maps betweenthe subcategories of left C∗-modules over B and A respectively. Conversely, ifF (—) = Y ⊗A — then F is a completely contractive functor from the cate-gory AOMOD to the category BOMOD, and F maps AHMOD to BHMOD,and also maps between the subcategories of left C∗-modules. Composing thesefunctors, and using the last theorem, and 3.4.10 and 3.4.6, we have for any leftoperator A-module W that

Y ⊗B (Y ⊗A W ) ∼= (Y ⊗B Y ) ⊗A W ∼= A ⊗A W ∼= W.

Thus G F = I, and similarly F G = I. Note that from this, and from the factthat G and F are completely contractive, it is easy to see from 1.2.7 that themap u → G(u) above is a complete isometry on BCB(Z1, Z2).

Thus if A and B are strongly Morita equivalent, then AOMOD and BOMODare equivalent as categories (as are also AHMOD and BHMOD, and as also arethe categories of left C∗-modules over A and B). Of course, a similar argumentgives analoguous results for the categories of right modules.

The above proves the part we will need later of the following result. Proofof the other parts may be found in [51]. We remark that this theorem is aC∗-algebraic version of Morita’s fundamental theorem from pure algebra (e.g.see [8, 368]).

Theorem 8.2.20 Two C∗-algebras A and B are strongly Morita equivalent ifand only if the categories AOMOD and BOMOD are equivalent (via completelycontractive functors). Moreover, if F : AOMOD → BOMOD is the equivalencefunctor, then X = F (A) is a strong Morita equivalence B-A-bimodule.

8.2.21 (The nonselfadjoint algebra case) In the light of 8.2.18, if A and Bare approximately unital nonselfadjoint operator algebras, then it is natural todefine A and B to be strongly Morita equivalent if there exists an operatorA-B-bimodule Y and an operator B-A-bimodule X , such that X ⊗hA Y ∼= Bcompletely B-B-isometrically, and Y ⊗hB X ∼= A completely A-A-isometrically.From this definition one can then proceed to develop a theory parallel to theC∗-algebra case. For example, 8.2.20 generalizes: A and B are strongly Moritaequivalent operator algebras if and only if the categories AOMOD and BOMODare equivalent via completely contractive functors. Similarly, there is a general-ization of C∗-modules to the nonselfadjoint situation. See the Notes section forreferences to the literature.


8.2.22 (Induced representations) Let Y be an equivalence A-B-bimodule.Thinking of modules W in BOMOD as nondegenerate representations of B onW , we saw in 8.2.19 that such representations of B induce representations of A(on Y ⊗B W ), and vice versa. Since L(Y ) is strongly Morita equivalent to B viathe equivalence bimodule Y ⊕c B (see 8.1.18), we see that there are one-to-onecorrespondences between the ‘representations’ of B, the ‘representations’ of A,and the ‘representations’ of L(Y ). In particular, by A.5.8 and the discussion in8.2.19 about HMOD, there are one-to-one correspondences between the nonde-generate ∗-representations of B on Hilbert space, and those of A, or of L(Y ).In fact we have seen part of this already in 8.2.8, although in a disguised form.There we took a nondegenerate ∗-representation π of L(Y ) on a Hilbert spaceH , and we saw that there were corresponding ∗-representations of A on a Hilbertspace H1, and B on H2. At first sight the spaces H1 and H2 defined in 8.2.8look different from the ones obtained via the correspondences above. We willnow show that they are the same, up to a unitary. First we claim that

H1∼= Y ⊗B H2, and H2

∼= Y ⊗A H1, (8.13)

unitarily. To see this, we will use the notation in 8.2.8, and we define a modulemap f : Y ⊗B H2 → H1 by f(y ⊗ ζ) = π12(y)(ζ). Using (8.10), we have

〈f(y ⊗ ζ), f(y′ ⊗ η)〉 = 〈π12(y′)∗π12(y)ζ, η〉 = 〈π22(〈y′|y〉ζ, η〉,

for y, y′ ∈ Y, ζ, η ∈ H . By (8.11), the last number equals 〈y⊗ ζ, y′⊗η〉. It followsfrom this that 〈f(ξ), f(ξ)〉 = 〈ξ|ξ〉, for any ‘finite rank’ tensor ξ ∈ Y ⊗B H2.Thus f is a well defined isometry. Also, we checked in 8.2.8 that f has denserange. Hence f is surjective. This yields the first relation in (8.13). The secondis similar.

By (8.12), (8.13), and 3.4.6, we have

(Y ⊕c B) ⊗B H2∼= (Y ⊗B H2) ⊕c (B ⊗B H2) ∼= H1 ⊕c H2

∼= H. (8.14)

This shows that the L(Y )-module induced from the B-module H2 via the equiv-alence bimodule Y ⊕c B, by the procedure of 8.2.19, is unitarily isomorphic toH . Thus, conversely, the B-module induced from the L(Y )-module H by theprocedure of 8.2.19, is unitarily isomorphic to H2.

The argument in and below (8.14) shows that any Hilbert B-module K in-duces a nondegenerate ∗-representation π of L(Y ) on (Y ⊗B K)⊕K. It is easy tocheck that π is corner-preserving, in the sense of 2.6.15. Its ‘four corners’ consistof a representation of A on B(Y ⊗B K), maps from Y to B(K, Y ⊗B K) and fromY to B(Y ⊗B K, K), and the given representation of B on K. Also, by 8.2.13(2), π is faithful if the given representation of B on K was faithful.

8.2.23 (Inducing universal representations) Suppose that Y is an equivalenceA-B-bimodule, and that K is a B-universal Hilbert module (see 3.2.7), or equiv-alently, a generator for BHMOD (see 3.2.8). By 8.2.19, Y induces an equivalence


of categories AHMOD ∼= BHMOD. By the simplest category theory, the in-duced representation of A on H = Y ⊗B K is A-universal. One corollary ofthis: we see from 3.2.12 and the double commutant theorem, that the secondcommutant of A in B(H), is completely isometrically isomorphic to A∗∗.

This construction is pleasantly functorial, and is therefore quite useful (forexample, it is key to the proof of the difficult implication in Theorem 8.2.20).

8.2.24 (Rieffel subequivalence) One of the pleasant consequences of purelyalgebraic Morita equivalence, is that if two rings A and B are Morita equivalentvia an A-B-bimodule Y , then there are one-to-one lattice isomorphisms betweenthe following lattices: (1) the two-sided ideals I of A, (2) the two-sided ideals Jof B, (3) the A-B-submodules X of Y , and (4) the two-sided ideals D of L(Y ).Rieffel showed that similar correspondences hold in the C∗-algebra setting, withthe word ‘norm-closed’ added. In fact this is quite easy to see: First, we replacethe inner products and module actions with concrete multiplication in the linkingalgebra, as in 8.1.19. Define a map from lattice (2) to lattice (3) by J → XJ = Y J .Conversely, define a map from lattice (3) to lattice (2) by X → J (X) = Y X ,for X in lattice (3). Using A.6.2, we have X = AX = Y Y X . Thus XJ (X) = X .Given J in lattice (2), a similar argument shows that J (XJ) = J . Thus indeedthe lattices (2) and (3) are lattice isomorphic. Similarly for (1) and (3). Notethat if X and J are in correspondence as above, then by A.6.2,

J = JJ = (Y X)Y X = XY Y X = XX.

Since J = XX , X is a full right C∗-module over J . By symmetry, if I is the idealin lattice (1) associated with X , then X is a a full left C∗-module over I. ThusX is a equivalence I-J-bimodule, and I and J are strongly Morita equivalent.The linking C∗-algebra L(X) of X may be taken to be the ∗-subalgebra D ofL(Y ) with corners I, X, X, and J , by the unicity of the C∗-norm on a ∗-algebra(see A.5.8). It is easy to check that D is an ideal of L(Y ).

Now we shall check that the lattices (2) and (4) are isomorphic, via thecorrespondence J → D above. By 8.1.18, we may use the lattice isomorphismswe have already verified, but with Y and A replaced by Y ⊕c B and L(Y ). Thusan ideal J of B corresponds to a L(Y )-B-submodule W of Y ⊕c B, namely thesubmodule W = (Y ⊕c B)J = X ⊕c J . Also, we obtain a corresponding idealI ′ = WW of L(Y ). However it is easily checked from facts in the previousparagraph that WW = D.

8.2.25 (Rieffel quotient equivalence) Let X be a closed A-B-submodule of anequivalence A-B-bimodule Y . Via the correspondences in 8.2.24, let I be thecorresponding ideal in A, and let J be the corresponding ideal in B. Let D bethe subset of L(Y ) with four corners I, X, X and J . We saw above that X isan equivalence I-J-bimodule, and D ∼= L(X) ∗-isomorphically. Moreover, D isa closed ideal in L(Y ). We consider the quotient map π : L(Y ) → L(Y )/D. By2.6.15, the canonical four corners of L(Y ) induce corners of L(Y )/D, and π iscorner-preserving. Write π = [πij ], as in 2.6.15. It is straightforward to check

322 Triples, and the noncommutative Shilov boundary

that the 1-2-corner W of L(Y )/D is a TRO, and also is an equivalence bimoduleover π11(A) and π22(B). For example, to see that W W = π22(B), observe that

π22(〈z|y〉) = π21(z)π12(y) ∈ W W, y, z ∈ Y.

It follows that π22(B) ⊂ W W , and the converse inclusion is clear.Next, note that the ‘1-2-corner’ π12 of π is a complete contraction from Y to

W , with kernel X . Since there is a completely contractive projection from L(Y )onto its 1-2-corner c(Y ), it follows easily that π12 is a complete quotient map.Indeed, to see that it is a quotient map, suppose that we are given an elementw ∈ W of norm < 1. Then since π is a quotient map, there exists an element w′

of L(Y ) of norm < 1 which π maps to w. The 1-2-corner of w′ is in Y , has norm< 1, and is mapped by π12 to w.

Thus W is completely isometrically isomorphic to Y/X , the latter in the senseof 1.2.14. Similarly the 1-1-corner of L(Y )/D is ∗-isomorphic to A/I, and the2-2-corner of L(Y )/D is ∗-isomorphic to B/J . Thus we see that A/I and B/Jare strongly Morita equivalent. The equivalence (A/I)-(B/J)-bimodule may betaken to be Y/X (or equivalently, W ). One easily sees that with respect to theseidentifications, the B/J-valued inner product on Y/X , for example, is simply:

〈y + X |y′ + X〉 = 〈y|y′〉 + J, y, y′ ∈ Y.

Thus the quotient Y/X has (a) a canonical C∗-module structure (discussedabove), and (b) a quotient operator space structure (from 1.2.14); and very fortu-nately these two structures are compatible. That is, the canonical operator spacestructure on the C∗-module Y/X equals its quotient operator space structure.

8.3 TRIPLES, AND THE NONCOMMUTATIVE SHILOV BOUNDARY

8.3.1 (Triple systems) In the literature, the word ternary is often used in placeof our usage of the word triple, to avoid confusion with the use of that wordin the JB∗-triple literature. Since we will not discuss JB∗-triples, we will usethe word ‘triple’ consistently, with apologies to those for whom it has differentconnotations. In 4.4.1 we discussed the ‘triple product’ [x, y, z] = xy∗z on aTRO. It is convenient for us to define a triple system to be an operator space Ypossessing a map [·, ·, ·] : Y × Y × Y → Y (again called a triple product), suchthat Y is complete isometric to a TRO Z via a linear map θ : Y → Z which is atriple morphism, that is, θ([x, y, z]) = [θ(x), θ(y), θ(z)] for all x, y, z ∈ Y . In factby 4.4.6, it is clear that an operator space Y can have at most one such tripleproduct, and thus this triple product is uniquely determined by the norms onMn(Y ), for n ≥ 1. We will therefore write xy∗z for this unique triple producton Y , without there being too much danger of confusion. We are not aware of asimple formula for the triple product in terms of the matrix norms; however wemention in passing that there is a remarkable intrinsic characterization of triplesystems in terms of these norms due to Neal and Russo [289].


In any case, by the above, we may simply define a triple system to be anoperator space which is linearly completely isometric to a TRO. By a tripleepimorphism we mean a surjective triple morphism, whereas a triple isomorphismis a one-to-one surjective triple morphism. By a subtriple of a triple system, wemean a closed subspace which is closed under the triple product. Thus TROsmay be defined to be the subtriples of B(K, H), or of a C∗-algebra. Any C∗-module Y , with its canonical operator space structure, is a triple system. If Y isan equivalence bimodule, then this triple product is just x〈y|z〉 = [x|y]z; whichwe simply write xy∗z (see also 8.1.19).

From the perspective of this section, TROs, C∗-modules, equivalence bimod-ules, and triple systems, are essentially the same thing. That is, we may use thesewords interchangeably in the statements of most results below. Thus althoughwe often restrict our attention to TROs, such results will immediately implycorresponding results for triple systems or C∗-modules.

It is clear that if Y is a triple system, then so is Mn(Y ). In particular, ifY = B(K, H), then the triple product on Mn(Y ) simply corresponds to theobvious triple product on B(K(n), H(n)).

Lemma 8.3.2 (Harris–Kaup) Let θ : Y → W be a triple morphism betweenTROs. Then:(1) θ is contractive, and indeed completely contractive.(2) θ is completely isometric if and only if it is one-to-one.

Proof (1) The amplification θn : Mn(Y ) → Mn(W ) is also a triple morphismbetween TROs (see 8.3.1). Thus it is enough to prove the first statement, orequivalently that ‖θ(y)∗θ(y)‖ ≤ ‖y∗y‖ for any y ∈ Y , or equivalently the fol-lowing containment of spectra: σ(θ(y)∗θ(y)) ⊂ σ(y∗y) ∪ 0. The latter followsimmediately from the following Claim: A nonzero scalar λ is in σ(y∗y) if andonly if there does not exist a z ∈ Y such that

(λI − yy∗)z(λI − y∗y) = y. (8.15)

To prove this claim, note that if λ /∈ σ(y∗y) then z = y(λI − y∗y)−2 satisfies(8.15). However z ∈ Y , since (λI − y∗y)−2 is, by spectral theory, a limit ofpolynomials in y∗y. Conversely, suppose that λ ∈ σ(y∗y). By basic spectraltheory, we can find a sequence bn ∈ Y Y with (λI − y∗y)bn → 0, but bn itselfdoes not converge to 0. Hence y∗ybn, and therefore also ybn, cannot converge to0. From this, it is easy to see that there can exist no z satisfying (8.15).

(2) Again, it suffices to prove that if θ in (1) is one-to-one, then it is isometric.This will follow if σ(θ(y)∗θ(y))∪0 = σ(y∗y)∪0, in the notation of (1). How-ever if λ ∈ σ(y∗y)\σ(θ(y)∗θ(y)), with λ = 0, then by Urysohn’s lemma there ex-ists a continuous nonnegative function f on σ(y∗y), which is zero on σ(θ(y)∗θ(y))and at 0, but is nonzero at λ. Clearly f(y∗y) = 0. Since f may be approximateduniformly by polynomials pn with no constant term, we have that f(y∗y) ∈ Y Y .For any z ∈ Y and any polynomial p, θ(zp(y∗y)) = θ(z)p(θ(y)∗θ(y)). Replacing


p by pn and letting n → ∞, we have θ(zf(y∗y)) = θ(z)f(θ(y)∗θ(y)) = 0. Thuszf(y∗y) = 0. Since this is true for all z ∈ Y , by 8.1.4 (1) we have f(y∗y) = 0.This is a contradiction.

We can rephrase part of 8.2.25 in the language of triple systems as follows:

Proposition 8.3.3 Suppose that Y is an equivalence A-B-bimodule, and thatX is a closed A-B-submodule of Y . Then the quotient operator space Y/X is atriple system. Indeed if q : Y → X is the canonical quotient map, then the tripleproduct on Y/X is given by [q(x), q(y), q(w)] = q(xy∗w), for x, y, z ∈ Y . Thus qis a triple epimorphism.

If X is a closed linear subspace of a triple system Y , such that xy∗z and zy∗xare in X for all x ∈ X and y, z ∈ Y , then we say that X is a triple ideal of Y . Inthis case, the quotient Y/X is a triple system, by 8.3.3.

Corollary 8.3.4 Let θ : Y → Z be a triple morphism between TROs. Then:(1) Ker(θ) is a triple ideal in Y .(2) Ran(θ) is closed, and is a subtriple of Z.(3) θ is a complete quotient map onto its range.(4) The induced map θ : Y/Ker(θ) → Z is a completely isometric triple mor-

phism onto Ran(θ).

Proof Item (1) is obvious. The induced map θ : Y/Ker(θ) → Z is, by 8.3.3,a one-to-one triple morphism onto Ran(θ). By Lemma 8.3.2, θ is completelyisometric, which gives (2), (3), and (4).

Corollary 8.3.5 (Hamana) A linear map θ : Y → W between full C∗-modulesor TROs, is a triple morphism if and only if θ is the 1-2-corner of a corner-preserving ∗-homomorphism π : L(Y ) → L(W ) between the Morita linking C∗-algebras. In this case, θ is one-to-one (resp. surjective) if and only if π is one-to-one (resp. surjective).

Proof The first ‘if’ is clear. For the converse, we may suppose that θ is atriple morphism between TROs. Define ρ(

∑nk=1 xky∗

k) =∑n

k=1 θ(xk)θ(yk)∗, forx1, . . . , xn, y1, . . . , yn ∈ Y . Now θ(Y ) is a TRO in W , and so by 8.1.4 (1) usedtwice, and by 8.3.4 (3), we have∥∥∥∑

k

θ(xk)θ(yk)∗∥∥∥ = sup

∥∥∥∑k

θ(xk)θ(yk)∗w∥∥∥ : w ∈ Ball(θ(Y ))

= sup

∥∥∥∑k

θ(xk)θ(yk)∗θ(z)∥∥∥ : z ∈ Ball(Y )

≤ sup

∥∥∥∑k

xky∗kz∥∥∥ : z ∈ Ball(Y )

=∥∥∥∑

k

xky∗k

∥∥∥.

Hence ρ is well-defined, and extends to a contraction, which we still call ρ, fromY Y to WW . It is easy to see that ρ is a ∗-homomorphism. Similarly one gets a


canonical ∗-homomorphism σ : Y Y → W W . It is now clear how to define thecorner-preserving ∗-homomorphism π : L(Y ) → L(W ), whose 1-2-corner is θ.

It is clear that if π is one-to-one (resp. surjective) then so is θ. If θ is one-to-one, then it is isometric. In this case, the one inequality in the centered equationsabove is also an equality. It follows that ρ is isometric, hence one-to-one. Simi-larly, σ is one-to-one. Since π is corner-preserving, it is now easy to see that πis also one-to-one. If θ is surjective, then ρ is surjective, since it has dense rangein W W . Similarly σ, and hence also π, is surjective.

8.3.6 (Inner products on triple systems) By 4.4.6, a completely isometric linearisomorphism between two equivalence bimodules is a triple morphism. Thus, it isthe 1-2-corner of a ∗-isomorphism π between the Morita linking C∗-algebras, asin 8.3.5. The 1-1- and 2-2-corners of π are ∗-isomorphisms between the algebrasacting on the given bimodules.

If Y is a triple system, then Y has a canonical equivalence bimodule structure.Namely, let A be the subspace of B(Y ) densely spanned by the maps z → [x, y, z],and let B be the subspace densely spanned by the maps z → [z, y, x], for x, y ∈ Y .If Y is a TRO, then it is clear that A is just the copy of Y Y in B(Y ). Thatis, A = KY Y (Y ); and by 8.1.14, Y is a left C∗-module over A. Hence it followsthat if Y is a triple system, then A is a C∗-algebra in the product of B(Y ), andY is a left C∗-module over A in a canonical fashion. Similarly, B is a C∗-algebrawith the reversed product of B(Y ), and Y is a right C∗-module over B. Clearly,Y is an equivalence A-B-bimodule. We will see in 8.4.2 that A is a ∗-subalgebraof Al(Y ) in the notation of 4.5.7, and B ⊂ Ar(Y ).

Next we characterize the possible right C∗-module actions on a triple systemwhich are compatible with the underlying operator space structure (via equation(8.6)). First we note that if Y is a full right C∗-module over a C∗-algebra C,and if σ is a faithful ∗-homomorphism from C onto an ideal of a C∗-algebraD, then we may make Y into a right C∗-module over D as follows. We defineyd = yσ−1(d) and 〈〈y|z〉〉 = σ(〈y|z〉), for d ∈ σ(C), and y, z ∈ Y . It is easy tocheck that Y is a right C∗-module over σ(C), and hence by 8.1.4 (4), over D.We denote this C∗-module as Yσ. Moreover, the operator space structure on Yσ

given by (8.6), is easily seen to coincide with the former one. We claim that everyright C∗-module action on Y over any C∗-algebra D, which is compatible withthe given operator space structure, arises in precisely this way. To see this, writeY ′ for Y viewed as a C∗-module over D. Thus the identity map I from Y to Y ′

is a complete isometry, and a triple isomorphism. Hence by the first paragraphof 8.3.6, there is associated a ∗-isomorphism σ from C onto the ideal I of D. Itis easy algebra to check that Y ′ = Yσ (using the fact that σ and I are corners ofa ∗-isomorphism between the Morita linking C∗-algebras (see 8.3.5)).

By the last two paragraphs, it follows that if a triple system Y is a right C∗-module over a C∗-algebra D, such that the norms from (8.6) equal the given ma-trix norms on Y , then this C∗-module equals Yσ for a faithful ∗-homomorphismσ from B onto an ideal of D. Here B is as in the second last paragraph. This givesanother ‘picture’ of the C∗-modules over a C∗-algebra D, as the triple systems


Y , together with a faithful ∗-homomorphism σ : B → D as above.We may now extend the notation introduced in 8.1.19 to triple systems. Thus

products such as y1y∗2 , y1y

∗2y3y

∗4y5, and so on, make sense for elements (yi) in

a triple system. Indeed y1y∗2 represents the operator z → [y1, y2, z] in the C∗-

algebra A above, whereas y1y∗2y3y

∗4y5 = [[y1, y2, y3], y4, y5] ∈ Y . As in 8.1.19, any

such expressions may be interpreted as products in the Morita linking C∗-algebraof Y , where Y is regarded as an equivalence A-B-bimodule as above.

8.3.7 (Nondegenerate triple morphisms) If θ : Y → B(K, H) is a triple mor-phism defined on a TRO Y , then we say that θ is nondegenerate if (i) θ(Y )K isdense in H , and (ii) θ(Y )∗H is dense in K. Using the fact that Y is the normclosure of Y Y Y , it is easy to see that (i) is equivalent to saying that the corre-sponding ∗-homomorphism from Y Y to B(H) given by 8.3.5, is nondegenerate.Similarly, (ii) is equivalent to saying that the corresponding ∗-homomorphismfrom Y Y to B(K) is nondegenerate. Also, (ii) is equivalent to saying that∩y∈Y Ker(θ(y)) = 0.

Thus (using 8.2.8 if necessary), we see that θ is nondegenerate if and onlyif the corresponding ∗-homomorphism from L(Y ) to B(H ⊕ K) from 8.3.5 isnondegenerate. Note that this implies that if, further, θ is one-to-one, then wemay view M(L(Y )) ⊂ B(H⊕K), as in Section 2.6. If θ is not nondegenerate thenwe may ‘cut it down’ to be nondegenerate, by replacing H with H ′ = [θ(Y )K],and K with K ′ = [θ(Y )∗H ].

8.3.8 (The noncommutative Shilov boundary) We next discuss the noncom-mutative Shilov boundary, or ‘triple envelope’, of a nonunital operator space. Toa certain extent this will parallel the development in Section 4.3, to which thereader may want to refer back to periodically.

Suppose that X is an operator space. If i : X → Y is a linear completeisometry into a triple system Y , such that Y is the smallest subtriple of Ycontaining i(X), then we say that (Y, i) is a triple extension of X . Notice thatin this case, by the argument for (4.7),

Y = Spani(x1)i(x2)∗i(x3) · · · i(x2n)∗i(x2n+1) : n ∈ N, x1, . . . , xn ∈ X

(notation as in 8.3.6). If Y is a full C∗-module or equivalence bimodule, then it iseasy to see that this is the same as saying that the copy of i(X) supported in the1-2-corner of the Morita linking algebra L(Y ), generates L(Y ) as a C∗-algebra.We say that two triple extensions (Y, i) and (Y ′, i′) are X-equivalent if thereexists a triple isomorphism θ : Y → Y ′ such that θ i = i′.

We define a triple envelope or noncommutative Shilov boundary of X to beany triple extension (Y, i) with the universal property of the next theorem.

Theorem 8.3.9 (Hamana) If X is an operator space, then there exists a tripleextension (Y, i) of X with the following universal property: Given any tripleextension (Z, j) of X there exists a (necessarily unique and surjective) triplemorphism θ : Z → Y , such that θ j = i.


8.3.10 (Remarks on the universal property) Before we begin the proof, we makea series of important but simple remarks stemming from the universal propertyof the theorem. These remarks parallel the discussion in 4.3.2, and provide partof the justification for the use of the term ‘noncommutative Shilov boundary’.We omit the proofs, which are almost identical to 4.3.2.

First, suppose that (Y, i) is a triple extension with the universal propertyof the theorem. Then there exists no triple ideal W of Y such that qW i is acompletely isometry on X , where qW : Y → Y/W is the canonical quotient triplemorphism. This follows by applying the universal property with j = qW i. Thesecond remark is that the set of triple extensions (Y, i) satisfying the universalproperty of the theorem, is one entire equivalence class of the relation we calledX-equivalence defined above 8.3.9. Third, if (Z, j) is any triple extension of X ,and if θ : Z → Y is the triple epimorphism provided by the universal property,and if W = Ker(θ), then (Z/W, qW j) is clearly X-equivalent to (Y, i). Thusby the second remark, (Z/W, qW j) may be taken to be a triple envelope ofX . Here again, qW : Z → Z/W is the quotient morphism. Putting these remarkstogether we obtain our fourth remark, namely that the triple envelope of X maybe taken to be any triple extension (Y, j) of X for which there exists no closedtriple ideal W of Y such that qW j is completely isometric on X .

8.3.11 (Proof of Theorem 8.3.9) For an operator space X we define T (X) andJ as in 4.4.7. Note that T (X) is simply the subtriple of I(X) generated by J(X).

Suppose that (Z, j) is a triple extension of X . We may suppose, by 8.3.6, thatZ is an equivalence C-D-bimodule. Let C1 and D1 be the unitizations of C andD respectively, and let L1(Z) be the ‘unitized linking C∗-algebra’ of 8.1.17:

L1(Z) =[

C1 ZZ D1

].

Inside L1(Z) there is a canonical copy S1 of the Paulsen system S(X). Moreover,it is easy to see that S1 generates L1(Z) as a C∗-algebra. Similarly, consider theC∗-algebra C∗

e (S(X)) generated by S(X) in I(S(X)), using the notation of 4.4.2.This is a C∗-envelope of S(X) (see the proof of 4.3.1). By 4.3.1, there exists a∗-homomorphism π : L1(Z) → C∗

e (S(X)), such that π extends the canonical mapfrom S1 to S(X). By 2.6.15, we see that π takes each of the four corners in L1(Z)to the matching corner in C∗

e (S(X)). Let θ be the 1-2-corner of π. As in 8.3.5, θis a triple morphism, and it is surjective since π is surjective.

We will henceforth write (T (X), J) for any triple envelope of X .

8.3.12 (Properties of the triple envelope)(1) As one would expect, if Z is a triple system, then Z is a triple envelope

of itself. Indeed, applying the universal property of the theorem to the identitymap j : Z → Z, we obtain a triple epimorphism θ : Z → T (Z) with θ j = i.Thus θ = i, and so θ is a triple isomorphism.

(2) If u : X1 → X2 is a surjective linear completely isometry between operatorspaces, then one may ‘extend’ u to a triple isomorphism between any triple

328 C∗-module maps and operator space multipliers

envelopes (T (X1), J1) and (T (X2), J2). Indeed a routine ‘diagram chase’ showsthat (T (X2), J2 u) is a triple envelope for X1.

(3) The triple envelope shares many of the properties of the injective envelopethat we met in Chapter 4. For example, any triple envelope of X is both a rigidand an essential extension of X , in the sense of 4.2.3. The proof is the same asthat of 4.3.6.

(4) Another useful fact is that T (Mmn(X)) ∼= Mmn(T (X)) completely iso-metrically isomorphically (or equivalently, by 4.4.6 and 8.3.2, triple isomorphi-cally) for any operator space X , and for m, n ∈ N. More generally it is true thatT (KI,J(X)) ∼= KI,J(T (X)) for arbitrary cardinals I, J . One may deduce suchrelations from the analoguous assertion for the injective envelope (see 4.2.10or 4.6.12), and the definition of the triple envelope given in the proof of 8.3.9.Similarly, although we shall not need this, T (⊕∞

i Xi) ∼= ⊕∞i T (Xi) triple iso-

morphically. Another way to prove such relations is to use the fourth remark in8.3.10 (see Appendix A in [53] for details).

(5) If A is a unital operator space, or an approximately unital operatoralgebra, then one can show that any C∗-envelope of A is a triple envelope of A.Most of this was observed at the end of 4.4.7. For the rest, see [53].

As a sample application, we give another proof of an earlier result (see 4.5.13):

Corollary 8.3.13 (A Banach–Stone theorem) If A and B are unital operatoralgebras, and if v : A → B is a linear surjective complete isometry, then thereexists a unital completely isometric isomorphism π from A onto B, and a unitaryU ∈ ∆(B), such that v = Uπ(·).Proof By (5) above, we may take T (A) = C∗

e (A), and similarly for B. By8.3.12 (2), one may ‘extend’ v to a triple isomorphism from C∗

e (A) to C∗e (B),

which we still write as v. If v(1) = U and v(a) = 1, then

U∗U = v(1)∗v(1) = v(a)v(1)∗v(1) = v(a) = 1.

Similarly UU∗ = 1, so that U is a unitary in C∗e (B). Then π = U−1v(·) satisfies

the hypotheses of 1.3.10, and hence it is a ∗-isomorphism. Thus

U−1v(a2) = π(a2) = π(a)2 = U−1v(a)U−1v(a) = U−2.

Hence U∗ = U−1 = v(a2) ∈ B, and so U ∈ ∆(B). Thus the restriction of π to Ais a unital completely isometric homomorphism onto B.

8.3.14 (The Shilov inner product) It is often convenient to take the tripleenvelope T (X) of X to be an equivalence bimodule. In this case we call therestriction to X of the associated C∗-module inner products on T (X), the Shilovinner products on X . We have already met this concept in 4.4.8.

8.4 C∗-MODULE MAPS AND OPERATOR SPACE MULTIPLIERS

In the previous section, we saw that the noncommutative Shilov boundary of anoperator space may be viewed as a C∗-module. This is pleasant, since then we


may hope to apply C∗-module methods directly to the study of operator spaces.For such applications, the multiplier algebras of an operator space considered inChapter 4, are often useful, since they are intimately connected with C∗-moduletheory, in several ways. The reader might turn to Section 4.5 for the definitionsof Ml(X) and Al(X). We now describe these algebras in C∗-module terms.

8.4.1 (Multipliers and the triple envelope) For any operator space X , let(T (X), J) be a triple envelope of X , which we may take to be a right C∗-moduleover a C∗-algebra F . The space of bounded right module maps on T (X) is anoperator space, as we mentioned after 8.2.2. We temporarily write LM(X) forthe subspace consisting of those module maps leaving J(X) invariant. We willnow show that Ml(X) ∼= LM(X) as operator algebras. That is,

Ml(X) ∼= a ∈ CBF (T (X)) : aJ(X) ⊂ J(X) (8.16)

completely isometrically isomorphically. If a ∈ LM(X), then since (iv) implies(i) in 4.5.2, a|X is a left multiplier of X . (A direct proof of this may also be given,using the last paragraph in 8.1.19.) Indeed by 4.5.2, the map a → a|X gives theisometric isomorphism in (8.16). Now Mn(CBF (T (X)) ∼= BF (Cn(T (X))), by8.2.3 (6). Thus if a = [aij ] ∈ Mn(CBF (T (X))), then a may be regarded asa module map on Cn(T (X)). By 8.3.12 (4), Cn(T (X)) is a triple envelope ofCn(X). We deduce from the isometric case of (8.16), that

Ml(Cn(X)) ∼= a ∈ CBF (Cn(T (X))) : aJn,1(X) ⊂ Jn,1(X) ∼= Mn(LM(X))

isometrically. By (4.11), we now have (8.16) completely isometrically.We next show that

Al(X) ∼= a ∈ BF (T (X)) : aJ(X) ⊂ J(X), a∗J(X) ⊂ J(X) (8.17)

∗-isomorphically. This follows from (8.16), and the fact that each side of (8.17)is just the diagonal algebra of the matching sides of (8.16) (see 4.5.7 and 8.2.4).Indeed, the diagonal algebra may be defined to be the span of the Hermitian ele-ments. The Hermitian (in this case, selfadjoint) elements of the algebra LM(X)above, clearly are in the set on the right side of (8.17). Conversely, if a is in thelatter set, then writing a = a+a∗

2 + ia−a∗2i , we see that a ∈ ∆(LM(X)).

By 8.2.4 and the last paragraph of 8.1.19, we also may regard (as subalgebras)

Al(X) ⊂ Ml(X) ⊂ KF(T (X))∗∗ ⊂ L(T (X))∗∗.

Corollary 8.4.2 Suppose that Y is a right C∗-module over a C∗-algebra B.Then Ml(Y ) = CBB(Y ) and Al(Y ) = BB(Y ). Similar assertions hold for theright multiplier algebras, indeed Mr(Y ) ∼= RM(I) and Ar(Y ) ∼= M(I), where Iis as in 8.1.4.

Proof The first two assertions are immediate from the proofs of (8.16) and(8.17) above, and using 8.3.12 (1). The second two also use 8.2.4 and the ‘other-handed version’ of facts from 8.1.14–8.1.16.

330 C∗-module maps and operator space multipliers

Corollary 8.4.3 Suppose that A is a C∗-algebra. Any nondegenerate left op-erator A-module X is a closed A-submodule, of a B-rigged A-module Z (see8.2.9), for some C∗-algebra B. One may take Z = T (X). Conversely, a closedA-submodule of a B-rigged A-module Z is a left operator A-module.

Proof If X is a left operator A-module, then by 4.6.2 (2) there is an asso-ciated ∗-homomorphism θ : A → Al(X). Using (8.17), θ may be viewed as a∗-homomorphism into BF (T (X)). By (4.7) we have θ(et)(z) → z, for z ∈ Z.That is, T (X) is an F-rigged A-module. Clearly X is an A-submodule of T (X).

Conversely, by 8.2.10 any B-rigged A-module Z is a nondegenerate left op-erator A-module; and therefore so is any A-submodule.

The following adds to the picture of adjointable multipliers on operator spacesthat we began in 4.5.8:

Theorem 8.4.4 Let X be an operator space. For a map T : X → X, the follow-ing are equivalent:(i) T ∈ Al(X).(ii) There exists a linear complete isometry σ from X into a C∗-algebra, and a

map R : X → X, such that σ(T (x))∗σ(y) = σ(x)∗σ(R(y)), for x, y ∈ X.

(iii) There exists a map R : X → X, such that

〈T (x) | y 〉 = 〈x |R(y) 〉, x, y ∈ X.

The inner product in (iii) is a Shilov inner product for X (see 8.3.14).

Proof That (i) implies (ii) follows from 4.5.8 (1), taking R = σ−1(S∗σ(·)).Suppose that T satisfies (ii), and that (T (X), J) is a triple envelope of X ,

with T (X) a full C∗-module over a C∗-algebra F . The subtriple Y generatedby σ(X) is a triple extension of X in the sense of 8.3.8. By 8.3.9, there exists atriple epimorphism θ : Y → T (X) such that θσ = J . By the proof of 8.3.5 thereis an associated ∗-homomorphism from Y Y to F , taking z∗

1z2 to 〈θ(z1)|θ(z2)〉,for z1, z2 ∈ Y . Applying this ∗-homomorphism to the equation in (ii), we obtainthe equation in (iii).

To see that (iii) implies (i), we define Bl(X) to be the set of maps T on Xsatisfying (iii). Set T ∗ = R, where R is as in (iii). It is easy to check, just asin 8.1.7, that Bl(X) is a closed subalgebra of B(X), and that ∗ is an isometricinvolution on Bl(X) satisfying the C∗-identity. Thus Bl(X) is a C∗-algebra.

Next we note that Al(X) is the range in B(X) of the isomorphism in (8.17).Explicitly, this map takes an a in the set on the right of (8.17), to the operatorTa = J−1(aJ(·)) on X . We have Ta ∈ Bl(X). Indeed a proof similar to theproof that (i) implies (ii) above, shows that a → Ta is a ∗-homomorphism intoBl(X). We therefore will be done if we can show that the range of this faithful∗-homomorphism equals Bl(X). To do this, it suffices to show that if U is aunitary in Bl(X), then U is in the range of the map above (since any unitalC∗-algebra is spanned by its unitary elements). For such U , we have


〈Ux|Uy〉 = 〈U∗Ux|y〉 = 〈x|y〉, x, y ∈ X.

Now T (X) = XF , by the centered equation in 8.3.8. Define a map taking theelement y =

∑nk=1 xkbk, to

∑nk=1 U(xk)bk, for x1, . . . , xn ∈ X, b1, . . . , bn ∈ F .

To see that this map is well defined and bounded, we compute:⟨ n∑k=1

U(xk)bk|n∑

k=1

U(xk)bk

⟩=

n∑i,j=1

b∗i 〈xi|xj〉bj = 〈y|y〉.

Hence the map above extends to a map a ∈ BF (T (X)). Clearly Ta = U .

The C∗-algebra Bl(X) in the last proof is another useful description of Al(X).It also shows why the name adjointable is appropriate for these maps.

8.4.5 (Comparisons with C∗-module maps) There are very many quite strikingparallels between multipliers (resp. adjointable multipliers) on operator spaces,and bounded module maps (resp. adjointable maps) on C∗-modules. We haveseen some of these already. We mention a few more: for example, from 4.7.4 and4.7.1 we know that for a dual operator space X , Al(X) is a W ∗-algebra, and anyu ∈ Al(X) is w∗-continuous. In the next section we shall see that such resultsplay an important role for W ∗-modules. We shall also briefly mention there someconnections with the one-sided M -ideals of Section 4.8. Indeed, a good deal ofthe results in the noncommutative M -ideal theory follow by applying C∗-moduletechniques. Many more such parallels may be found discussed in [56, 73].

In 8.1.10 we remarked that a linear map u on a right C∗-module Y is acontractive module map if and only if the map u ⊕ IY on C2(Y ) is contractive.This of course is the analogue for C∗-modules of condition 4.5.2 (ii), whichcharacterizes operator space multipliers. Beginning from this fact from 8.1.10,and using basic facts about C∗-modules met early in this chapter, one may giveanother development of the theory of operator space multipliers that we saw inChapter 4, but avoiding many of the technical details about the injective envelopeused in Section 4.5. See [56] for details. Such an approach is close to the originaldevelopment of the operator space multiplier theory (see [53]). For example, wegive a quick proof that (ii) implies (iv) in Theorem 4.5.2. Condition (ii) theresays that u ⊕ IX : C2(X) → C2(X) is completely contractive. By 1.2.11, I(X)is ‘completely complemented’ in B(H) for some H . Thus there is a completelycontractive ∞2 -module map projection C2(B(H)) ∼= B(H, H(2)) → C2(I(X)).Hence, by 3.6.2, we can extend u ⊕ IX to a completely contractive ∞2 -modulemap C2(I(X)) → C2(I(X)) ⊂ C2(B(H)). Such a map is necessarily of the formu ⊕ v. By the ‘rigidity’ property of I(X), v = IX . Since I(X) is a C∗-module,by the result at the start of this paragraph, u is a contractive module map onI(X). By (4.7), it restricts to a contractive module map on T (X).

8.5 W ∗-MODULES

The theory of W ∗-modules may be thought of as a ‘dual variant’ of the theoryof C∗-modules. Indeed our development in this section is parallel, to some ex-

332 W ∗-modules

tent, to the pattern of Sections 8.1 and 8.2 above. However, we shall see thatW ∗-modules are quite a bit simpler than C∗-modules. For example, W ∗-modulesbehave much more like Hilbert spaces, and there is a very powerful ‘stable iso-morphism theorem’ (8.5.28 below) valid for all W ∗-modules, which is very usefulfor operator space applications.

Throughout this section, M and N are W ∗-algebras.

8.5.1 (The definitions) We say that a right C∗-module Y over a C∗-algebra Ais selfdual if every bounded A-module map u : Y → A is of the form u(·) = 〈z|·〉,for some z ∈ Y . We say that Y is a right W ∗-module if Y is a selfdual rightC∗-module over a W ∗-algebra.

If Y is a right C∗-module over a W ∗-algebra M , then we will consistentlywrite Iw for the w∗-closure in M of the span of the range of the M -valued innerproduct on Y (recall from 8.1.4 that I is the norm closure of this span). It iseasy to see, using simple w∗-approximation arguments (e.g. see 2.7.4 (4)), thatIw is a w∗-closed ideal in M . We say that Y is w∗-full if Iw = M .

In the following remarks, Y is a selfdual right C∗-module over A.(1) It follows for example from 8.1.11 (2) and 8.2.15, that

CBA(Y, A) ∼= Y completely isometrically. (8.18)

(2) If Z is another right C∗-module over A, then

BA(Y, Z) = BA(Y, Z), and BA(Y ) = BA(Y ). (8.19)

Indeed, the fact that any u ∈ BA(Y, Z) is adjointable follows by considering theA-valued map 〈z|u(·)〉, for fixed z ∈ Z.

(3) The adjoint C∗-module Y (see 8.1.1) is a selfdual left C∗-module overA. We leave the details as an exercise.

(4) Selfduality is an operator space invariant. That is, if Z is another C∗-module over a possibly different C∗-algebra B, say, and if Y and Z are linearlycompletely isometric, then Z is selfdual as a B-module. To see this, one first usesthe next result to see that we can assume that Y and Z are both full. By 4.4.6,Y and Z are triple isomorphic. The rest is a pleasant algebraic exercise.

Lemma 8.5.2 Let Y be a right C∗-module over a C∗-algebra A, and let I be asin 8.1.4. Then BA(Y, A) = BC(Y, D) as sets, for any C ∈ A, A1, M(A), I andD ∈ A, A1, M(A), I, M(I). Hence Y is selfdual as an A-module if and only ifY is selfdual as a D-module, for any D ∈ A1, M(A), I, M(I).Proof Suppose that u ∈ BI(Y, D). By Cohen’s theorem A.6.2, we may writeany y ∈ Y as y = y1a1a2 for a1, a2 ∈ I. Hence u(y) = u(y1)a1a2 ∈ AI ⊂ I.Moreover, if c ∈ C then u(yc) = u(y1a1(a2c)) = u(y1)a1(a2c) = u(y)c. Thus

BI(Y, D) ⊂ BC(Y, I) ⊂ BI(Y, I) ⊂ BI(Y, D).

Thus BC(Y, D) = BI(Y, D) = BI(Y, I), for any C, D as above.The last assertion follows immediately from the first.


Proposition 8.5.3 Suppose that Y is a right C∗-module over a W ∗-algebra M .Then Iw is the multiplier algebra of I.

Proof Clearly I is a w∗-dense ideal in Iw . Fix a faithful unital w∗-continuous∗-representation π of Iw on H , say. This is easily seen, using 2.1.9, to be anondegenerate ∗-representation of I. By Section 2.6, we may identify M(I) withT ∈ B(H) : Tπ(I) ⊂ π(I), π(I)T ⊂ π(I). In fact the latter algebra clearlycontains π(Iw), and is a subalgebra of π(I)′′ by 2.6.5. On the other hand,

π(I)′′ = π(I)w∗

⊂ π(Iw),

by the double commutant theorem. Thus π(Iw) = M(I).

Lemma 8.5.4 Let Y be a right C∗-module over a W ∗-algebra M . Then:(1) Y is a W ∗-module over M if and only if Y has a Banach space predual with

respect to which the inner product on Y is separately w∗-continuous.If Y is a W ∗-module, then:(2) Y has a unique Banach space predual with respect to which the inner product

on Y is separately w∗-continuous.(3) With respect to the w∗-topology induced by the predual in (2), a bounded net

(xt)t converges to x in Y if and only if 〈y|xt〉 → 〈y|x〉 in the w∗-topology ofM , for all y ∈ Y .

(4) Let W = M∗⊗M Y (see Section 3.4). Then W is an operator space predual

of Y inducing the w∗-topology in (2) and (3) above.

Proof We remark that (4) and its proof, and the proof of (2), are simpler if onereplaces M∗

⊗M Y by the (quite analoguous, but much simpler) Banach module

version of the projective tensor product. To understand the argument better, thereader may want to consider this variant first.

If Y is a W ∗-module then by 3.5.10 and the other-handed version of (8.18),

(M∗⊗M Y )∗ ∼= MCB(Y , M) ∼= Y,

completely isometrically. Unraveling these isomorphisms yields a completely iso-metric surjection ρ : Y → W ∗, given by

ρ(x)(ψ ⊗ y) = ψ(〈y|x〉), ψ ∈ M∗, x, y ∈ Y.

By the norm density of the finite rank tensors from M∗⊗Y in W , it is clear thatwith respect to the w∗-topology on Y induced by W , the ‘if and only if’ conditionin (3) holds. Thus by A.2.5, the inner product is separately w∗-continuous withrespect to this topology.

Define θ : W → Y ∗ by θ(ψ ⊗ y)(x) = ψ(〈y|x〉), for ψ ∈ M∗ and x, y ∈ Y . Itis easy to check that θ = (ρ∗)|W . Suppose that Y∗ is a predual of Y , regarded asa subspace of Y ∗, for which the inner product on Y is separately w∗-continuous.

334 W ∗-modules

Then θ maps into Y∗. Viewing θ as a map into Y∗, θ∗ corresponds to the mapρ above. Hence θ is a (completely) isometric surjection onto Y∗, and ρ is ahomeomorphism for the w∗-topologies. We have proved (2)–(4).

The ‘only if’ in (1) follows from the proof of (4). On the other hand, supposethat Y has a Banach space predual, and that the M -valued inner product is sep-arately w∗-continuous. Let u : Y → M be a bounded M -module map. By 8.1.23,we may choose a cai (et)t for KM (Y ), with terms of the form

∑nk=1 |xk〉〈xk| for

some xk ∈ X . For x ∈ X , we have

u(et(x)) =n∑

k=1

u(xk)〈xk|x〉 =⟨ n∑

k=1

xku(xk)∗∣∣∣x⟩ = 〈wt|x〉, (8.20)

where wt =∑n

k=1 xku(xk)∗ (which depends on t). It follows from 8.1.11 (2) that‖wt‖ = ‖u et‖ ≤ ‖u‖. Thus (wt)t is a bounded net in Y , and so it has a w∗-convergent subnet, with limit w say. Replace the net with the subnet. By thehypothesis, 〈wt|x〉 → 〈w|x〉. Since u(et(x)) → u(x) in norm, by (8.20) we haveu(x) = 〈w|x〉, for all x ∈ Y . Thus Y is selfdual over M .

We will henceforth use the phrase the w∗-topology of a W ∗-module Y , for the(unique) topology in (2)–(4) above.

Corollary 8.5.5 Suppose that Y is a right W ∗-module over M . Then:(1) BM (Y ) = BM (Y ), and this is a W ∗-algebra.(2) A bounded net (Ti)i in BM (Y ) converges in the w∗-topology to T ∈ BM (Y )

if and only if Ti(y) → T (y) in the w∗-topology of Y , for all y ∈ Y . Indeed,Y

⊗M W is a predual for BM (Y ), where W is as in 8.5.4 (4).

Proof By 3.5.10, 8.5.4 (4), and (8.19), we have

(Y⊗M W )∗ ∼= CBM (Y, W ∗) = CBM (Y ) = BM (Y ).

The latter space is a C∗-algebra, and it is therefore a W ∗-algebra by the theo-rem of Sakai mentioned at the start of Section 2.7. The assertion involving netconvergence is proved similarly to the analoguous statements in 8.5.4 (4).

There is a stronger variant of the following, which assumes only a Banachspace predual. The version here will suffice for most operator space applications.Our proof showcases the multipliers from Chapter 4, and has the advantage ofgeneralizing to the nonselfadjoint algebra situation. See the Notes for details.

Theorem 8.5.6 (Zettl, Effros–Ozawa–Ruan) Let Y be a full right C∗-moduleover a C∗-algebra A, and suppose that Y has an operator space predual. If M isM(A) then M and BA(Y ) are W ∗-algebras, and Y is a w∗-full W ∗-module overM . Moreover, Y has a unique operator space predual, the space in 8.5.4 (4).

Proof Let Y∗ be a fixed operator space predual of Y . We will use the fact from8.4.2 that Al(Y ) = BM (Y ), and Ar(Y ) = M(A) = M . By 4.7.4 we know that


Al(Y ) and Ar(Y ) are W ∗-algebras, and hence so are BM (Y ) and M . By 4.7.5,the canonical trilinear map BM (Y ) × Y × M → Y is separately w∗-continuous.

We will now check that the inner product on Y is separately w∗-continuous,with respect to the w∗-topology determined by Y∗. To this end, suppose that wehave a bounded net (yt)t converging to y in the w∗-topology of Y . Fix x, w inY . By the above, the ‘rank one’ operator |w〉〈x| is w∗-continuous on Y . Hencew〈x|yt〉 → w〈x|y〉 in the w∗-topology. Let (〈x|ytµ〉) be a w∗-convergent subnetof the bounded net (〈x|yt〉), converging to b ∈ M say. By the last paragraph, wehave that w〈x|ytµ 〉 → wb in the w∗-topology. Hence wb = w〈x|y〉. Since this istrue for all w ∈ Y , it follows from 8.1.4 (1) that b = 〈x|y〉. Hence 〈x|yt〉 → 〈x|y〉in the w∗-topology. By A.2.5 (2), the inner product is separately w∗-continuous.By 8.5.4 and 8.5.3, Y is a w∗-full W ∗-module over M , and we have the otherconsequences stated in 8.5.4 and 8.5.5. The uniqueness follows easily from 8.5.4(2), and from basic operator space duality principles (see Section 1.4).

Corollary 8.5.7 Let Y be a right C∗-module over a W ∗-algebra M . Then Y isa W ∗-module if and only if Y has an operator space predual. In this case theoperator space predual is unique.

Proof The ‘only if’ follows from 8.5.4 (4). For the other direction, by 8.5.3,M(I) = Iw. Thus if Y has an operator space predual, then, by 8.5.6, Y is aW ∗-module over Iw. By 8.5.2, Y is also selfdual over M . The uniqueness wasproved in 8.5.6.

Corollary 8.5.8 A bounded module map u : Y → Z between W ∗-modules overM , is w∗-continuous.

Proof By (8.19), u is adjointable. We will apply A.2.5 (2). If (yt)t is a boundednet converging to y ∈ Y in the w∗-topology of Y , then

〈u(yt)|z〉 = 〈yt|u∗(z)〉 −→ 〈y|u∗(z)〉 = 〈u(y)|z〉, z ∈ Y.

By 8.5.4 (3), u(yt) → u(y) in the w∗-topology. Thus u is w∗-continuous.

We separate one other interesting fact, which follows easily from 8.2.3 (1),and the first paragraph of the proof of 8.5.6, for example.

Corollary 8.5.9 A right W ∗-module Y over a W ∗-algebra M is a normal dualoperator BM (Y )-M -bimodule in the sense of 3.8.2.

8.5.10 (The linking W ∗-algebra) If Y is a right W ∗-module over M , then wedefine the linking W ∗-algebra of Y to be Lw(Y ) = BM (Y ⊕c M). This equalsBM (Y ⊕c M) by (8.19), since Y ⊕c M is a selfdual M -module, as is easily veri-fied. As in 8.1.17, by considering the adjointable inclusion and projection mapsbetween Y ⊕c M and its two summands, it is clear that Lw(Y ) may be viewed asa 2×2 matrix algebra with corners BM (Y ), Y, Y and M . Thus any W ∗-module isa corner eN(1− e), for a W ∗-algebra N and a projection e ∈ N . The converse isalso true, namely that if e is a projection in a W ∗-algebra N , then Y = eN(1−e)

336 W ∗-modules

is a right W ∗-module over (1−e)N(1−e). This may be seen by using 8.5.4 if nec-essary. Thus we obtain another ‘picture’ of W ∗-modules, namely as the cornersof W ∗-algebras. This should be compared with 8.1.19.

The linking W ∗-algebra of a W ∗-module is very useful when it comes tocalculations, because the inner product and module actions have been replacedby multiplication in the W ∗-algebra, just as we saw in 8.1.19. As one illustrationof this principle, we invite the reader to check that the proofs in 8.2.24 and 8.2.25may be adapted in an obvious way to give the analoguous results for w∗-closedideals and quotients. In fact the W ∗-module versions of these results are simpler,due to the well-known correspondence between two-sided w∗-closed ideals in aW ∗-algebra and central projections.

8.5.11 (W ∗-closed TROs) Via the linking W ∗-algebra, one may now view W ∗-modules as the w∗-closed subtriples of B(K, H), or of a W ∗-algebra. Indeed, aswe just saw, we may write any W ∗-module Y over M , as a corner eN(1− e) of aW ∗-algebra N , with M ∼= (1− e)N(1− e). Suppose that N has been representedfaithfully as a von Neumann subalgebra of B(H) say. Then the projection edetermines a splitting H = H1⊕H2 say, and it is evident that M corresponds toa von Neumann algebra in B(H2), and Y corresponds to a w∗-closed subtriple,and an M -submodule, of B(H2, H1). Another useful way to represent Y , is tosuppose that M is a von Neumann algebra in B(K), and to consider the isometryΦ: Y → B(K, Y ⊗M K) from 8.2.14 satisfying Φ(y)∗Φ(z) = 〈y|z〉, for y, z ∈ Y ;with Y unitarily isomorphic to the TRO Φ(Y ). We leave it as an exercise thatΦ is a w∗-homeomorphism onto Φ(Y ), which is w∗-closed. (Hint: use A.2.5 and8.5.4 (3), and simple net arguments of the kind found in 8.5.36, for example.)

Conversely, using 8.5.4, for example, it is easy to check that any w∗-closedsubtriple Z of B(K, H), or of a W ∗-algebra, is a w∗-full right W ∗-module overthe w∗-closure of ZZ. Indeed such a space is a W ∗-equivalence bimodule in thesense described next:

8.5.12 (Weak Morita equivalence) W ∗-equivalence M -N -bimodules are definedanaloguously to the equivalence bimodules in 8.1.2, with the words ‘C∗-module’replaced by ‘W ∗-module’, and ‘full’ by ‘w∗-full’. If there exists such a bimoduleover M and N , then we say that M and N are weakly Morita equivalent. It isnot hard to see that weak Morita equivalence is an equivalence relation coarserthan ∗-isomorphism of W ∗-algebras (see the Notes to this section).

Corollary 8.5.13 If Y is a W ∗-equivalence M -N -bimodule, then M ∼= BN (Y )∗-isomorphically.

Proof By the ‘left-handed’ version of 8.5.3, and by 8.1.15, M ∼= M(KN (Y )).Now the result follows by 8.1.16 (3), and (8.19).

8.5.14 (W ∗-modules are W ∗-equivalence bimodules) Analoguously to 8.1.14,any right W ∗-module Y over a W ∗-algebra N is a w∗-full left W ∗-module overBN (Y ). That it is selfdual as a left module follows, for example, from 8.5.7. Tosee that it is w∗-full, suppose that p is the support projection in BN (Y ) for the


w∗-closed ideal generated by the ‘rank one’ operators. For all x, y, z ∈ Y , wehave 0 = (1− p)|y〉〈z|(x) = (1 − p)y〈z|x〉. By 8.1.4 (2), 1− p = 0, so that p = 1.

Consequently, if Y is a w∗-full right W ∗-module over N , then Y is a W ∗-equivalence BN (Y )-N -bimodule. More generally, if Y is a right W ∗-module overN then Y is a W ∗-equivalence BN (Y )-Iw-bimodule, where Iw is as above.

Conversely, if Y is a W ∗-equivalence M -N -bimodule, then by 8.5.3 and theproof of 8.5.13, we have M ∼= M(KN (Y )) and N ∼= M(I). Since Y is an equiv-alence KN (Y )-I-bimodule (see 8.1.14), it follows from 8.1.20 that M and Nhave isomorphic centers. Also, as in 8.1.18, we have that if Y is a w∗-full rightW ∗-module over N , then N and Lw(Y ) are weakly Morita equivalent.

8.5.15 (W ∗-summands) If Y is a right W ∗-module over M then by 3.8.11,Cw

I (Y ) is a dual operator M -module for any cardinal I. It is easy to see thatCw

I (Y ) is also a W ∗-module. For example, if Y is represented, as in 8.5.11, bothas an M -submodule, and as a w∗-closed subtriple, of B(K, H), with M acting asa von Neumann subalgebra of B(K), then (if necessary, by simple arguments ofthe kind in 3.8.10–3.8.12) Cw

I (Y ) may be identified with an M -submodule, andw∗-closed subtriple, W of B(K, H(I)). By 8.5.4 (1), W is a W ∗-module over M ,with inner product S × T → S∗T . It is then easy to see that the correspondinginner product on Cw

I (Y ) is given by

〈(yi)|(zi)〉 =∑i∈I

〈yi|zi〉, (yi), (zi) ∈ CwI (Y ),

where the convergence of the sum is in the w∗-topology of M . The columnswith a finite number of nonzero entries are w∗-dense in Cw

I (Y ). Indeed if PJ isthe projection from Cw

I (Y ) onto the set of ‘columns supported on CJ(Y )’, fora finite subset J of I, then (PJ )J converges in the w∗-topology of BM (Cw

I (Y ))to the identity map on Cw

I (Y ), by 8.5.5 (2) and 1.6.3. From this it is not hardto show that BM (Cw

I (Y )) ∼= MI(BM (Y )) as W ∗-algebras. Since we shall not usethis fact, we omit the proof. In [73], the interested reader will find the proof ofan operator space generalization of this fact. Namely, Al(Cw

I (X)) ∼= MI(Al(X))as W ∗-algebras, for any dual operator space X .

Similar assertions hold for RwI (X), for a left W ∗-module X over M .

We say that a submodule Y of a W ∗-module Z is w∗-orthogonally comple-mented in Z, if Y is orthogonally complemented in the sense of 8.1.9, and if Yis a w∗-closed subspace of Z. In this case Y is a W ∗-module too, by 8.5.4.

Lemma 8.5.16 Suppose that Z is a right W ∗-module over a W ∗-algebra M ,and that Y is a subspace of Z. The following are equivalent:

(i) Y is an orthogonally complemented M -submodule of Z,(ii) Y is a w∗-orthogonally complemented M -submodule of Z,(iii) Y is a w∗-closed M -submodule of Z,(iv) Y is a right M -summand of Z, in the sense of 4.8.1.

338 W ∗-modules

Proof Clearly (ii) implies (i) and (iii). If P is a projection in BM (Y ), thenP is w∗-continuous by 8.5.8. Thus Ran(P ) is w∗-closed. Therefore (i) implies(ii). By definition, and by 4.5.15 (iii) and 8.4.2, the right M -summands of Z areprecisely the ranges of the adjointable projections on Z. Thus (iv) is equivalent to(i). Finally, given (iii), it follows from 8.5.4 (1) that Y is selfdual as an M -module.Hence the inclusion map from Y into Z is adjointable by (8.19). Its adjoint iseasily checked to be an orthogonal projection onto Y , giving (i).

Proposition 8.5.17 If Y is a right C∗-module over a C∗-algebra B, then Y ∗∗

with its canonical B∗∗-module action (see 3.8.9), is a W ∗-module over B∗∗. IfY is an equivalence A-B-bimodule, then Y ∗∗ is a W ∗-equivalence bimodule overA∗∗ and B∗∗.

Proof First suppose that Y is full over B. We recall that Y ∼= pL(1 − p), asin (8.3). Here L is the linking C∗-algebra of Y , and p is a projection in M(L).By replacing Y with pL(1 − p), we may assume that Y is a subtriple of L, andthat B = Y Y ⊂ L. Let Z be the w∗-closure of Y in pL∗∗(1 − p). Clearly,Z = pL∗∗(1 − p), a w∗-closed subtriple of L∗∗. By A.2.3, Y ∗∗ ∼= Z completelyisometrically, and w∗-homeomorphically. Similarly, if N = (1−p)L∗∗(1−p), thenB∗∗ ∼= N , as W ∗-algebras. We have B = Y Y ⊂ ZZ ⊂ N . Thus the w∗-closureof ZZ contains the w∗-closure of B in L∗∗, namely N . Thus the w∗-closureof ZZ equals N . By 8.5.11, Z is a w∗-full right W ∗-module over N . We maytransfer these structures, to make Y ∗∗ a w∗-full right W ∗-module over B∗∗. ThisB∗∗-module action on Y ∗∗ coincides with the canonical second dual action from3.8.9. This is because the product map Z ×N → Z is separately w∗-continuous,and extends the canonical map Y × B → B. A similar argument yields the lastassertion of the Proposition.

If Y is not full over B then we consider the ideal I from 8.1.4. By the above,Y ∗∗ is a right W ∗-module over I∗∗. However I∗∗ is an ideal in B∗∗, and so Y ∗∗

is a W ∗-module over B∗∗, by 8.5.2 and 8.1.4 (4).

8.5.18 (Dual triple systems) Appropriate weak* versions of the theory of TROsand triple systems presented in the first half of Section 8.3, also go throughwithout difficulty. In fact the theory becomes simpler, due to the correspondencebetween w∗-closed two-sided ideals in a W ∗-algebra and central projections.

By 8.5.6 and the remarks above it, a TRO Y which has a predual, is a w∗-fullW ∗-module over M(Y Y ). Putting this together with 8.5.11 and 8.5.14, we seethat such ‘dual TROs’; the triple systems which have a predual; W ∗-modules;and W ∗-equivalence bimodules, are essentially the same thing, in a sense similarto the discussion in 8.3.1.

A w∗-continuous triple morphism u : Y → Z between w∗-closed TROs, hasrange which is a w∗-closed TRO. Indeed, let W = Ker(u), which is a w∗-closedtriple ideal in Y . By the last several lines of 8.5.10, and the argument for 8.3.3,there is a central projection e such that W = eY . Let W ′ = (1−e)Y , then uW ′ isa one-to-one w∗-continuous triple morphism with Ran(u) = Ran(uW ′). By 8.3.2,uW ′ is isometric, so that by A.2.5 its range is w∗-closed.


If Y is a TRO, then Y ∗∗ is a TRO and a W ∗-equivalence bimodule, as inthe proof of the last Proposition, for example. Note that if u : Y → B(K, H) isa triple morphism, then by routine w∗-approximation arguments, the canonicalw∗-continuous extension u : Y ∗∗ → B(K, H) (see A.2.2) is a triple morphism too.As one application of this, we deduce that any subtriple Y of B(K, H) satisfiesa ‘Kaplansky density theorem’, namely that Ball(Y ) is w∗-dense in the unit ballof the w∗-closure of Y . This follows from the above, 8.3.4 (3), and A.5.10.

A completely isometric surjective linear map (or equivalently, a surjectivetriple isomorphism) between dual TROs, is automatically w∗-continuous. Thisfollows from the uniqueness of the predual of a W ∗-module (see 8.5.7).

Theorem 8.5.19 The right M -ideals (see 4.8.1) in a right Hilbert C∗-moduleare exactly the closed right submodules.

Proof Let Z be a right C∗-module over a C∗-algebra B. If Y is a right M -idealof Z, then the w∗-closure W of Y in Z∗∗ is a right M -summand of Z∗∗. By8.5.16, and using 8.5.17, we see that W is a B∗∗-submodule of Z∗∗. Viewed assubsets of Z∗∗, we have

Y B ⊂ (WB∗∗) ∩ Z ⊂ Y ⊥⊥ ∩ Z = Y,

using A.2.3 (4). Thus Y is a B-submodule of Z.Conversely, if Y is a B-submodule of Z, then its w∗-closure W in Z∗∗, is a

B∗∗-submodule. Indeed this follows from the fact that the B∗∗-module action onZ∗∗ is separately w∗-continuous, and A.2.1. By 8.5.16, W is a right M -summandof Z∗∗, so that Y is a right M -ideal of Z.

8.5.20 (C∗-modules and M -ideals) In fact one may view the theory of one-sidedM -ideals in operator spaces, introduced briefly in Section 4.8, as a generalizationof the behaviour of submodules of C∗-modules. See [56, 57, 73] for details.

In connection with the last result, we note that the classical M -ideals (see4.8.1) in an equivalence A-B-bimodule are exactly the A-B-submodules. Onedirection of this is not hard. For example, if Y is an A-B-submodule of anequivalence A-B-bimodule Z, then by 8.5.19, Y is both a left and a right M -ideal of Z. By 4.8.4, Y is a complete M -ideal, and hence an M -ideal. The reversedirection seems to be harder. Suppose that Y is an M -ideal in Z. If a ∈ Asa,consider the map Tz = az, for z ∈ Z. It is easy to see that T ∈ Her(B(Z)) (seeA.4.2), and so by [195, Corollary I.1.25], aY ⊂ Y . Thus Y is a left A-submodule,and similarly it is a right B-submodule. Hence the M -ideals in a TRO Y are the(Y Y )-(Y Y )-submodules.

8.5.21 (Partial isometries in C∗-modules) We say that an element u in a rightC∗-module Y over M is a partial isometry if p = 〈u|u〉 is an orthogonal projectionin M . This element p is called the initial projection of u. Note that it followsthat up = u (since 〈u− up|u− up〉 = 0). Thus |u〉〈u| is an orthogonal projectionin the W ∗-algebra BM (Y ).

340 W ∗-modules

We say that two partial isometries u and v in Y are orthogonal if 〈u|v〉 = 0.In this case the orthogonal projections |u〉〈u| and |v〉〈v| are mutually orthogonal.

Lemma 8.5.22 Suppose that Y is a right W ∗-module over M , and that y ∈ Y .Then y = u|y|, where |y| = 〈y|y〉 1

2 ∈ M , and u is a partial isometry in Y whoseinitial projection is the range projection of |y| in the von Neumann algebra sense(see 2.2.7 in [320]).

Proof We view y as an element of the linking W ∗-algebra of Y (see 8.5.10),and take its polar decomposition there, as in 2.2.9 of [320]. It is easy to see fromthe formula given there for u, that u ∈ Y , that u∗u ∈ M , and that the latter isthe range projection of |y| in M .

Lemma 8.5.23 (Paschke) Let Y be a right W ∗-module. Then Y has an or-thonormal basis. That is, there exists a set xii∈I in Y consisting of mutu-ally orthogonal nonzero partial isometries, such that x =

∑i xi〈xi|x〉 in the

w∗-topology of Y , for all x ∈ Y . In particular,∑

i∈I |xi〉〈xi| converges in thew∗-topology of BM (Y ), to IY .

Proof We consider the subsets B of Y consisting of mutually orthogonal nonzeropartial isometries in Y , ordered by inclusion. At least one such set exists by8.5.22, and by Zorn’s lemma we may choose a maximal such set, xi : i ∈ Isay. We first claim that if 〈xi|x〉 = 0 for all i ∈ I, then x = 0. To see this, writex = u|x| for a partial isometry u ∈ Y as in 8.5.22. Then 〈xi|u〉|x| = 0. If p isthe initial projection of u, then u = up, and so 〈xi|u〉p|x| = 0. Since p is therange projection for |x|, we see that 〈xi|u〉p = 0 = 〈xi|u〉. This contradicts themaximality above, if u = 0.

By the remarks before 8.5.22, TJ =∑

i∈J |xi〉〈xi| is an orthogonal projectionin BM (Y ), for any finite subset J ⊂ I. We therefore have

〈TJx|x〉 =∑i∈J

〈x|xi〉〈xi|x〉 ≤ 〈x|x〉, x ∈ X.

The increasing net (TJ )J converges in the w∗-topology of BM (Y ) to an operatorT say, with 0 ≤ T ≤ I. By 8.5.5 (2), for any x ∈ X the sum

∑i xi〈xi|x〉 converges

in the w∗-topology of Y , to T (x). Taking the inner product with xj , for a fixedj ∈ I, we have 〈xj |T (x)〉 = 〈xj |x〉, by the w∗-continuity of the inner product.Hence T (x) = x by the first part of the proof.

8.5.24 (The Parseval identity) Suppose that xi : i ∈ I is an orthonormalbasis for Y , as in 8.5.23. It follows from the proof above, and 8.5.4 (1), that∑

i∈I

〈y|xi〉〈xi|y〉 = 〈y|y〉, y ∈ Y,

the convergence in the w∗-topology. Since∑

i |xi〉〈xi| = I, we see using (8.7) ifnecessary, that (xi) is an element of Rw

I (Y ), and has norm there equal to 1.As a consequence, we obtain another characterization of W ∗-modules, as

exactly the w∗-closed submodules of CwI (M):


Corollary 8.5.25 A Banach module Y over a W ∗-algebra M is a right W ∗-module over M if and only if Y is isometrically M -isomorphic to an orthogonallycomplemented submodule of Cw

I (M), for some cardinal I.

Proof The ‘only if’ follows from 8.5.16 and the remark above it. Conversely, ifY is a right W ∗-module over M , let xi : i ∈ I be an orthonormal basis for Yas in 8.5.23. Define α : Y → Cw

I (M) by α(y) = (〈xi|y〉)i, for y ∈ Y . By 8.5.24, αis an isometry. Also, α is w∗-continuous with respect to the w∗-topology of Y ,by 1.6.3 (2) and 8.5.4 (1), and is an M -module map. By A.2.5, the range of α isw∗-closed. Now apply 8.5.16.

8.5.26 (The ultraweak direct sum) We define the ultraweak direct sum ⊕wci∈I Yi

of a family Yi : i ∈ I of right W ∗-modules over a W ∗-algebra M , to be the setof (yi)i∈I ∈

∏i∈I Yi, such that the finite partial sums of the series

∑i∈I 〈yi|yi〉

are uniformly bounded above. Equivalently, it is the set of (yi)i∈I such that∑i∈I 〈yi|yi〉 converges in the w∗-topology of M . It is easy to check, using the

polarization identity (1.1), that for (yi) and (zi) in ⊕wci∈I Yi, the finite partial

sums of∑

i∈I〈yi|zi〉 converge in the w∗-topology of M . We write 〈(yi)|(zi)〉 forthe w∗-limit. Most of the conditions in the definition of a C∗-module are easy tocheck for ⊕wc

i∈I Yi, and all will follow from considerations in the next paragraph.Note that if Y is a right W ∗-module, then the W ∗-module Cw

I (Y ) met in 8.5.15equals the ultraweak direct sum of I copies of Y .

Although we shall not use the general ultraweak direct sum much, we mentionsome of its properties. For these, it is helpful to view Y = ⊕wc

i∈I Yi in a slightlydifferent way. We begin with a faithful normal representation of M on a Hilbertspace K. We suppose that for each i ∈ I, Yi is represented (as in 8.5.11 say)as a w∗-closed M -submodule, and a subtriple, of B(K, Hi), for a Hilbert spaceHi. Set H = ⊕i Hi, and let Pi be the projection from H onto Hi. We also setW = T ∈ B(K, H) : PiT ∈ Yi for all i ∈ I, and equip W with its canonicalinner product S × T → S∗T . For S, T ∈ W we have S∗T =

∑i S∗PiPiT , which

is a w∗-convergent sum in M . Hence this inner product is valued in M . It is easyto see that W is a w∗-closed M -submodule, and subtriple, of B(K, H). From8.5.4 (1), it follows that W is a right W ∗-module over M . Writing yi = PiT ,for T ∈ W , it becomes evident that W corresponds precisely to the definitionof ⊕wc

i∈I Yi above. In other words, W is unitarily M -isomorphic to this sum. Itfollows that ⊕wc

i∈I Yi is a right W ∗-module over M .One may deduce from the above description, and the associativity property

for Hilbert space sums, that the ultraweak direct sum is associative. Thus, forexample, ⊕wc

i∈I(⊕wcj∈JYij) ∼= ⊕wc

j∈J (⊕wci∈I Yij) ∼= ⊕wc

i,j Yij unitarily, for right W ∗-modules Yij over M . Also, one can easily see that the set of tuples in an ultraweakdirect sum which are zero except in a finite number of entries, is w∗-dense.

We shall not use this, but it can be deduced from 8.5.23 that the right W ∗-modules over a W ∗-algebra M , are exactly the ultraweak direct sums of w∗-closedright ideals pM of M . See [302,421].

342 W ∗-modules

8.5.27 (Second duals of C∗-module sums) If (Yi) is a family of C∗-modules overa C∗-algebra B, then (⊕c

i Yi)∗∗ ∼= ⊕wci Y ∗∗

i unitarily as B∗∗-modules. We merelysketch the proof. Let Y = ⊕c

i Yi, let Z = ⊕wci Y ∗∗

i , and let L be the ‘augmentedlinking algebra’ KB(Y ⊕c B). Let p0 be the projection of Y ⊕c B onto 0 ⊕ B,and similarly let pi be the projection of Y ⊕c B onto the copy of Yi. We regardpi ∈ M(L) ⊂ L∗∗. In the latter W ∗-algebra one can show that

∑i pi = (1 − p0).

As in the proof of 8.5.17, we have Yi∼= piLp0, and Y ∼= (1 − p0)Lp0, unitarily

as right B-modules. Also as in the proof of 8.5.17, we have Y ∗∗i

∼= piL∗∗p0 andY ∗∗ ∼= (1 − p0)L∗∗p0, unitarily as right B∗∗-modules. However it is not hardto check, using basic facts about w∗-limits of increasing nets of projections in aW ∗-algebra, that

⊕wci Y ∗∗

i∼= ⊕wc

i piL∗∗p0∼= (1 − p0)L∗∗p0,

unitarily as right B∗∗-modules. This proves the result.

A powerful tool associated with W ∗-modules, and indeed operator spaces, isthe following weak* variant of the stabilization result in 8.2.6.

Theorem 8.5.28 Let Y be a w∗-full right W ∗-module over a W ∗-algebra N .Then there exists a cardinal I such that Cw

I (Y ) ∼= CwI (N) unitarily (as right

N -modules). Also, MI(Y ) is linearly completely isometrically isomorphic (via aright N -module map) to the W ∗-algebra MI(N).

Proof We prove this very similarly to our arguments for 8.2.6, which the readershould follow along with (another argument is sketched in the Notes). By 8.5.25and 8.5.16, there exists a cardinal I and a w∗-closed N -submodule W of Cw

I (N),such that Y ⊕c W ∼= Cw

I (N) unitarily (as right N -modules). By set theory wemay assume that I2 = I. It follows from this, and (1.59) for example, thatCw

I (CwI (N)) ∼= Cw

I (N). Using the latter fact, and using ‘associativity’ of theultraweak sum (see 8.5.26), the ‘Eilenberg swindle’ works, similarly to the proofof 8.2.6 (1), to give:

CwI (N) ∼= Cw

I (Y )⊕c CwI (W ) ∼= Cw

I (Y )⊕c CwI (Y )⊕c Cw

I (W ) ∼= CwI (Y )⊕c Cw

I (N),

unitarily as N -modules. We may assume by 8.5.14 that Y is a W ∗-equivalenceM -N -bimodule. By (the ‘other-handed variant’ of) Lemma 8.5.23, and since N isisomorphic to the algebra of left M -module maps on Y (see 8.5.14), there existsa cardinal J and a subset zj : j ∈ J of Y , such that 1N =

∑j〈zj |zj〉 in the

w∗-topology of N . Thus (zj) ∈ CwJ (Y ), which permits us, exactly as in the proof

of 8.2.6 (3) (and in fact a little more easily), to define maps showing via 8.1.21that N is unitarily isomorphic to an orthogonally complemented N -submoduleof Cw

J (Y ). Thus by 8.5.16, there exists a w∗-closed submodule P of CwJ (Y ) such

that N ⊕c P ∼= CwJ (Y ), By the argument in the first paragraph of this proof, we

obtain that CwJ (Y ) ∼= Cw

J (N)⊕c CwJ (Y ) unitarily. By set theory, we may suppose

that I = J . Thus CwJ (Y ) ∼= Cw

J (N).


The last assertion follows since

MI(Y ) ∼= RwI (Cw

I (Y )) ∼= RwI (Cw

I (N)) ∼= MI(N),

using (1.20).

The following follows at once from 8.5.28 and 8.5.17:

Corollary 8.5.29 If Y is any full right C∗-module over a C∗-algebra B, thenthere is a cardinal I such that Cw

I (Y ∗∗) ∼= CwI (B∗∗) completely B∗∗-isometrically.

Also, MI(Y ∗∗) is completely B∗∗-isometric to the W ∗-algebra MI(B∗∗).

8.5.30 (W ∗-algebra ‘covers’ of an operator space) In operator space applica-tions one sometimes applies the preceding result, with Y an injective or tripleenvelope of an operator space X (see Sections 4.4 and 8.3). These are triplesystems, and may be taken to be C∗-modules, as discussed in those sections. By8.5.29 we see that for some cardinal I, MI(Y ∗∗) is a W ∗-algebra.

Thus for any operator space X , there is a useful and essentially canonicalW ∗-algebra ‘containing’ MI(X).

Corollary 8.5.31 (The stable isomorphism theorem for W ∗-algebras) TwoW ∗-algebras M and N are weakly Morita equivalent if and only if there exists acardinal I such that MI(M) ∼= MI(N) ∗-isomorphically.

Proof The ‘only if’ may be proved similarly to 8.2.7, but replacing the use of8.2.6 (5) with the last assertion of 8.5.28 (and the ‘other-handed’ variant of thatassertion). Another argument is sketched towards the end of the Notes for 8.5.The other direction is easier. For example, one may appeal to the later result8.5.38, and the fact that the commutant of M ⊗MI is M ′ ⊗ 1.

8.5.32 (A basic construction) Suppose that Y is a full right C∗-module over aC∗-algebra B. We may suppose that Y is an equivalence A-B-bimodule. Considerthe Morita linking C∗-algebra L(Y ), and identify Y with a corner pL(Y )(1−p) asusual (see (8.3)). Fix a faithful nondegenerate representation of L(Y ) on a Hilbertspace. As we saw in 8.2.8, this Hilbert space may be taken to be H ⊕ K, whereH and K are two Hilbert spaces on which respectively A and B are faithfullyand nondegenerately represented. Since A = Y Y , we have [Y K] = H . Similarly,[Y H ] = K. Using these facts it is easy to explicitly compute the commutantL(Y )′ in B(H ⊕ K). Indeed, a simple calculation shows the following facts.First, L(Y )′ is a set of diagonal matrices, whose 1-1 entry R is in A′, and whose2-2 entry S is in B′. Second, these entries are mutually dependent, and thisdependence is given by the equation Ry = yS for all y ∈ Y . This last equationprovides a map π : B′ → A′, defined by π(S) = R. That is,

π(S) y = y S, y ∈ Y, S ∈ B′. (8.21)

In fact π is a ∗-isomorphism from B′ onto A′. One (perhaps too slick) way tosee this proceeds as follows. Recall from 8.2.19 that the strong Morita equiva-lence of A and B gives a category isomorphism between the categories of mod-ules over A and B respectively. This isomorphism is implemented by a pair

344 W ∗-modules

of functors F and G, given by F (W ) = Y ⊗B W , and G(V ) = Y ⊗A V . Bybasic algebra, the map T → F (T ) = IY ⊗ T yields a surjective isomorphismBB(W ) ∼= AB(F (W )). Indeed this map is clearly contractive (by 8.2.12 (1)),but one may exhibit a contractive inverse via the other functor G, just as inpure algebra (e.g. see [8, Proposition 21.2]). Thus the last isomorphism is alsoisometric. By 8.2.22, F (K) = Y ⊗B K is unitarily A-isomorphic to H above.Setting W = K and appealing to the above facts, we have BB(K) ∼= AB(H)isometrically isomorphically. This says precisely that B ′ ∼= A′; and it may beverified that this isomorphism is exactly our map π above. We have by 1.2.4that π is a ∗-isomorphism. We remark in passing that this proves that stronglyMorita equivalent C∗-algebras naturally have Hilbert space representations inwhich their commutants are ∗-isomorphic.

Summarizing, we saw that the commutant of L(Y ) is the set of matrices[π(S) 0

0 S

], S ∈ B′.

It is now easy to see that the second commutant L(Y )′′ is the set of matrices[c ef∗ d

], c ∈ A′′; d ∈ B′′; e, f ∈ E,

where E = T ∈ B(K, H) : TS = π(S)T for all S ∈ B′. In other words,E = B′B(K, H), where we are viewing H as a B′-module via π. Also, we haveE = pL(Y )′′(1 − p). Thus E is a subtriple of B(K, H), and E has a B′′-valuedinner product defined by 〈T1|T2〉 = T ∗

1 T2, for T1, T2 ∈ B(K, H). Indeed we have

B = Y Y ⊂ EE ⊂ B′′,

the last inclusion following from the definition of E and a direct calculation.Taking w∗-closures in B(K) in the last equation, and using the double com-mutant theorem, we deduce that EE is w∗-dense in B′′. From 8.5.4 (1), wededuce that E is a w∗-full right W ∗-module over B′′. Hence, by symmetry, E isa W ∗-equivalence A′′-B′′-bimodule.

By the double commutant theorem, L(Y ) is w∗-dense in L(Y )′′. It is clearfrom this, and from the fact that E = pL(Y )′′(1 − p), that Y is a w∗-denseA-B-submodule of E. We will see some applications of this momentarily, and inthe Notes section.

8.5.33 (Universal representations) We mention in passing that if we beginthe construction in 8.5.32 by choosing a ‘universal representation’ (see 3.2.7) ofL(Y ) (or equivalently, by 8.2.23 and 8.1.18, a representation of L(Y ) inducedfrom a universal representation of B, say), then that construction allows one torecover 8.5.17. In fact 8.5.32 yields much more information in this case, such asthe fact that Y ∗∗ ∼= B′B(K, H), for appropriate Hilbert modules H and K. Italso allows us to treat representations of B∗∗, or Y ∗∗, in a functorial way that isoften important in applications.


8.5.34 (Normal rigged W ∗-modules) We define a normal N -rigged M -moduleto be a right W ∗-module Z over N for which there exists a (unital) normal∗-homomorphism θ : M → BN (Z). This name is due to Rieffel, who does nothowever insist on all of the conditions above. In the literature, they are oftencalled M -N -correspondences.

The following shows again that the ‘interesting objects’ in the theory fallwithin the operator module framework:

Proposition 8.5.35 A W ∗-module over N is a normal N -rigged M -module ifand only if it is a normal dual operator M -N -bimodule in the sense of 3.8.2.

Proof If Z is a normal N -rigged M -module, then by 8.5.9 it is a normaldual operator BN (Z)-N -bimodule. Since the left action comes from a normal∗-homomorphism θ : M → BN (Z), Z is a normal dual M -N -bimodule.

Conversely, if Z is a normal dual operator M -N -bimodule, then as in theproof of 4.7.6, there is a normal ∗-homomorphism θ : M → Al(Z). HoweverAl(Z) = BN (Z) by 8.4.2.

8.5.36 (Inducing normal representations) Suppose that H is a Hilbert spaceon which M is normally represented on. If Y is a right W ∗-module over M , thenit follows from 8.2.13 that Y ⊗M H is a Hilbert space. Note, further, that if Yis a normal M -rigged N -module, then Y ⊗M H is a normal Hilbert N -module(in the sense of 3.8.5). To see this, suppose that (at)t is a bounded net in Nconverging in the w∗-topology to an a ∈ N . Then by 8.5.35, aty → ay in thew∗-topology of Y , for any y ∈ Y . By 8.5.4 (3), if ζ, η ∈ H and y, z ∈ Y , then

〈aty ⊗ η, z ⊗ ζ〉 = 〈〈z|aty〉η, ζ〉 −→ 〈〈z|ay〉η, ζ〉 = 〈ay ⊗ η, z ⊗ ζ〉.

Since finite sums of ‘rank one tensors’ are norm dense in Y ⊗M H , it is evidentthat aty⊗η → ay⊗η weakly in the Hilbert space Y ⊗hM H . For the same reason,we may now deduce that atξ → aξ weakly, for any ξ ∈ Y ⊗M H . Thus by A.2.5(2), Y ⊗M H is a normal Hilbert N -module.

By 8.2.13, if, further, M is faithfully represented on H , and if the canonicalmap from N into BM (Y ) is one-to-one, then N is faithfully represented on theHilbert space Y ⊗M H . We shall not use this, but it follows easily from A.6.2that if I is as in 8.1.4, then Y ⊗M H = Y ⊗I H .

Corollary 8.5.37 (Rieffel) Suppose that π : M → B(K) is a faithful normalrepresentation, and let R = π(M)′, the commutant in B(K).(1) If Y is a W ∗-module over M , then there exists a Hilbert space H on which

R is normally represented (namely Y ⊗M K), such that Y ∼= RB(K, H)completely isometrically, and w∗-homeomorphically.

(2) Conversely, if H is a Hilbert space on which R is normally represented,then the w∗-closed right π(M)-submodule RB(K, H) of B(K, H), with itscanonical B(K)-valued inner product, is a right W ∗-module over π(M).

In fact the isomorphism in (1) is a unitary M -module map.

346 W ∗-modules

Proof (2) This is clear by a direct computation, using the double commutanttheorem, and 8.5.4 (1).

(1) We will use 8.5.32 (Rieffel’s proof is sketched in the Notes section).First assume that Y is w∗-full over M . Since Y is a W ∗-module, the normalrepresentation of M on K induces, by 8.5.36, a faithful normal representationof BM (Y ) on H = Y ⊗M K. Similarly, using also the definition of Lw(Y ) in8.5.10, we have a faithful normal ∗-representation of Lw(Y ) = BM (Y ⊕c M) onthe Hilbert space (Y ⊕c M) ⊗M K. The latter space, as in (8.14), is unitarilyequivalent to (Y ⊗M K) ⊕ K = H ⊕ K. Just as in the last paragraph of 8.2.22,the associated normal representation of Lw(Y ) on H ⊕ K is one-to-one (andhence completely isometric, by 1.2.4) and corner-preserving, and its ‘1-2-corner’is a map from Y to B(K, H). Clearly the latter map is completely isometricand w∗-continuous, and hence (using also A.2.5 (3)) Y may be regarded as aw∗-closed subspace of B(K, H). If I is as in 8.1.4 then, as we saw in the proof of8.5.3, I acts nondegenerately on K, and M(I) = π(I)′′ = π(M). By 8.2.13 (2),A = KM (Y ) acts nondegenerately on H . Now we are in a position to apply thearguments in 8.5.32. By the last facts in 8.5.32, we have E = Y . Since Y is anM -submodule of E, the last assertion of the Corollary is also clear in this case.

In the general case, we use the fact that a w∗-closed ideal in a W ∗-algebra isof the form pM for a central projection p. Thus Iw = pM for such a projection p.Let K ′ = π(p)K, and apply the previous case to the canonical representation ofIw on K ′. Write θ for this last representation, and let N = θ(Iw)′. We obtain aHilbert space H on which N is normally represented, such that Y ∼= NB(K ′, H).However, there is a canonical normal ∗-homomorphism r → π(p)r, from R ontoN . Thus K ′ and H may be viewed as R-modules, and it is easily checked thatRB(K, H) ∼= NB(K ′, H) unitarily as M -modules, via the map T → T|K′.

Corollary 8.5.38 (Connes) W ∗-algebras M and N are weakly Morita equiv-alent if and only if there exist faithful normal representations π : M → B(K)and ρ : N → B(H), with π(M)′ ∼= ρ(N)′ ∗-isomorphically. Moreover, in thiscase, writing R for π(M)′ and for ρ(N)′, the TRO RB(K, H) in 8.5.37 (2) is aW ∗-equivalence N -M -bimodule.

Proof If Y is a W ∗-equivalence N -M -bimodule, and if π is as in 8.5.37, thenthe proof of (1) of that result, together with the paragraph after (8.21), showsthat if ρ is the induced representation of N on Y ⊗M K, then π(M)′ ∼= ρ(N)′. Aquick proof of the converse is given in the Notes, however it does not yield thefinal assertion. So instead, suppose that π, ρ, R are as in the statement of 8.5.38.If Y = RB(K, H) then, as in 8.5.37 (2), it is easy to see that Y is a TRO, which isa right and a left W ∗-module over π(M) and ρ(N) respectively. We need to showthat these W ∗-modules are full. The w∗-closure Iw of Y Y is a w∗-closed idealin π(M). Thus there is a central projection e in π(M) with Iw = eπ(M). SinceY = Y Iw (e.g. see 8.1.4 (2)), we have Y = Y e. Since e ∈ π(M), (1 − e)K is anR-module. Let P be 1− e, viewed as a map from K onto (1− e)K. Since e ∈ R′,P is an R-module map. Since H is a cogenerator of RNHMOD (see 3.8.6), if


(1 − e)K = (0) then there exists a nonzero R-module map T : (1 − e)K → H .Thus TP ∈ Y , and so TP = TPe, which is absurd. Thus e = 1, and Y is w∗-fullon the right. A similar argument shows that Y is w∗-full on the left.

8.5.39 (Correspondences) Another important ‘picture’ of W ∗-modules is re-lated to the standard form L2(M) of a W ∗-algebra M (see 3.8.5, or for full details,see, for example, [175] or [408, Chapters VIII and IX]). One reason why the stan-dard form is of importance here, is that it is a normal faithful Hilbert space rep-resentation of M such that M ′ ∼= Mop. Thus in Corollary 8.5.37 we may replaceR = M ′ by Mop. We view a left M op-module action on a Hilbert space, as a rightM -module action of M . In particular, by 8.5.37 (1), if Y is a right W ∗-moduleover M , then there exists a Hilbert space H = Y ⊗M L2(M) on which Mop is nor-mally ∗-represented, or equivalently on which M is normally represented on theright of H , such that Y ∼= BM (L2(M), H). Conversely, by 8.5.37 (2), any Hilbertspace H on which M is normally represented as a right action (that is, any nor-mal ∗-representation of M op), gives rise to a right W ∗-module over M , namelyBM (L2(M), H). One may show that BM (L2(M), H) ⊗M L2(M) ∼= H unitarily.Indeed it is easy to see that the canonical map from BM (L2(M), H)⊗M L2(M)to H is isometric. That it has dense range, and is thus surjective, follows frommodular theory (see [27, Theorem 2.2], and references therein). Thus we see thatthere is a bijective correspondence between such Hilbert spaces H , and rightW ∗-modules over M .

It is easily seen that the bijection above restricts to a bijective correspondencebetween the Hilbert spaces H on which M is normally represented on the right,and on which another W ∗-algebra N is normally represented on the left; andthe class of normal M -rigged N -modules (see 8.5.34). A correspondence betweenN and M is a Hilbert space H which is a normal N -M -bimodule as above.Thus, such Hilbert spaces are in a bijective relation with the normal M -riggedN -modules. We do not have space to even touch on this extremely importanttopic in detail here, but refer the reader to [101, Section V.B], [408, ChaptersVIII and IX], and [346,6, 27, 119], for example, and references therein.

8.5.40 (The W ∗-module tensor product) This is a ‘W ∗-module version’ ofthe interior tensor product discussed in Section 8.2. We write it as Y ⊗MZ,for a right W ∗-module Y over M , and a normal N -rigged M -module Z. Justas the C∗-module interior tensor product is just the module Haagerup tensorproduct, the W ∗-module tensor product ⊗M coincides with the module weak*Haagerup tensor product ⊗w∗hM which we discussed briefly in 3.8.14. Someof the benefits of knowing that these tensor products coincide are that, first,we get useful expressions for elements in this tensor product as w∗-convergentsums

∑i∈I xi ⊗ yi, with a convenient description of the norm of such a sum.

This facilitates easy computations. Second, as in 8.2.12, we can appeal to theuseful properties of this tensor product to show that this product is functorial,associative, commutes with the ultraweak sum, and so on. One may deduce that,for example, analoguously to 8.2.15, we have

348 A sample application to operator spaces

W ⊗MZ ∼= CBM (Z, W ) completely isometrically.

We omit the proof of this result, which is similar to the proof of 8.2.15, and is avariant of a result originally from [119]. See [48] for a complete discussion of thistensor product, which parallels our earlier development of the C∗-module tensorproduct. Indeed there are precisely analoguous W ∗-module versions of results8.2.11–8.2.19. We refer the reader to [361, 48] for a development of this theorywhich parallels our earlier discussion. To avoid this already lengthy chapter be-coming completely unwieldy we have omitted these results. However, the readerwho has followed the discussion till now, should at least have no difficulty statingappropriate W ∗-module versions of 8.2.11–8.2.19.

8.6 A SAMPLE APPLICATION TO OPERATOR SPACES

Because of space limitations, we will only list one of the very many applicationsof the preceding theory to operator spaces. See the Notes for references to theliterature for other applications.

8.6.1 (Injectivity and semidiscreteness) We return to the notions of ‘OS-nuclearity’, ‘OS-semidiscreteness, and the ‘WEP’, introduced in Section 7.1. Fora general operator space, the relationships between these concepts, and otherproperties such as ‘injectivity’, are quite interesting. Some of these are not dif-ficult, such as the result from 7.1.5 that any OS-nuclear operator space X hasthe WEP, or the fact from 7.1.9 that an OS-semidiscrete dual operator space isinjective. In fact, for a finite-dimensional operator space X , all of the followingproperties are equivalent: injectivity, OS-nuclearity, OS-semidiscreteness, andthe WEP. These are also equivalent to X being a triple system, and also equiva-lent to saying that for some n ∈ N, X is a completely contractively complementedsubspace of Mn (that is, there exists a complete isometry from X onto a subspaceW of Mn, and a completely contractive projection from Mn onto W ). Most ofthese equivalences are quite trivial to see. Indeed, if X is injective then it is aTRO by 4.4.2. For any finite-dimensional TRO X , the linking C∗-algebra L(X)is finite-dimensional. By (8.3), X is completely contractively complemented inL(X). However any finite-dimensional C∗-algebra is completely contractivelycomplemented in some Mn, so that X is completely contractively complementedin Mn too. We leave the remaining implications to the reader.

Theorem 8.6.2 (Effros, Ozawa, and Ruan) If X is a dual operator space thenthe following are equivalent:

(i) X is injective,(ii) X is OS-semidiscrete,(iii) X has the WEP,(iv) X is completely isometrically isomorphic and w∗-homeomorphic to a ‘cor-

ner’ pM(1 − p), for an injective W ∗-algebra M , and a projection p ∈ M .


Proof We saw in 7.1.3 that (i) is equivalent to (iii). We saw in 7.1.9 that(ii) implies (i). Using the fact that an injective W ∗-algebra is OS-semidiscrete(see [99, 419]), it is clear that (iv) implies (ii). Finally, if X is an injective dualoperator space then by 4.4.2, X may be regarded as an TRO, and hence anequivalence bimodule. By 8.5.6, X is a W ∗-equivalence M -N -bimodule, overW ∗-algebras M and N say. By the proof of 4.2.10, MI(X) is injective for anycardinal I. By 8.5.28, we deduce that MI(N) is injective. Let L be the Moritalinking W ∗-algebra of X (see 8.5.10). By the last remark in 8.5.14, N and L areweakly Morita equivalent. By 8.5.31, we deduce that MI(L), and consequentlyalso L, is injective. Since X is the 1-2-corner of L, we have proved (iv).

8.6.3 It is known (see [149]) that an operator space X is OS-nuclear if andonly if it is locally reflexive (see 6.6.7 for one definition of this term), and X∗∗ isOS-semidiscrete. This result should be compared with 6.6.8. Putting it togetherwith 8.6.2, we see that an operator space X is OS-nuclear if and only if X islocally reflexive and X∗∗ is injective.

In light of the finite-dimensional case discussed in 8.6.1, and the fact thatany OS-semidiscrete operator space is injective, and hence is a triple system, itis tempting to suppose that any OS-nuclear locally reflexive operator space is atriple system. We will argue below that this is not the case.

8.6.4 (The commutative case) It is interesting to consider the ‘commutativecase’ of certain topics in this section, and chapter. If Y is a triple system withthe property that xy∗z = zy∗x for all x, y, z ∈ Y , then we say that Y is acommutative triple system. These objects have been throughly analyzed in theJB∗-triple literature (e.g. see [225,165,379] and references therein). In fact, thefollowing turns out to be essentially the only example of a commutative triplesystem. Suppose that Ω is a compact space, and that we have a continuous actionof the circle T on Ω. Then

H = f ∈ C(Ω) : f(α · w) = αf(w) for all w ∈ Ω, α ∈ T,

is a triple subsystem of C(Ω), which is commutative in the sense above. Sucha space H is called a Cσ-space. Conversely, every commutative triple system iscompletely isometric to a Cσ-space (this may be deduced, for example, from a factabout minimal operator spaces mentioned in the Notes for Section 8.3). Thereare a host of characterizations of Cσ-spaces scattered throughout the Banachspace and JB∗-triple literature. We can add the following to the known list:

Proposition 8.6.5 Let Y be a triple system. Then the following are equivalent:(i) Y is a commutative triple system,(ii) Y is a minimal operator space (see 1.2.21),(iii) Y = Y op (see 1.2.25 for this notation),(iv) Y ∗∗ is completely isometrically isomorphic, and w∗-homeomorphic, to an

L∞ space,(v) Y Y and Y Y are commutative algebras.


Proof It is clear that (ii) implies (iii). Suppose (iii). If Y is a subtriple of aC∗-algebra A, we may view Y op as a subtriple of Aop. Since the identity mapfrom Y to Y op is a complete isometry, it follows from 4.4.6 that we have (i).

Supposing (i), then (v) follows by simple algebra. Also, we saw in 8.5.17 thatY ∗∗ is a triple system. It is easy to check by routine w∗-density arguments thatZ = Y ∗∗ is a commutative triple system. It is well known that a commutativetriple system Z with a predual is isometric to an L∞ space. The usual waythat this is proved, is to show first that any extreme point u of Ball(Z) is aunitary. That is, uu∗z = zu∗u = z, for any z ∈ Z. Then the map z → u∗z isa complete isometry from Z onto ZZ. Thus the commutative C∗-algebra ZZhas a predual, and is therefore an L∞ space. This yields (iv).

It is obvious that (iv) implies (ii). Finally, given (v), we view Y as anequivalence A-B-bimodule in the usual way, with A = Y Y and B = Y Y .If y, x1, x2, x3, x4 ∈ Y , then (using the notation of 8.1.19)

yx∗1x2x

∗3x4 = x2x

∗3yx∗

1x4 = x2x∗1x4x

∗3y = x4x

∗3x2x

∗1y.

Hence if θ is as in 8.1.20, then θ(x∗1x2x

∗3x4) = x4x

∗3x2x

∗1 ∈ A. Thus for any x ∈ Y

we have θ(x∗x)2 = (xx∗)2, and so θ(x∗x) = xx∗. By the polarization identity(1.1), θ(x∗y) = yx∗, for all x, y ∈ Y . This yields (i).

That there exist OS-nuclear operator spaces which are not triple systems,may now be seen as follows. We will use the well-known fact from Banach spacetheory that every Banach space, and hence every minimal operator space (see1.2.21), is locally reflexive. Consequently, by 8.6.3, and using the fact from 1.4.12that Min(E)∗∗ = Min(E∗∗), saying that a minimal operator space X = Min(E)is OS-nuclear is the same as saying that X∗∗ = Min(E∗∗) is injective as anoperator space. By (4.5), this is equivalent to E∗∗ being an injective Banachspace. However the injective Banach spaces are commutative C∗-algebras (see4.2.11). Thus a minimal operator space X is OS-nuclear if and only if X∗∗ ∼= L∞.Using 1.4.12, the latter is equivalent to X∗ being an ‘L1-space’. By 8.6.5 and theremark above it, if E is an ‘L1-predual’ which is not a Cσ-space, then Min(E)is a OS-nuclear operator space which is not a triple system. In fact there areextensive studies, in the Banach space literature, of interesting L1-preduals whichare not Cσ-spaces (e.g. see [242] and references therein).


C∗-modules over commutative C∗-algebras were introduced by Kaplansky [220].The general case surfaced in the early 1970s in the work of Paschke (e.g. see[300, 302, 303]) and Rieffel (e.g. see [359–361, 363]). Sometimes they are called‘Hilbert C∗-modules’, or simply ‘Hilbert modules’ (the latter term of course hasa quite different meaning in our book). Their use has become quite ubiquitousin C∗-algebra theory. In the 1980s, Connes, Kasparov, and many others, usedC∗-modules as a tool at quite a sophisticated level. Thus some of the results in


this chapter are essentially folklore, well known to the experts for a long time,but hard to date or attribute precisely. For more on the history of the subject,and for more detailed citations, see the texts cited at the start of this chapter.Frank has compiled an extensive listing of papers related to C∗-modules in [161].

Wittstock was the first to explicitly consider the canonical operator spacestructure on a C∗-module [432]. In the late 1980s and early 1990s, this con-nection was developed by Hamana and Kirchberg from the TRO viewpoint. In-deed Kirchberg used the operator space structure of TROs in his amazing workfrom that period (see his papers cited in the reference list below, and referencestherein). The next contributions to the ‘operator space view’ of C∗-modulescame in the early 1990s with the project [65], which led to the papers [46–52,64]and others. At much the same time, and independently, Magajna and Tsui be-came interested in C∗-modules from this perspective (see [264, 412], and refer-ences therein). Around 1999, interest in TROs picked up with the importantpaper [141]. As evidenced by the number of recent papers using them, it seemsthat TRO and C∗-module methods are playing an increasingly central role inoperator space theory at the present time. Many of these papers are listed below.See also, for example, an amazing series of papers recently undertaken by Junge(e.g. see [207]), which among other things apply W ∗-module techniques and freeprobability to study the operator Hilbert space and related topics.

8.1: Most of the results in Section 8.1 not attributed below to others, aredue to Paschke or Rieffel. Our presentation here and in some later sections,benefits from a frequent use of Cohen’s factorization theorem, and the linkingalgebra. Item (4) in 8.1.1 follows from (1)–(3), and the polarization identity(1.1). The definition given in 8.1.2 of a strong Morita equivalence is not quiteRieffel’s original one; 8.1.2 is essentially the formulation in [77]. Sometimes strongMorita equivalence is called Morita–Rieffel equivalence. It is easy to check thatstrong Morita equivalence is an equivalence relation (using, for example, 8.2.18together with 8.2.12 (2) and (4)), and that ∗-isomorphic C∗-algebras are stronglyMorita equivalent (using the Bσ construction in 8.3.6). In connection with 8.1.4(2), if y ∈ Y then there is a z ∈ Y with y = z〈z|z〉 (see [38, 387]). Result8.1.5 is due to Paschke [302]. The observation 8.1.8 seems to appear first in[239]. It seems likely that simple results such as 8.1.6, 8.1.21, and 8.1.22, werewell known to a few experts well before they appeared in [47]. Frank noticedindependently a variant of the proof of 8.1.6 at about the date of [47], butphrases it in terms of quasimultipliers [160]. We are grateful to Paschke forcommunicating to us the argument in 8.1.12, based on [302, Lemma 4.1]. We arenot sure exactly when 8.1.12 dates to. The result 8.1.13 was certainly known atthe beginning of the 1970s (see the cited papers of Paschke), although some ofthis is no doubt much older. Kasparov and Lin established 8.1.16 (3) in [221,254].If one applies this result in the case that Y is a C∗-algebra B, and use the factfrom 8.1.11 that KB(B) ∼= B, then we deduce that BB(B) ∼= M(B). This relationis more or less the well-known ‘double centralizer’ description of the multiplierC∗-algebra. In connection with 8.1.10, there are some other characterizations of


bounded module maps in [56, 383]. The linking C∗-algebra is due to Brown etal. [76]; see also [77]. Strongly Morita equivalent commutative C∗-algebras areisomorphic: indeed the map θ in 8.1.20 restricts to a ∗-isomorphism between Aand B (see the last part of the proof of 8.6.5). From 8.1.22, it follows easily thatC∗-module direct sums are commutative and associative. Corollary 8.1.24 is arestatement of a common tool. Theorem 8.1.26 appeared first in [47]; a simplerproof communicated to us by Kirchberg appears in [65, p. 41].

Theorem 8.1.27 appears in the standard sources on C∗-modules, and is quiteuseful for operator spaces. For example, it may be applied in the study of finitelygenerated operator modules (e.g. see [53, 54]). The following folklore results areoften found together with the result 8.1.27: (a) Any algebraically finitely gen-erated C∗-module is projective in the sense of pure algebra (see [203, SectionIV.3]); (b) any two algebraically finitely generated C∗-modules which are alge-braically A-isomorphic, are unitarily isomorphic; (c) any algebraically finitelygenerated and projective module Y over a C∗-algebra A, possesses an A-valuedC∗-module inner product 〈·|·〉; and (d) if Y is as in (c), and if we write Z forY with the inner product in (c), then any other such inner product on Y maybe written as 〈u(·)|u(·)〉, for a bicontinuous module automorphism u of Z (infact u and u−1 may be chosen in BA(Y )+). We give quick proofs of these results(a)–(d): For (a), note that as in the proof of 8.1.27 (2), Y is algebraically finitelygenerated over A if and only if it is also finitely generated over A1. Thus by8.1.27 (1) we have that Y is a complemented A1-submodule of Cn(A1), fromwhich it follows by algebra that it is projective as an A-module (in the sense ofthe algebra texts). For (b), we may suppose that A is unital, and, by 8.1.27 (1),that Y and Z are orthogonally complemented submodules of Cn(A) and Cm(A)respectively, and that g : Y → Z is a surjective one-to-one A-module map. Weobtain an induced A-module map on Cn(A) which factors through g. Howeverany A-module map from Cn(A) to Cm(A) may be viewed as left multiplicationby an m × n matrix of A-module maps on A, and is therefore bounded. Thus gis bounded. Applying [423] 15.3.8 again, we obtain a unitary from Y onto Z. For(c), note that a finitely generated projective A-module, in the purely algebraicsense, may be taken to be simply a submodule Y of Cn(A), which is the rangeof an idempotent module map P : Cn(A) → Cn(A). Any such P may be viewedas left multiplication by a matrix of A-module maps on A, and therefore P isbounded by A.6.3. So Y is a closed submodule of Cn(A). Therefore Y may in-herit the inner product from Cn(A). Finally, for (d), by (b) any other such innerproduct on Y equals 〈u(·)|u(·)〉, with u and u−1 continuous by 8.1.27 (2). Also,〈u∗u(·)|·〉 = 〈|u|(·)||u|(·)〉. By basic spectral theory in the C∗-subalgebra of B(Z)generated by u∗u, we may ensure that |u| and |u|−1 are positive.

See, for example, [162,355] for extensions of some parts of the last mentionedresults. We remark that if two C∗-modules are A-isomorphic via a map f withf, f−1 adjointable, then they are unitarily isomorphic. This follows easily fromthe polar decomposition [423].

A pre-C∗-module is a right (say) A-module Y with an A-valued inner product


satisfying (1)–(4) in 8.1.1. The completion of a pre-C∗-module in the associatednorm is a C∗-module. We omit the easy details. We did not need this constructionin our presentation of the theory, although it is an important tool.

There is a bimodule version of the multiplier algebra, which has been studiedby Echterhoff and Raeburn [132], and others. This is often useful in generalizingresults that are valid for modules over unital algebras. Recall that if Y is a fullright C∗-module over B, or an equivalence A-B-bimodule, and if L(Y ) is thelinking C∗-algebra, then we observed in 8.1.17 that M(L(Y )) may be taken tobe BB(Y ⊕c B). We define the multiplier bimodule M(Y ) to be the 1-2-cornerof this C∗-algebra. Clearly M(Y ) may be identified with BB(B, Y ) (recall thatby 8.1.11 and 8.1.17, Y is identified canonically with K(B, Y )). Also M(Y ) is aright C∗-module over BB(B) = M(B), and similarly M(Y ) is a left C∗-moduleover B(Y ) ∼= M(A)). One sees that Y is an M(A)-M(B)-submodule of M(Y ). Itcan be also shown M(Y ) ∼= BA(Y , A), and that if L(Y ) is represented nondegen-erately on H ⊕K as in 8.2.8, then M(Y ) may be viewed as a subset of the weakoperator topology closure of Y in B(K, H) (which also equals the w∗-closureof Y ). Many facts about C∗-modules which we develop in this chapter, may beeasily adapted to its multiplier bimodule. E.g. see [132,355].

8.2: The tensor products here are originally due to Rieffel. The first part of8.2.2 dates to Wittstock’s early study [432] of C∗-modules as operator spaces.The papers [46–52, 64, 65], particularly [47] and [65], are the source of the op-erator space aspects of most other results in this section. Some of these papersdevelop analogues for nonselfadjoint algebras of C∗-modules and strong Moritaequivalence, as mentioned in 8.2.21, and they use decisively most of the theoryin the first half of the present text. Unfortunately we are not able to discussthis topic further here. We refer the interested reader to these papers for furtherresults and references. See, for example, [166] for an application of the operatorspace viewpoint of the interior tensor product.

A version of 8.2.4 was first proved by Paulsen, in connection with [65]. See[65, 46, 355] for some generalizations of the material in 8.2.5–8.2.7. These ledto our presentation here, which is not very novel. Parts of 8.2.6 are due toKasparov; all of it is intimately related to the advances in [75, 76]. This resultis closely tied to the subject of frames [162, 355]. Indeed frames have been usedin C∗-module theory since the beginning (e.g. see [75]), and are usually calledquasibases. Note that any C∗-algebra A is strongly Morita equivalent to K∞(A)(via the equivalence bimodule and TRO C(A), for example). Thus the converseassertion in 8.2.7 follows from the facts, mentioned in the Notes for 8.1, thatstrong Morita equivalence is an equivalence relation coarser than ∗-isomorphism.One may avoid the use of 4.5.13 in 8.2.7 by using the ∗-isomorphisms

K∞(A) ∼= K∞(KB(Y )) ∼= KB(C(Y )) ∼= KB(C(B)) ∼= K∞(B),

(using 8.1.15, 8.2.6 (4), and the second paragraph of 8.2.16). We believe that thisargument is due to Blackadar. Similar remarks apply to the proof of 8.5.31. Wecould only find 8.2.12 (3) in [47] in this generality. The operator space insights in


8.2.25 were certainly known in the 1980s to Hamana and others. See also [285].The correspondences in 8.2.24 and 8.2.25 may also be viewed as a special caseof the ‘inducing procedure’ discussed in 8.2.19 and 8.2.22. For example, a closedideal J of B may be regarded as an object in BOMOD, and the induced leftA-module Y ⊗B J can be shown to be unitarily A-B-isomorphic to XJ = Y J .We may deduce from this, for example, that Y ⊗B (B/J) ∼= Y/XJ . For muchmore on induced representations, particularly in settings involving groups, see,for example, [362, 356], and references therein.

Strong Morita equivalence may also be characterized in terms of a functorialequivalence between the categories of C∗-modules. A proof may be found in [47];another was shown to us by Skandalis. Beer showed in [30] that unital C∗-algebras are strongly Morita equivalent if and only if they are Morita equivalentin the sense of algebra. See also [11, 51, 52].

As we mentioned in 8.2.14, if Y is a right C∗-module over a nondegenerateC∗-subalgebra B of B(K), then the structure of Y is closely linked to the normon Y ⊗B Kc. We give formulae for this norm which avoid mention of the innerproduct. Namely, if z ∈ Y ⊗ Kc, then

‖z‖Y ⊗BKc = sup(∑

k

‖ζk‖2) 1

2 β(y1, . . . , yn) : z =∑

k

yk ⊗ ζk

,

the supremum taken over finite sums z =∑

k yk ⊗ ζk, for yk ∈ Y, ζk ∈ Kc, where

β(y1, . . . , yn) = sup∥∥∑

k

u(yk)∗u(yk)∥∥ 1

2

= sup∥∥∑

k

ykbk

∥∥,the last two suprema being taken, respectively, over all u in Ball(BB(Y, B)), and(bk) in Ball(Cn(A)). See [47] for more details.

8.3: The main source of results in this section are the papers of Hamana(e.g. see [189–191,193]), and the papers of Kirchberg in the reference list below.See also [154]. Triple systems are particular examples of JB∗-triples, itself a vastfield in its own right. See the survey [379], for example. Various parts of 8.3.2 arein [197,198,225]. In connection with 8.3.5, Solel has proved that a surjective linearisometry between TROs extends to an isometry between the linking algebras,one of a tractable form [395]. Note that if π : A → B(H) is a ∗-homomorphismand if U is a unitary, then θ = Uπ(·) is a triple morphism. From 8.3.5 it is easyto prove a converse to this, and characterize triple morphisms from a C∗-algebrainto a TRO (or into a C∗-algebra, or into B(K, H)). See, for example, [58, 226]for variants on this. From such a result, and 4.4.6, one may recover quickly someof the Banach–Stone theorems we have encountered in this text. See also thenotes to Section 6.7 in [158].

Some of the unattributed material in this section, for example parts of 8.3.12,is from [53]. See that paper for more on the triple envelope and its properties. Forexample, it is shown there that if X is a Banach space then the triple envelopeof Min(X) is triple isomorphic to the Cσ-space (see Section 8.6)


h ∈ C(S) : h(αψ) = αh(ψ) for all ψ ∈ S, α ∈ T,

where S is the w∗-closure of the extreme points in Ball(X∗). In this case, Theorem8.3.9 may be rephrased in terms of the existence of a ‘Shilov boundary linebundle’ for any Banach space. See also [445] for the finite-dimensional case. Mostof 8.3.12 (5) also comes from the latter paper. For a subspace X of Mm,n, withm, n finite, the triple envelope (which coincides in this case with the injectiveenvelope) is easy to describe. Indeed, analoguously to 4.3.7 (2), the subtriple ofMm,n generated by X is triple isomorphic to a finite direct sum of ‘rectangularmatrix blocks’ (e.g. see [392]). Some of these blocks are redundant, just as in4.3.7 (2). If one eliminates such blocks, then the remaining direct sum of blocksis the triple envelope of X .

As we said earlier, [141] sparked much recent interest in TROs (e.g. see[222, 226, 289, 290, 293, 295, 376, 395], and [154], which was earlier). In many ofthese papers, facts about TROs are deduced via the linking C∗-algebra. Thenoncommutative Shilov boundary, or triple envelope, has been used extensivelyin [55, 56, 59], for example.

As an illustration of the use of the techniques in this chapter in operator spacetheory, we briefly discuss OS-nuclearity (see Section 7.1) of TROs. Most of whatwe say next is due to Kirchberg, and Kaur and Ruan (see [226]), although we havechanged the proofs. We first give an illustrative proof of an old result from [30],namely that Morita equivalence preserves nuclearity of C∗-algebras. We use ideasfrom [46, 131]. Suppose that Y is an equivalence A-B-bimodule, and that B isan OS-nuclear C∗-algebra. Hence B factors through finite-dimensional matrixalgebras, as in 7.1.1. By 8.1.24, we obtain a nets of maps factoring Y throughfinite-dimensional matrix algebras over B. It follows from these facts, and usinga doubly-indexed net, that Y is OS-nuclear. These nets easily yield completelycontractive nets of maps factoring A ∼= KB(Y ) through finite-dimensional matrixalgebras over B (using, for example, 8.2.15 (1), and 8.2.12 (1), as in [46, p. 391]).It follows that A is OS-nuclear. By the same principle, and by 8.1.18, the linkingalgebra L(Y ) is OS-nuclear. Conversely, it is clear that if L(Y ) is OS-nuclearthen so is Y . Finally, suppose that Y is OS-nuclear. We remarked in 8.1.25 thatB factors through finite-dimensional matrix algebras over Y . Hence B is nuclearby the same arguments.

8.4: This material from [53] was part of the original development of theoperator space multiplier theory (see the Notes to Section 4.5). One may varythe definition of Ml(X) in 4.5.1 by considering the set of linear maps u : X → X ,such that there exists a linear complete isometry σ : X → Z into a right C∗-module Z over a C∗-algebra B, and an a ∈ BB(Z) such that σ(u(x)) = aσ(x),for all x ∈ X . This new set of maps coincides precisely with the set in 4.5.1.Indeed the new set clearly contains all the maps considered in 4.5.1. Conversely,given such σ : X → Z and a ∈ BB(Z), we may view a as an element u (supportedon the ‘1-1-corner’) of BB(Z ⊕c B). The latter, by 8.1.16 (3) and 8.1.17, is justLM(L(Z)). There is a canonical copy of X in (the 1-2-corner of) L(Z). Note


that u is a left multiplier of L(Z), and hence is in Ml(X), by 4.5.9.The operator space centralizer algebra is the set Al(X)∩Ar(X) of operators

on X . This is a commutative C∗-algebra which is discussed thoroughly in [66,Section 7]. Multipliers between two different operator spaces are discussed in [56].

8.5: Selfdual modules over a W ∗-algebra were first defined and investigatedby Paschke [302]. Unfortunately, some of their theory has been relegated to folk-lore, which is a pity because they are a very powerful tool. The main sourcesfor this section are [302,303,361], from which most of the results in this sectioncome, which are not specifically attributed below to others. Our approach hereuses the linking W ∗-algebra extensively, and the operator space framework inves-tigated in [48]. See the latter paper for some additional information. For furtherapplications of the linking W ∗-algebra to operator space theory, e.g. see [226].

Much of the proof in 8.5.4, 8.5.5, and 8.5.6 is unchanged if the module tensorproduct here is replaced by the Banach module projective tensor product. Thelatter was used in Paschke’s original approach [302]. Additional ideas for theproofs of these results come from [48,54, 383,444]. Results 8.5.6 and 8.5.7 are aweaker variant of a result of Zettl and Effros et al. [444, 141]; see also the paperin preparation cited in the Notes to 4.7. The latter paper reverses the order, anduses the ingredients of these results to prove deep facts about operator spacemultipliers, and also to generalize W ∗-modules to nonselfadjoint algebras. Notethat 8.5.6 quickly follows from 4.4.10 (see [141, Theorem 2.6]).

Weak Morita equivalence was called ‘Morita equivalence’ by Rieffel; some ofthe postulates in his definition have been dropped over time. See, for example,[30] for more on this topic. One can see easily, using 8.5.38 for example, that weakMorita equivalence is an equivalence relation coarser than ∗-isomorphism of W ∗-algebras. Indeed, suppose that Y is a W ∗-equivalence M2-M1-bimodule. If K isa faithful Hilbert M1-module, then by the first paragraph of the proof of 8.5.38,the induced representation of M2 on Y ⊗M1 K is a faithful M2-module in whichthe commutant of M2 is isomorphic to the commutant of M1 in B(K). By thesame principle, if W is a W ∗-equivalence M3-M2-bimodule, then W⊗M2Y ⊗M1 Kis a faithful Hilbert M3-module for which the commutant of M3 is isomorphic tothe commutant of M2, and hence also to the commutant of M1 in B(K). Nowapply 8.5.38 to see that M1 is weakly Morita equivalent to M3.

For more on dual TROs, or 8.5.18, see for example [198, 226, 376]. The ‘Ka-plansky density theorem’ in 8.5.18 is due to Harris [198]. By considering thelinking algebra, it is clear that the w∗-closure of a subtriple of B(K, H) coin-cides with its WOT-closure [226]. We imagine that parts of 8.5.16, 8.5.17, 8.5.28,8.5.29, and 8.5.31, may be described as folklore. That is, they do not appear inthe older literature as far as we know, but nonetheless seem to be well known tosome experts. For example, we could not find 8.5.17 explicitly in the C∗-moduleliterature, but is somewhat implicit in [77]. It also corresponds to a well knownresult in the JB∗-triple literature (see [379,28], and references therein); and ap-pears in a more explicit form in [53] (see also [141]). Item 8.5.19 is from [57];and 8.5.20 should be compared with [28]. The ultraweak direct sum, and 8.5.22–


8.5.25, are due to Paschke. A special case of 8.5.27 may be found in [56].Results 8.5.28 and 8.5.29 are sketched in [53, Lemma 5.8]. Also, 8.5.31 is

mentioned without proof in various places in the literature; in [303] Paschkeproved the related result that if M and N are weakly Morita equivalent, thenthere is a cardinal I and projection e in MI(N) with central cover 1, such thatM ∼= eMI(N)e ∗-isomorphically. A more recent exposition of the separable caseof 8.5.31 may be found in [376]. The W ∗-algebra ‘cover’ in 8.5.30 has hithertobeen useful because it can often help reduce a general operator space problem to avon Neumann algebra problem. More concretely, the following kind of principleis sometimes useful. Suppose that P and Q are properties that an operatorspace X (resp. an operator on X) may or may not have. Suppose that if aW ∗-algebra (resp. a map on a W ∗-algebra) has property P then it has propertyQ. Suppose that property Q descends to subspaces (resp. invariant subspaces).Finally, suppose that P ‘lifts’ to injective (or triple) envelopes, second duals, and‘amplifications’ MI(X), for any cardinal I. Then it follows from the constructionin 8.5.30 that if X (resp. a map on X) has property P then it has property Q.This technique was used in [53, 57].

The construction in 8.5.32 is adapted from Magajna’s explanation from [264]of some results from [361]. It follows from 8.5.32, taking A to be a W ∗-algebra M ,that every C∗-module Y over M is a w∗-dense M -submodule of a W ∗-module Eover M . This latter W ∗-module is called the selfdual completion of Y (see [302,361]). A variation on the proof of 8.5.37 shows that E ∼= MB(Y , M) unitarily.We sketch a slight variant of Rieffel’s proof of 8.5.37 (1) (see [361, Section 6]):Define Φ: Y → B(K, Y ⊗M K) as in 8.5.11, so that Φ(y)∗Φ(z) = 〈y|z〉, fory, z ∈ Y , and Φ is w∗-continuous. Since Iw is a w∗-closed ideal in M , there is acentral projection e in M , and therefore also in R = M ′, such that Iw = eM .Let K ′ = eK, then the canonical map from B(K) to B(K ′) restricts to a normal∗-homomorphism from M ′ onto the commutant of the copy of Iw in B(K ′).As in 8.5.32, the latter commutant is ∗-isomorphic to a certain commutant inB(H), where H = Y ⊗M K = Y ⊗Iw K ′. Composing, we obtain a normalrepresentation of R on H . By 8.5.37 (2), W = RB(K, H) is a right W ∗-moduleover M . It is easily checked that Ran(Φ) is an M -submodule of W , which is w∗-closed by A.2.5. If Ran(Φ) = W , then by 8.5.16, Ran(Φ) = pW , for a nontrivialorthogonal projection p. One may view p ∈ B(H), leading to the contradictionthat [Φ(Y )K] = H . So Φ is a unitary M -module map onto W .

Facts in 8.5.35 and 8.5.40 are from [48]. See [226] for other applications. TheW ∗-module tensor product is due to Rieffel (e.g. see [361]), as is 8.5.36. Theargument there shows that any normal M -rigged N -module Y gives rise to afunctor MNHMOD → NNHMOD, just as in 8.2.19. If Y is a W ∗-equivalenceN -M -bimodule, then it is easy to see that we obtain an equivalence of categoriesMNHMOD ∼= NNHMOD. Conversely, Rieffel shows that if these categoriesare equivalent, then M is weakly Morita equivalent to N . Observation 8.5.38 isdue to Connes (unpublished), as is the theory of correspondences (see 8.5.39).

As is the case in much of this chapter, there are several alternative routes


through many of the results in this section. We have presented the route whichwe guess would be most useful for operator space applications. We sketch anotherattractive alternative route through some main results in this section: Supposethat Y is a w∗-full right W ∗-module over a W ∗-algebra N ; or equivalently, sup-pose that Y is a W ∗-equivalence M -N -bimodule. By 8.5.37 and its proof, we maytake Y = RB(K, H) for faithful normal Hilbert modules K and H over N and Mrespectively. Here the commutants of N and M , in B(K) and B(H) respectively,are ∗-isomorphic (which may be proved in many ways), and we have written Rfor these commutants. Since R acts faithfully, by 3.8.6 and the remark after it,H and K are both ‘universal for NHMOD’ in the sense of Chapter 3. By a simplevariant of 3.2.11 (4), there is a unitary R-module map U : K (J) → H(J), for somecardinal J . Since the commutant of MJ (M) is I⊗R, the map taking x ∈ MJ (M)to U∗xU is a ∗-homomorphism from MJ (M) into (I ⊗R)′ = MJ(N ′′) = MJ (N).By symmetry of the construction, this is a ∗-isomorphism MJ(M) ∼= MJ (N).This yields 8.5.31, and also easily gives one direction of 8.5.38. We are indebtedto Weaver for communicating this argument to us. Also,

CwJ (Y ) ∼= RB(K, H(J)) ∼= RB(K, K(J)) ∼= Cw

J (RB(K, K)) = CwJ (R′) = Cw

J (N),

giving 8.5.28, and the Corollary preceding it.For a sample of the literature on W ∗-modules and correspondences, see [101],

[6,27,119,168,287,346], and references therein. Anantharaman-Delaroche, Kraus,Magajna, and Pop, have worked on questions concerning both operator spacesand correspondences. E.g. see [6,234,268,345], and references therein. There is anotion of Morita equivalence of ‘correspondences’ (see e.g. the recent papers ofMuhly and Solel), and many results generalize to such contexts.

Much of the theory of W ∗-modules generalizes to selfdual C∗-modules overmonotone complete C∗-algebras, as has been notably developed in [192]. Thiswill be useful in operator space theory because injective C∗-algebras (and inparticular the algebra I22(X) studied in Sections 4.4 and 4.5) are monotonecomplete but are not usually W ∗-algebras.

8.6: Theorem 8.6.2 is from [141], but the proof here highlights W ∗-moduleresults. A proof avoiding ‘stabilization’ may be found in [54]. In [68] it is shownthat any injective operator space is a selfdual C∗-module. It follows that if Yis a right C∗-module over a C∗-algebra B, equipped with its canonical operatorspace structure, and if Y is injective as an operator space, then Y is selfdual overB, and BB(Y ) is an injective C∗-algebra. These results are related to results ofHamana and Lin [192, 255]. Equivalence bimodules over two commutative C∗-algebras are well understood (e.g. see [356], and references therein). The lastpart of the proof of 8.6.5 we have seen in [387]. With a little more work, one mayremove the word ‘completely’ in (iv) of 8.6.5. Preduals of L1-spaces are calledLindenstrauss spaces, and they have an extensive literature (e.g. see [237], andreferences therein). The example of an OS-nuclear operator space which is nota TRO is due to Rosenthal (private communication).

Appendix

A.1 OPERATORS ON HILBERT SPACE

We begin by reviewing a few basic facts about operators on Hilbert space, whichmay be found in almost any book on functional analysis.

A.1.1 If S, T are contractive linear operators between Hilbert spaces such thatST = I, then it follows that S = T ∗, and T is an isometry and S a coisometry.If in addition TS = I, then T is a unitary. If P is an idempotent operator on aHilbert space then P is a projection (i.e. P = P ∗) if and only if ‖P‖ ≤ 1.

A.1.2 If H is a Hilbert space then the space S∞(H) of compact operators is anorm closed (two-sided) ideal in B(H). We write S1(H) for the usual trace class,a (two-sided) ideal in S∞(H) (and also in B(H)), and which is a Banach spacewith respect to the trace class norm ‖T ‖1 = tr|T |. The trace tr is a contractivefunctional on S1(H), and via the dual pairing (S, T ) → tr(ST ) it is well-knownthat S∞(H)∗ ∼= S1(H) and S1(H)∗ ∼= B(H) isometrically.

From this it is evident that the product on B(H), viewed as a map fromB(H) × B(H) to B(H), is separately w∗-continuous. That is, if St → S in thew∗-topology on B(H), then StT → ST and TSt → TS in the w∗-topology too.The w∗-topology on B(H) is also called the σ-weak topology. A linear functionalon B(H) is σ-weakly continuous if and only if it is of the form

∑∞k=1〈 · ζk, ηk〉,

for ζk, ηk ∈ H with∑∞

k=1 ‖ζk‖2 and∑∞

k=1 ‖ηk‖2 finite. By such considerations,the involution ∗ on B(H) may also be seen to be w∗-continuous.

A.1.3 More generally if H, K are Hilbert spaces, we let S∞(H, K) denote thecompact operators from H into K. For any 1 ≤ p < ∞, we let Sp(H, K) denotethe Schatten p-class of compact operators T : H → K such that |T |p belongs

to S1(H). This is a Banach space for the norm ‖T ‖p =(tr|T |p

) 1p . The following

ideal property holds: for any T ∈ Sp(H, K), V1 ∈ B(K), V2 ∈ B(H), the operatorV1TV2 belongs to Sp(H, K) and ‖V1TV2‖p ≤ ‖V1‖‖T ‖p‖V2‖.A.1.4 We write WOT for the weak operator topology. This topology makes themap T → 〈Tζ, η〉 continuous on B(H), for all ζ, η ∈ H . On bounded sets theWOT and σ-weak topologies coincide. Thus a bounded net in B(H) convergesin the WOT topology if and only if it converges in the w∗-topology.

A.1.5 A subspace X ⊂ B(H) is said to be reflexive if

X = T ∈ B(H) : Tζ ∈ [Xζ] for all ζ ∈ H.

360 Duality of Banach spaces

We write T∞ for the operator T ⊕T ⊕· · · on H (∞), and X(∞) = T∞ : T ∈ X.If W is a w∗-closed subspace of B(H), then W (∞) is reflexive in B(H(∞)). Tosee this, suppose that Tζ ∈ [W (∞)ζ] ⊂ [B(H)(∞)ζ], for all ζ ∈ H(∞). By firstsetting ζ = (0, · · · , 0, η, 0, · · · ), and then ζ = (η, · · · , η, 0, · · · ), it is easy to arguethat T = S∞ for some S ∈ B(H). Let ϕ ∈ W⊥. By A.1.2, there exist vectorsζ, η ∈ H(∞) such that ϕ(R) = 〈R∞ζ, η〉, R ∈ B(H). Then ϕ(S) = 〈Tζ, η〉 = 0,since Tζ ∈ [W (∞)ζ], and ϕ ∈ W⊥. Thus S ∈ (W⊥)⊥ = W , so that T ∈ W (∞).(E.g. see [354,102,108] for more on this topic.)

A.2 DUALITY OF BANACH SPACES

In this section and the next, E and F are Banach spaces. We write iE : E → E∗∗

for the canonical embedding. However we often suppress this map and simplyconsider E as a subspace of E∗∗.

Lemma A.2.1 (Goldstine) Ball(E) is w∗-dense in Ball(E∗∗).

Lemma A.2.2 Let u : E → F ∗ be a bounded linear map. Then there exists aunique w∗-continuous u : E∗∗ → F ∗ extending u. Moreover ‖u‖ = ‖u‖.Proof Set u = i∗F u∗∗ where iF : F → F ∗∗ is the canonical isometry.

Lemma A.2.3 Let E be a closed linear subspace of a Banach space F .(1) As subsets of F ∗∗ we have E

w∗= E⊥⊥.

(2) The second dual of the inclusion map E → F is an isometry from E∗∗ ontoE⊥⊥. Thus E∗∗ ∼= E⊥⊥ isometrically, via this canonical isometry.

(3) (F/E)∗∗ ∼= F ∗∗/E⊥⊥ isometrically, and w∗-w∗-homeomorphically, via the‘transpose’ of the canonical isomorphism (F/E)∗ → E⊥. This is the sameas the map obtained from q∗∗, where q : F → F/E is the canonical quotientmap, by factoring out Ker(q∗∗) = E⊥⊥.

(4) F ∩ E⊥⊥ = E.

Proof Items (1), (2), and (3) are in the standard sources. For (4), notice thatE ⊂ E⊥⊥ ⊂ F ∗∗. If y ∈ F but y /∈ E, choose ϕ ∈ F ∗ such that ϕ(E) = 0, butϕ(y) = 0. If y ∈ E⊥⊥ there exists by (1) a net (xt)t in E with xt → y in thew∗-topology of F ∗∗. Hence 0 = ϕ(xt) → ϕ(y) = 0, a contradiction.

We will also (silently) use the following simple principle many times:

Lemma A.2.4 If T : E → F is a w∗-continuous map between dual Banachspaces, and if W is a w∗-closed subspace of Ker(T ), then the induced map fromE/W to F is w∗-continuous.

Theorem A.2.5 (Krein–Smulian)(1) Let E be a dual Banach space with predual E∗, and let F be a linear subspace

of E. Then F is w∗-closed in E if and only if Ball(F ) is closed in the w∗-topology on E. In this case F is also a dual Banach space, with predualE∗/F⊥, and the inclusion of F in E is w∗-continuous.

Appendix 361

(2) If u ∈ B(E, F ), where E and F are dual Banach spaces, then u is w∗-continuous if and only if whenever xt → x is a bounded net converging inthe w∗-topology in E, then u(xt) → u(x) in the w∗-topology.

(3) Let E and F be as in (2), and u : E → F a w∗-continuous isometry. Thenu has w∗-closed range, and u is a w∗-w∗-homeomorphism onto Ran(u).

Proof Items (1) and (2) may be found in the standard texts; (2) is often statedfor functionals ϕ but the result as stated here follows from this by consideringϕ u. For (3), note that it is easy to check using (1) that Ran(u) is w∗-closed inF . Thus the restriction of u to Ball(E) takes w∗-closed (and thus w∗-compact)sets to w∗-compact (and thus w∗-closed) sets in Ran(u). Thus the inverse of urestricted to the ball is w∗-continuous, so u−1 is w∗-continuous by (2).

A.3 TENSOR PRODUCTS OF BANACH SPACES

We review a few facts about tensor products of Banach spaces E and F , whoseproofs may be found in many texts (see [118,121,324,407], for example).

A.3.1 If (xk)k and (yk)k are finite families in E and F respectively, then onemay define for z =

∑k xk⊗yk in the algebraic tensor product E⊗F , the quantity∥∥∥∑

k

xk ⊗ yk

∥∥∥∨

= sup∣∣∣∑

k

ϕ(xk)ψ(yk)∣∣∣ : ϕ ∈ Ball(E∗), ψ ∈ Ball(F ∗)

.

This is a norm on E ⊗ F . The completion of E ⊗ F in this norm is called theinjective tensor product and written as E⊗F . This tensor norm gets its namefrom the fact that it has the injective property. Namely, if ui : Ei → Fi areisometries for i = 1, 2, then the corresponding map u1 ⊗ u2 : E1⊗E2 → F1⊗F2

is an isometry too. More generally if u1, u2 are contractive then so is u1 ⊗ u2.We remark that the definition and facts in the last paragraph have obvious

variants for the N -fold injective product X1⊗ · · · ⊗XN of any N -tuple of Banachspaces. There is an associativity law: for example, if N = 3 then we have thatX1 ⊗ X2 ⊗ X3 = (X1 ⊗ X2) ⊗ X3 = X1 ⊗ (X2 ⊗ X3).

We may identify any element z =∑

k xk ⊗ yk as above with a boundedoperator u : F ∗ → E, namely u(ψ) =

∑k ψ(yk)xk for any ψ ∈ F ∗. We say that

u is associated with z. Under this identification, E ⊗ F coincides with the spaceof all finite rank and w∗-to-norm continuous operators from F ∗ into E. Clearly

E⊗F → B(F ∗, E) isometrically. (A.1)

Of course E⊗F ⊂ B(E∗, F ) isometrically too. Likewise if F is a dual space withpredual F∗, we may identify E ⊗ F with finite rank operators from F∗ into Eand we have E⊗F → B(F∗, E) isometrically.

A.3.2 Let Ω be a compact space. We let C(Ω; E) denote the Banach space of allcontinuous functions f : Ω → E. Equip C(Ω; E) with the supremum norm, thatis, ‖f‖ = sup‖f(t)‖E : t ∈ Ω. We simply write C(Ω) for C(Ω; C). We may

362 Tensor products of Banach spaces

identify C(Ω)⊗E with a subspace of C(Ω; E) by regarding any f =∑

k gk ⊗ xk

(with gk ∈ C(Ω) and xk ∈ E) as a function, f(t) =∑

k gk(t)xk. Then the normof f in C(Ω; E) is equal to sup|∑k gk(t)ϕ(xk)| : t ∈ Ω, ϕ ∈ Ball(E∗), henceis equal to its injective tensor norm by (A.1). Moreover C(Ω) ⊗ E is dense inC(Ω; E) (e.g. see [407, IV; 7.3] for a proof) hence we have

C(Ω)⊗E ∼= C(Ω; E) isometrically. (A.2)

Similarly if Ω is a locally compact space, we let C0(Ω; E) denote the Banachspace of all continuous functions from Ω to X vanishing at ∞, equipped withthe supremum norm. Then (A.2) extends to the relation C0(Ω)⊗E ∼= C0(Ω; E).

A.3.3 A bounded bilinear map T : E ×F → Z is a bilinear map for which thereis a constant C such that ‖T (x, y)‖ ≤ C‖x‖‖y‖, for all x ∈ E, y ∈ F . The leastsuch C is written as ‖T ‖. We say that T is contractive if ‖T ‖ ≤ 1.

The Banach space projective tensor product E⊗F is the completion of thealgebraic tensor product E ⊗ F in a certain norm. We do not need to explic-itly write down this norm, instead we will simply state the universal propertyof E⊗F , namely that it linearizes bounded bilinear maps. More precisely, thecanonical map ⊗ : E × F → E⊗F is a contractive bilinear map, and for anybounded bilinear T : E × F → Z, the associated linear map E ⊗ F → Z iscontinuous with respect to the just mentioned norm, and extends to a boundedlinear map T : E⊗F → Z with ‖T‖ = ‖T ‖. From this it is easy to see that

B(E⊗F, Z) ∼= B(E, B(F, Z)) ∼= B(F, B(E, Z)) isometrically. (A.3)

In particular, via the obvious isomorphisms,

(E⊗F )∗ ∼= B(E, F ∗) ∼= B(F, E∗) isometrically. (A.4)

A.3.4 A bounded operator u : E → F is said to be 2-summing if I2 ⊗u extendsto a bounded operator from 2⊗E into 2(F ). We set

π2(u) =∥∥I2 ⊗ u : 2⊗E −→ 2(F )

∥∥.It is not hard to check that π2(· ) is a (complete) norm on the space Π2(E, F ) of2-summing operators from E into F . We also note that π2(u) is the supremumof the norms of the mappings I2n

⊗ u : 2n⊗E → 2

n(F ), for n ∈ N.

A.3.5 We review three tensor products related to Hilbert space factorization.Let (ek)k denote the canonical basis of 2. For z ∈ E ⊗ F define

γ2(z) = inf∥∥∥ n∑

k=1

ek ⊗ xk

∥∥∥2⊗E

∥∥∥ n∑k=1

ek ⊗ yk

∥∥∥2⊗F

(A.5)

where the infimum is over all finite families (xk)nk in E and (yk)n

k in F such thatz =

∑nk=1 xk ⊗ yk. The quantity γ2 is a norm on E ⊗ F , and we let E ⊗γ2 F

denote the resulting completion.

Appendix 363

Consider∑n

k=1 ek ⊗ xk for x1, . . . , xn ∈ E as above, and let u : E∗ → 2 bethe associated linear map. According to (A.1),

∥∥∑nk=1 ek ⊗ xk

∥∥2⊗E

is equal tothe usual operator norm of u. Analoguously, we define

∥∥∑nk=1 ek ⊗ xk

∥∥Π2(E∗,2)

to be π2(u). Then for z ∈ E ⊗ F we define

g2(z) = inf∥∥∥ n∑

k=1

ek ⊗ xk

∥∥∥Π2(E∗,2)

∥∥∥ n∑k=1

ek ⊗ yk

∥∥∥2⊗F

(A.6)

where the infimum is over ways to write z =∑n

k=1 xk ⊗yk, with xk ∈ E, yk ∈ F .As above, g2 is a norm, and we let E ⊗g2 F denote the resulting completion.

Next we let

γ∗2(z) = inf

∥∥∥ n∑k=1

ek ⊗ xk

∥∥∥Π2(E∗,2)

∥∥∥ n∑k=1

ek ⊗ yk

∥∥∥Π2(F∗,2)

, (A.7)

where again the infimum over all ways to write z =∑n

k=1 xk ⊗ yk in E ⊗ F .Again, γ∗

2 is a norm and we let E ⊗γ∗2

F denote the resulting completion.It is not hard to see that ‖ · ‖∨ ≤ γ2(·) ≤ g2(·) ≤ γ∗

2 (·) ≤ ‖ · ‖∧ on E ⊗ F .

A.3.6 Let H, K be Hilbert spaces. Then the space K⊗H of finite rank operatorsfrom H to K is dense both in S∞(H, K) and in S1(H, K). It is clear from (A.1)that S∞(H, K) ∼= K⊗H, and it turns out that

S1(H, K) ∼= K⊗H ∼= K ⊗γ∗2

H. (A.8)

A.4 BANACH ALGEBRAS

We refer the reader in this section to [74, 106, 297] for any omitted proofs ofassertions below, or for more background.

A.4.1 A C-Banach algebra is a Banach space A which is also an algebra suchthat ‖ab‖ ≤ C‖a‖‖b‖ for all a, b ∈ A. If C = 1 then we simply say Banachalgebra. We say that a Banach algebra A is unital if it has a unit (i.e. identity) ofnorm 1. A bounded approximate identity is a bounded net (et)t with eta → a andaet → a. This is a contractive approximate identity (cai) if moreover ‖et‖ ≤ 1for all t. If A possesses a cai we say that A is approximately unital.

A.4.2 Let A be a unital Banach algebra. A state on A is a contractive unitalfunctional on A. An element h ∈ A is said to be Hermitian if ϕ(h) ∈ R forevery state ϕ on A. Note that by the Hahn–Banach theorem we may replacestates on A here by contractive unital functionals on Span 1, h. Equivalently,h is Hermitian if ‖ exp(ith)‖ ≤ 1 for all t ∈ R. We write Her(A) for the setof Hermitian elements of A. By the first definition of Hermitians above, it isevident that if u : A → B is a unital contractive linear map between unitalBanach algebras, then u(Her(A)) ⊂ Her(B).

364 C∗-algebras

It is well-known that if ϕ(h) = 0 for all states ϕ on A, then h = 0. Also, if Ais a unital C∗-algebra, then Her(A) is exactly the set Asa of selfadjoint elements.

A nonzero homomorphism on a unital Banach algebra is a state, and is calleda character. The maximal ideal space MA of a commutative unital Banach algebraA is the set of characters of A, together with the w∗-topology inherited from A∗.

A.4.3 Suppose that A is an approximately unital Banach algebra. We maydefine a unitization of A by considering the canonical ‘left regular representation’λ : A → B(A), and identifying A + C1 with the span of λ(A) + CIA, which iseasy to see is a unital Banach subalgebra of B(A). Thus if a ∈ A and α ∈ C then

‖a + α1‖ = sup‖ac + αc‖ : c ∈ A, ‖c‖ ≤ 1

. (A.9)

We write this unitization as A1 if A is nonunital.It is occasionally useful that there are some other equivalent expressions for

the quantity above. For example, if (et)t is a cai for A then

‖a + α1‖ = limt

‖aet + αet‖ = supt

‖aet + αet‖. (A.10)

To see that this limit exists and that these quantities are the same, let β bethe quantity on the right-hand side of (A.9), let ε > 0 be given, and choosec ∈ Ball(A) with ‖ac + αc‖ > β − ε. Then ‖aetc + αetc‖ → ‖ac + αc‖, so thatthere is a t0 with ‖aetc + αetc‖ > β − ε for t ≥ t0. Then

β − ε < ‖aetc + αetc‖ ≤ ‖aet + αet‖ ≤ β.

This proves what we asserted. One can see that one does not change the quan-tities in (A.9) and (A.10) by considering expressions ca + αc or eta + αet.

As a consequence of (A.10), if π : A → B is a contractive homomorphismbetween Banach algebras, such that (π(et))t is a cai for B for some cai (et)t forA, then π extends uniquely to a contractive unital homomorphism π betweenthe unitizations. To see this define π(a + α1) = π(a) + α1, for a ∈ A and α ∈ C,and appeal to formula (A.10) twice to see that π is contractive too. If, further,π is isometric, then by (A.10) it follows that π is isometric too.

A.5 C∗-ALGEBRAS

We refer the reader in this section to any book on C∗-algebras for any omittedproofs of assertions below, or for more background.

A.5.1 A concrete C∗-algebra is a closed ∗-subalgebra of B(H) for a Hilbert spaceH . A von Neumann algebra is a concrete C∗-algebra which is closed in the w∗-topology on B(H), and which contains IH . By A.1.2 it follows that the producton a von Neumann algebra is separately w∗-continuous in each variable, andthat the involution is also w∗-continuous. An (abstract) C∗-algebra is a Banachalgebra A with a conjugate linear involution ∗ : A → A such that (a∗)∗ = a and(ab)∗ = b∗a∗, for all a, b ∈ A, which also satisfies the C∗-identity: ‖a∗a‖ = ‖a‖2,for a ∈ A. A C∗-subalgebra of a C∗-algebra is a closed selfadjoint subalgebra.

Appendix 365

A.5.2 An element a in a C∗-algebra A is positive if a = b∗b for some b ∈ A.We write A+ for the set of such elements, and write a ≤ b if b − a ∈ A+, and ifa and b are selfadjoint. We will assume familiarity with the basic properties ofthis ordering and the continuous functional calculus for normal operators.

A C∗-algebra is approximately unital, and indeed has a positive increasingcai. The unitization A1 in A.4.3 of a C∗-algebra A is a C∗-algebra.

A functional ϕ ∈ A∗ is called a state if it is ‘positive’ (i.e. ϕ(a) ≥ 0 if a ≥ 0)and has norm 1. This is equivalent to other definitions of states elsewhere inthis book. Indeed there is a host of equivalent definitions of states, or indeed ofelements in A+. For example,

a ≥ 0 ⇔ ϕ(a) ≥ 0 for all states ϕ of A (or of A1). (A.11)

A ∗-homomorphism (resp. ∗-isomorphism) is a homomorphism (resp. isomor-phism) satisfying π(a∗) = π(a)∗ for all a. Such maps are automatically positive.

A.5.3 A primary result in the subject of operator algebras is the Gelfand–Naimark theorem, which states that every abstract C∗-algebra A is ∗-isomorphicto a concrete C∗-algebra. A major part of the proof of this result is the Gelfand–Naimark–Segal (GNS) construction, which shows that the positive functionals ϕon a C∗-algebra A are the functions of the form 〈π(·)ζ, ζ〉, for a ∗-homomorphismπ : A → B(H), and a vector ζ ∈ H , such that [π(A)ζ] = H . If ϕ is a state thenwe can take ‖ζ‖ = 1.

A W ∗-algebra is an abstract C∗-algebra with a Banach space predual, suchthat there exists a w∗-continuous ∗-isomorphism from A onto a von Neumannalgebra. By a theorem of Sakai, the last part of this definition is redundant, butwe will avoid using this deeper fact here.

A.5.4 The commutative C∗-algebras, are by a theorem of Gelfand, exactlythe spaces C0(Ω) mentioned in A.3.2. We will assume that the reader is familiarwith the basic correspondences between constructions in the category of compactspaces K, and the category of commutative unital C∗-algebras. For example,the correspondences between closed subsets of K, and ideals of C∗-algebras andtheir quotients. Or, more generally, the correspondence between continuous (resp.and one-to-one, and surjective) functions between compact spaces, and unital(resp. and surjective, and one-to-one ) ∗-homomorphisms between commutativeunital C∗-algebras. These correspondences follow easily from Gelfand’s theory.Recall that algebraic isomorphisms between C(K)-spaces (resp. between closedsubalgebras of C(K)-spaces containing constant functions), are ∗-isomorphisms(resp. isometric).

A.5.5 The universal representation πu : A → B(Hu) of a C∗-algebra A is con-structed by taking a direct sum of ∗-homomorphisms associated with all thestates on A by the GNS construction (see A.5.3 above). Thus πu has the prop-erty that for any state ϕ of A, there exists a unit vector ξ ∈ Hu such thatϕ = 〈πu(·)ξ, ξ〉.

366 C∗-algebras

Every nondegenerate ∗-homomorphism from A to B(H), for any Hilbert spaceH , is unitarily equivalent to the restriction to an invariant subspace of a directsum of sufficiently many copies of πu.

Theorem A.5.6 The second dual of a C∗-algebra A is a W ∗-algebra. IndeedA∗∗ is linearly isometric, via a w∗-continuous map, to a von Neumann algebra.

Proof Let πu : A → B(Hu) be the universal representation of A (see A.5.5).Let πu : A∗∗ → B(Hu) be the unique (contractive) w∗-continuous extension ofπu as in A.2.2 (thus πu = i∗ π∗∗

u , where i : S1(Hu) = B(Hu)∗ → B(Hu)∗ is thecanonical injection). Let ψ be a linear functional on A of norm 1. It is well-knownthat we may write ψ = 〈π(·)ξ, ξ′〉, for unit vectors ξ, ξ′ and a ∗-homomorphismπ (see 1.2.8 for an easy proof of this fact; also Zsido has recently shown us abeautiful simple proof that will be in [401]). For a ∈ A we have that

|ψ(a)|2 ≤ ‖π(a)ξ‖2 = 〈π(a∗a)ξ, ξ〉,

from which we see that the positive map ϕ = 〈π(·)ξ, ξ〉 has norm 1, and is there-fore a state on A. Thus by A.5.5, ϕ = 〈πu(·)η, η〉 for a unit vector η ∈ Hu. We nowhave |ψ(a)|2 ≤ 〈πu(a∗a)η, η〉 = ‖πu(a)η‖2. Thus the functional πu(a)η → ψ(a)on [πu(A)η] is well defined and contractive. By the Riesz representation theo-rem, there exists a vector ζ ∈ Hu such that ψ = 〈πu(·)η, ζ〉. Hence, since Ais w∗-dense in A∗∗, we have 〈ν, ψ〉 = 〈πu(ν)η, ζ〉 for ν ∈ A∗∗. We deduce that|〈ν, ψ〉| ≤ ‖πu(ν)‖, which implies that πu is an isometry. Since A is w∗-dense

in A∗∗, we have πu(A∗∗) ⊂ πu(A)w∗

. By the Krein–Smulian theorem A.2.5, the

range of πu is w∗-closed. Hence πu(A)w∗

⊂ πu(A∗∗), and so πu(A)w∗

= πu(A∗∗).

Thus A∗∗ is linearly isometric to the von Neumann algebra πu(A)w∗

.

A.5.7 By the last result together with an observation in A.5.1, we see that ifA is a C∗-algebra then A∗∗ possesses a product extending that of A, which isseparately w∗-continuous in each variable, and with respect to which A∗∗ is aW ∗-algebra. By Goldstine’s lemma A.2.1 such a product on A∗∗ must be unique.We call this the canonical W ∗-algebra structure on A∗∗.

Proposition A.5.8 Let π : A → B be a homomorphism between C∗-algebras.Then π is contractive if and only if π is a ∗-homomorphism. If these hold thenfirst, π has closed range, and induces a ∗-isomorphism between the C∗-algebrasA/Ker(π) and π(A). Second, π is isometric if and only if π is one-to-one.

Proof Most of these may be found in any book on C∗-algebras (and will begeneralized in 8.3.2). We merely sketch the part that cannot be so easily found,namely that a contractive homomorphism π is a ∗-homomorphism. If A is unitalthen we can assume that B is unital and that π(1) = 1 (otherwise replace Bby π(1)Bπ(1)). If a ∈ A+ and if ϕ is a state on B, then ϕ π is a state onA. Using (A.11) twice shows that ϕ(π(a)) ≥ 0, and that π(a) ≥ 0. Thus π ispositive, and is therefore also a ∗-homomorphism. In the nonunital case extend π

Appendix 367

to a contractive homomorphism between the unitization C∗-algebras (or simplyconsider π∗∗ : A∗∗ → B∗∗, and use the results on second duals above), and thenapply the ‘unital case’ above to obtain the result.

Theorem A.5.9 [409, 39] Let A be a C∗-algebra, B a Banach algebra, andπ : A → B a contractive homomorphism. Then π(A) is norm closed, and itpossesses an involution with respect to which it is a C∗-algebra. Moreover, π isthen a ∗-homomorphism into this C∗-algebra. If π is one-to-one then π is anisometry.

Proof We may assume that B is the closure of π(A). If A is unital then B isunital, and π(1A) = 1B. Since π(Her(A)) ⊂ Her(B) (see A.4.2),

π(A) = π(Her(A)) + iπ(Her(A)) ⊂ Her(B) + iHer(B) ⊂ B.

Hence Her(B)+iHer(B) is dense in B. By the Vidav–Palmer theorem [74,298] Bis a C∗-algebra (in fact Her(B) + iHer(B) is always norm closed [297, Theorem2.6.7]). Thus we may use A.5.8 if necessary to obtain the stated conclusions. If Ais not unital, we may conclude as in the proof of A.5.8 (here one may use A.4.3,for example, to extend π to suitable unitizations).

A.5.10 The following simple principle is useful in proving the Kaplansky densitytheorem, and variants of it. Namely, suppose that E is a closed subspace of a dualspace F ∗, and that we wish to prove that Ball(E) is w∗-dense in the unit ballof its w∗-closure. By A.2.2, there is a w∗-continuous contraction u : E∗∗ → F ∗

extending the inclusion map u : E → F ∗. Suppose that u takes the open unit ballof E∗∗ onto the open unit ball of Ran(u). This is often automatic, as in the casethat E is a ∗-subalgebra of a von Neumann algebra F ∗ (this follows by A.5.8;in this case u is a homomorphism by a standard w∗-density argument using theseparate w∗-continuity in A.5.7 and A.5.1). It is easy to check, by A.2.5 (1), thatRan(u) is w∗-closed. Thus E

w∗= Ran(u). Indeed, if z ∈ Ran(u), with ‖z‖ < 1,

then there exists η ∈ E∗∗ with u(η) = z and ‖η‖ < 1. By A.2.1, there is a net(et)t in Ball(E), converging in the w∗-topology to η. Thus et = u(et) → u(η) = zin the w∗-topology of F ∗. It is quite obvious that this implies that Ball(E) isw∗-dense in the unit ball of E

w∗.

A.6 MODULES AND COHEN’S FACTORIZATION THEOREM

Again, see [106,297,321] for omitted proofs, complementary results, and history.

A.6.1 Reflecting on the difference between algebras and rings, it is naturalthat this difference should be reflected in the modules over each. For us, a (left)module over an algebra A will always be a vector space X over C, which is a(left) module in the traditional sense, but we insist also that

(αa)x = a(αx) = α(ax), a ∈ A, α ∈ C, x ∈ X.

There is then a one-to-one correspondence between left A-modules, and repre-sentations of A on vector spaces (that is, homomorphisms π from A into the

368 Modules and Cohen’s factorization theorem

algebra of linear maps on X , for a vector space X). This correspondence is givenby the formula

π(a)(x) = ax, a ∈ A, x ∈ X.

We call π the canonical homomorphism associated with the module action.Now suppose that X is a (left, say) A-module over a Banach algebra A. We

say that X is a normed (resp. Banach) A-module if X is a normed (resp. Banach)space, and the module action A × X → X is a contractive bilinear map. This isthe same as saying that the associated homomorphism π : A → B(X) in the lastparagraph, is contractive. We shall not do so here, but for many results (such asthose in Chapter 5) one may want to weaken the last condition to allow boundedmodule actions.

An A-B-bimodule is a left A-module which is also a right B-module, suchthat the two actions commute. That is, a(xb) = (ax)b for a ∈ A, b ∈ B, x ∈ X .A Banach A-B-bimodule is a Banach space and a bimodule, which is both a leftand a right Banach module.

We say that a (left) Banach A-module X is a nondegenerate A-module, orthat A acts nondegenerately on X , if X equals the norm closure of the linearspan of the products ax for a ∈ A and x ∈ X . If A has a bounded approximateidentity (et)t then this is equivalent to saying that etx → x for all x ∈ X , andwe will see some other equivalent conditions in A.6.4 below. A bimodule will becalled nondegenerate if it is nondegenerate both as a left and a right module.

Theorem A.6.2 (Cohen’s factorization) Suppose that A is a Banach algebrawith a bounded approximate identity, and that X is a left Banach A-module.Then X is a nondegenerate A-module if and only if any x ∈ X may be writtenin the form x = ay for some a ∈ A, y ∈ X. In this case, if further A has a cai,and if x has norm < 1, then we may also choose a and y with norm < 1.

Corollary A.6.3 Let A, X be as in A.6.2. Then every A-module map f : A → Xis bounded.

Proof Take a sequence an → 0 in A. Let Y = c0(A), the set of sequencesconverging to 0 with terms in A. This is a right Banach A-module, which is nothard to see is nondegenerate. Applying Cohen’s theorem we may write an = bnb,for b, bn ∈ A with bn → 0. Thus f(an) = bnf(b) → 0.

A.6.4 If X is a (not necessarily nondegenerate) left Banach A-module over aBanach algebra A then we define the essential part of X to be the norm closureof the linear span of the products ax for a ∈ A and x ∈ X . If A has a boundedapproximate identity (et)t then this set is clearly exactly the set of x ∈ X suchthat etx → x. Clearly in this case the essential part is a nondegenerate A-module.Thus in this case (i.e. if A has a bounded approximate identity), it follows fromCohen’s theorem applied to the essential part, that the essential part of X equalsax : a ∈ A, x ∈ X, and also equals

∑nk=1 akxk : n ∈ N, ak ∈ A, xk ∈ X.

Thus our use of the notation AX in this book (see Section 1.1), at least in thiscase, is less ambiguous than it may seem.

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Index

2-summing, 362

Adjoint module, 297Adjoint operator space, X, 12, 60, 109Adjointable maps, B(Y, Z), K(Y, Z), 299Adjointable multipliers, Al(X), 172, 329Algebraically finitely generated, 308Amenability, 283, 285Amplification, 4, 13Approximate identity, 363Approximately unital, 49, 363Arens product, 78Arens regular, 79

Balanced bilinear map, 119Banach module (or bimodule), 368Banach space injective tensor product,

E⊗F , 361Banach space projective tensor product,

E⊗F , 362Banach–Stone theorems, 173, 191, 328Bimodule Paulsen system, 128Boundary representation, 150

Cσ-space, 349C*-algebra, 5, 364C*-cover, 50C*-envelope, C∗

e (·), 156C*-extension, 150C*-module, 297Cai (contractive approximate identity),

49, 363Centralizer algebra, Z(E), 131, 172CES-representation, 117Choquet boundary, Ch(E), 148Cogenerator, 111Commutative triple system, 349Complemented submodule, 274Complete order isomorphism or injection,

17Completely bounded bimodule maps,

ACBB(X, Y ), 124Completely bounded maps, CB(X, Y ), 9Completely bounded module maps,

ACB(X, Y ), CBB(X, Y ), 124Completely bounded multilinear map, 30

Completely bounded norm, ‖ ‖cb, 4Completely positive, 17Conditional expectation, 155, 166Corner of a map, 87Corner of an algebra, 87Corner-preserving, 87Correspondence, 345, 347Countably generated module, 311

Decomposable maps, DEC(A, B), 256Decomposable norm, ‖ ‖dec, 257Delta norm, δ, 248Delta tensor product, X ⊗δ B, 249Diagonal, ∆(·), 19, 50Direct limit, 67Direct sum, 3, 8, 26, 57, 105, 106, 110, 139Direct sum C∗-module, ⊕c

i Yi, 300Dirichlet algebra, 159Disc algebra, A(D), 59Double commutant property, 114Dual operator algebra, 88, 92Dual operator module (or bimodule), 136,

183Dual operator space, 10, 22, 39

Enveloping operator algebra, O(B), 71Equivalence bimodule, 297, 336Essential extension, 153Essential ideal, 85Essential module maps, ACBess(X, Y ),

CBessB (X, Y ), 124

Essential part of a module, 368Extension of an operator space, 153Extremal module actions, 177

Free product, A ∗ B, 75Full C∗-module, 297Function module, 131, 135Function multiplier algebra, M(E), 131,

172Function-extension, 148

Generator, 111

h-module (or bimodule), 103Haagerup tensor product, X ⊗h Y , 30

386 Index

Hardy space, H∞(D), 91Hermitian, 363Hilbert column space, Hc, 11Hilbert module, HMOD, 105, 109, 110Hilbert row space, Hr , 11Hilbert space factorization, 362

I11, I22, I(S(X)), 162Ideal, 66Induced representation, 319, 320, 345Injective bimodule, 129Injective envelope for bimodules, 179Injective envelope, I(X), 153, 154Injective operator space, 7Interior tensor product, Y ⊗A Z, 315Interpolation, 15, 66

Jointly completely bounded, 35

Left multiplier algebra, LM(A), 82, 84Left multipliers of an operator space,

Ml(X), 168, 171, 329Linking algebra, L(Y ), 303, 335Locally reflexive, 264Logmodular algebra, 159

M-ideal (or summand), 184M-projection, 134Matrix normed algebra, 68Matrix normed module (or bimodule),

104Maximal C∗-algebra, C∗

max(A), 69Maximal operator space, Max(E), 10Maximal tensor product, A ⊗max B, 233Minimal operator space, Min(E), 10Minimal tensor product, X ⊗min Y , 27, 57Module complementation property, 275Module Haagerup tensor product,

X ⊗hA Y , 119, 315Module operator space projective tensor

product, X⊗A Y , 119

Module tensor product, X ⊗A Y , 119Morita equivalence, 297, 318, 319Multiple of a Hilbert space, 3, 110Multiple of an operator or representation,

3, 110Multiplier algebra, M(A), 84Multiplier matrix norms, 170

Noncommutative H∞, 159Noncommutative Shilov boundary, 157,

164, 326Noncommuting variables, 200, 206Nondegenerate module, 368Nondegenerate representation, 51

Normal delta norm, σδ, 251Normal delta tensor product, X⊗σδM ,

251Normal dual bimodule, 137Normal Haagerup tensor product,

X ⊗σh Y , 41Normal Hilbert module, NHMOD, 138Normal minimal tensor product, X⊗Y , 39Normal spatial tensor product, M ⊗N , 90Normal tensor product, A ⊗nor M , 233Normal virtual h-diagonal, 287Nuclear C∗-algebra, 238, 259Nuclearity, 259, 269, 349

One-sided M -ideal, 184One-sided M -projection, 174One-sided M -summand, 184Operator algebra, 49, 62Operator module (or bimodule), OMOD,

102, 124Operator space, 5Operator space projective tensor product,

X⊗ Y , 35

Operator space structure, 5Operator system, 17Oplication, 175Opposite operator space, Xop, 12, 60, 108Orthogonally complemented C∗-module,

300, 305, 337Orthonormal basis, 340

p-summable sequences, p, Op, 209, 211Paulsen system, S(X), 21Prolongation, 107

Q-algebra, 215Q-space, 219Quasi-equivalent, 110Quasibasis, 311Quasimultipliers, 192Quotient map, 2Quotient module, 106, 109, 117, 139Quotient operator algebra, 66Quotient operator space, 8

Reducing property, 275Reducing submodule, 109Reflexive subspace, 360Rigged module, 313, 345Right multiplier algebra, RM(A), 84Right multipliers of an operator space,

Mr(X), 172Rigid, 153Ruan’s axioms, 7

Index 387

Schatten spaces, Sp, 3, 225, 359Schur product, 214Selfadjoint algebra, 269Selfdual C∗-module, 332Semi-invariant subspace, 109Semidiscrete W ∗-algebra, 238, 259Semidiscreteness, 260, 271, 348Semigroup operator algebra, O(G), 72Shilov boundary, ∂E, 149Shilov inner product, 165Similarity, 195Spatially equivalent, 110Stabilization, 312, 342Standard form, 138State, 57, 363, 365Subtriple, 161, 323

Ternary ring of operators (TRO), 103,161, 336

Trace class, 23, 359Triangular algebra, T n, T ∞, 242Triple envelope, T (X), 164, 326Triple extension, 326Triple ideal, 324Triple morphism, 161, 322Triple product, 161, 164Triple system, 322

Ultraproduct, 16, 62Ultraweak direct sum, ⊕wc

i∈I Yi, 341Uniform algebra, 59, 147Unital function space, 147Unital operator space, 16Unital-subalgebra, 4Unital-subspace, 4Unitary C∗-module map, 300Unitization, A1, 52, 55, 364Universal Hilbert module, 111Universal representation, πu, 70, 365

Virtual h-diagonal, 285Virtual diagonal, 283

W*-algebra, 89, 365W*-full, 332W*-module, 332Weak Morita equivalence, 336Weak expectation property (WEP), 270,

348Weak* Haagerup tensor product,

X∗ ⊗w∗h Y ∗, 42Weak* module Haagerup tensor product,

X ⊗w∗hM Y , 141, 347

X-projection, 152

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Operator Algebras and Their Modules: An Operator Space Approach (London Mathematical Society...

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