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6 INTERPOLATION IN SEMIGROUPOID ALGEBRAS

MICHAEL A. DRITSCHEL1, STEFANIA MARCANTOGNINI1 AND SCOTT MCCULLOUGH2

ABSTRACT. A seminal result of Agler characterizes the so-called Schur-Agler class of functions on thepolydisk in terms of a unitary colligation transfer function representation. We generalize this to the unitball of the algebra of multipliers for a family of test functions over a broad class of semigroupoids. Thereis then an associated interpolation theorem. Besides leading to solutions of the familiar Nevanlinna-Pickand Caratheodory-Fejer interpolation problems and their multivariable commutative and noncommuta-tive generalizations, this approach also covers more exotic examples.

1. INTRODUCTION

The transfer function realization formalism for contractive multipliers of (families of) reproducingkernel Hilbert spaces and Agler-Pick interpolation has been, starting with the work of Agler [3],generalized from the classical setting ofH∞(D) (D the unit disk), to many other algebras of functions.

In this paper we pursue realization formulæ and Agler-Pick interpolation in two directions. Firstwe consider an algebra of functions on a semigroupoidG. The precise definition of a semigroupoidis given below. In essence it can be thought of as an ordered unital semigroup, though perhaps withmore than one unit. For now the salient point is that the semigroupoid structure means that the algebraproduct generalizes both the pointwise and convolution products. This setting has the advantageof being fairly concrete and amenable to study using reproducing kernel Hilbert space ideas andtechniques while at the same time connecting with the theoryof graphC∗-algebras.

Secondly, we view the norm on the algebra as being determinedby a (possibly infinite) collectionΨ of functions onG, referred to as test functions. Results on Agler-Pick interpolation (in the classicalsense) for both finite and infinite collections of test functions with varying amounts of additionalimposed structure can be found in [6, 8] and this point of viewgoes back at least to [4]. A collectionof test functions determines a family of kernels, and vice versa. This duality between test functionsand kernels will have a familiar feel to those acquainted with Agler’s model theory [2]. The advantageof such an approach is that it allows us to consider interpolation problems on, for example, polydisksand multiply connected domains [22].

We should mention that Kribs and Power [29, 30] introduce a somewhat more restrictive notionof a semigroupoid algebra. These are related to so-called quiver algebras of Muhly [38], and arethe nonselfadjoint analogues of the higher rank graph algebras of Kumjian and Pask [31]. In thesepapers order is either imposed through the presence of a functor from the semigroupoid toNd, orby the assumption of freeness. In either case, the resultingobject is cancellative, and there is arepresentation (related to our Toeplitz representation oncharacteristic functionsχa; see section 1.3)in terms of partial isometries and projections on a generalized Fock space with orthonormal basislabelled by the elements of the semigroupoid. The algebras of interest in these papers are obtained as

2000Mathematics Subject Classification.47A57 (Primary), 47L55, 47L75, 47D25, 47A13, 47B38, 46E22 (Secondary).Key words and phrases.interpolation, transfer functions, semigroupoid, noncommutative function space, multiply con-

nected domains, Nevanlinna-Pick, Caratheodory-Fejer.1Research supported by the EPSRC.2Research supported by the NSF.

1

the weak operator topology closure of the algebras coming from the left regular representation (i.e.,the projections and partial isometries mentioned above), and so in a natural sense are the multiplieralgebras for these Fock spaces.

The Kribs and Power semigroupoid algebras include the noncommutative Toeplitz algebras firstintroduced in [44]. Pick and Caratheodory interpolation has been considered in this context by Ariasand Popescu [11] and Davidson and Pitts [20] (with some earlier work by Popescu on these and relatedinterpolation problems to be found in [42, 43, 45, 46]), and somewhat more generally by Jury andKribs [27]. See also [28]. In fact, while the commutant lifting theorem unifies the classical Pick andCaratheodory-Fejer interpolation problems, to our knowledge, Jury’s PhD dissertation [26] was thefirst to do so in terms of the positivity of kernels, and also the first to give a concrete realization formulafor the case of the semigroupN. Recently, realization formulæ in a noncommutative setting have alsobeen investigated in [17]. Muhly and Solel [39] have considered Nevanlinna-Pick interpolation fromthe vantage of what they call Hardy algebras, covering many of the examples mentioned above alongwith the statement of a realization formula.

Interpolation problems on domains other than the unit disk in C have been of long-standing interest.On multiply connected domains, the seminal work is that of Abrahamse [1], with further contributionsto be found in [16, 21, 34, 35, 36, 48]. Regarding domains inC

n, the fundamental paper of Agler[3] provides the foundation upon which most subsequent workis based. A sampling of papers ofparticular interest in this direction includes [5, 6, 7, 8, 9, 14, 15, 18, 19, 23, 24, 37].

In this paper we have for clarity restricted our attention toscalar valued interpolation (although westray a bit in the examples in Section 8). We do not anticipatethat the generalization to the matrixcase will provide any obstacles which cannot be overcome with what are by now standard techniques.Indeed we have ensured that none of the proofs found below depend on the commutativity of thecoefficients of our functions, and it appears likely that theresults will continue to hold when thecoefficients come from, say, a norm closed subalgebra of aC∗-algebra. This is left for later work.

A few words about the organization of the paper. The rest of Section 1 outlines the basic toolsused throughout: semigroupoids,⋆-products, Toeplitz representations, test functions and reproduc-ing kernels, theC∗-algebra generated by evaluations on the set of test functions and its dual, andtransfer functions. This is followed by a statement of the main results, which are the realization andinterpolation theorems.

In Section 2 we more closely study the⋆-product, especially with regards to inverses and positivity.Section 3 begins with a consideration of the semigroupoid algebra analogue of the Szego kernel,

and highlights the close connection between positivity of these kernels and complete positivity ofthe⋆-product map (a generalization of the Schur product map). Asnoted earlier in the introduction,multiplier algebras arising from a single reproducing kernel are too restrictive for us, so we detail howwe will handle families of kernels and the associated families of test functions. Cyclic representationsof the space of functions over certain finite sets (they should be “lower” with respect to the order onthe semigroupoid) which are contractive on test functions are shown to be connected to reproducingkernels. This plays a crucial role in the Hahn-Banach separation argument in the realization theorem.

Given a positive object, an analyst’s immediate inclination is to factor. The fourth section is devotedto a factorization result for positive kernels on the dual oftheC∗-algebra from Section 1, as well asmaking connections to representations of this algebra.

Two other key items needed in the proof of the realization theorem are taken up in Section 5. Thefirst is the cone of matricesCF . For the separation argument in the proof of the realizationtheorem towork, we must know thatCF is closed and has nonempty interior. Closedness requires a surprisinglydelicate argument, and so occupies the bulk of the section. We also show that certain sets of kernelsin the dual of theC∗-algebra mentioned above are compact.

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Sections 6 and 7 comprise the proof of the realization theorem and the interpolation theorem. Thefirst implication of the proof of the realization theorem is essentially the most involved part, but due toall of the preparatory work in Sections 3–5, is dispensed with quickly. Other parts involve variationson themes which will be familiar to those acquainted with recent proofs of interpolation results. Theseinclude an application of Kurosh’s theorem, a lurking isometry argument, and a fair amount of tediouscalculation. After the proof of the realization theorem, the proof of the interpolation theorem is almostan afterthought.

In Section 8 we turn briefly to a menagerie of examples, both old and new. Though we mention itin passing, we have postponed the application to Agler-Pickinterpolation on an annulus to a separatepaper for two reasons. First, the argument is fairly long andinvolves ideas and techniques unrelated tothe rest of this paper; and second, the underlying semigroupoid structure is that of Pick semigroupoid(which is essentially trivial) and as such the version of Theorem 1.3 which is needed does not requirethe semigroupoid overhead. In any case, this section barelyscratches the surface of what is possible!

We would like to thank Robert Archer for his careful reading of the paper, and the many usefulcomments and questions which have without a doubt improved it.

1.1. Semigroupoids. There is no standard name in the literature for the sort of object on whichwe want to define our function algebras. The names “small category” and “semigroupoid” are twocommonly used terms, though our definition differs somewhatfrom that standardly given for either ofthese. We have opted for the latter.

The term “semigroupoid” was originally coined by Vagner, asfar as we are aware [47]. Similarnotions are familiar from the theory of inverse semigroups (see for example, [32] or [40]), and havebeen explored in connection with the classification theory of C∗-algebras. The use of semigroupoidsin the study of nonselfadjoint algebras originates with Kribs and Power [29], though again, their useof the terminology is a bit different from ours.

So letG be a set with a functionX ⊂ G×G → G, called apartial multiplicationand writtenxyfor (x, y) ∈ X. We defineidempotentsas those elementse of G such thatex = x wheneverex isdefined andye = y wheneverye is defined. Note that these are commonly referred to asidentitiesinthe groupoid literature.

The following laws are assumed to hold:

(1) (associative law) If either (ab)c or a(bc) is defined, then so is the other and they are equal.Also if ab, bc are defined, then so is(ab)c.

(2) (existence of idempotents) For eacha ∈ G, there existe, f ∈ G with ea = a = af .Furthermore ife ∈ G satisfiese2 = e, thene is idempotent.

(3) (nonexistence of inverses) If a, b ∈ G andab = e wheree is idempotent, thena = b = e.(4) (strong artinian law ) For anya ∈ G the cardinality of the set{z, b, w : zbw = a} is finite.

Moreover there is anN <∞ such thatsupa,c∈G card{b ∈ G : cb = a} ≤ N .

Hereafter we refer to a setG with a partially defined multiplication with all of the properties so farlisted as asemigroupoid.

Since we have associativity, we can mostly forget parentheses. If we were to reverse the third law(so that every element has an inverse), then the first three rules would comprise the definition of agroupoid. The strong artinian law is related to the (partial) order which we eventually impose on oursemigroupoid. The first part of it ensures that the multiplication we will define for functions over thesemigroupoid is well defined, while the second part guarantees the existence of at least one collectionof test functions, or equivalently, that the associated collection of reproducing kernels is nontrivial.It does so by restricting how badly non-cancellative the semigroupoid can be. Alternately, the strong

3

artinian law could be replaced by the condition that for eacha ∈ G the set{z, b, w : zbw = a} isfinite and a hypothesis about the existence of a collection oftest functions (see Section 1.4).

There is one other rule which it is useful to state, though it follows from those already given:

(5) (strong idempotent law) If zaw = a, thenz andw are idempotents.

To see that this is a consequence of our other laws, first note that zaw = a means thatznawn = afor n ∈ N. The strong artinian law implies that only finitely many of thezn are distinct. In particular,there is anM > 0 such that(z2M

)2 = z2jfor somej ≤ M . Let h1 = z2j

, h2 = h21, and so on, with

hm = h2m−1 = z2M

(soh1 = h2m). Clearlyhjhk = hkhj for all j, k. Hence

(h1h2 · · ·hm)2 = h21 · · ·h2

m = h2 · · ·hmh1 = h1h2 · · ·hm,and soh1h2 · · ·hm = z2j+···+2M

is idempotent. Since there are no inverses, this implies that z isidempotent. Likewisew is idempotent. Note that (5) implies our assumption thate is idempotent ife2 = e.

If ea = a thene is unique, since ife′a = a, thena = ea = e(e′a) = (ee′)a, implying thatee′ isdefined. But then sincee ande′ are assumed to be idempotents,e = ee′ = e′.

From the definition we havee2 = e means thate is idempotent. On the other hand, ifa = ea thena = e(ea) = e2a, and soe2 is defined, and by uniqueness,e2 = e. Also if e andf are idempotentsandef is defined, thene = ef = f .

The productab exists if and only if there is an idempotentf such thataf , fb are defined. For ifsuch anf exists, then by associativity,(af)b = a(fb) = ab, while conversely, ifab is defined, thenthere is an idempotentf such thataf = a and so(af)b = a(fb) is defined and sofb is defined.

Based on these observations, it is common to view a set with a partial multiplication verifying thefirst two rules as a sort of generalized directed graph with the vertices representing the idempotents,though because we have not assumed any cancellation properties, this analogy is imperfect.

We define subsemigroupoids in the obvious way. In particular, a subsetH of a semigroupoidGwill be a subsemigroupoid if whenevera, b ∈ H andab makes sense inG thenab ∈ H, and for alla ∈ H the idempotentse, f such thatea = a = af are also inH.

We put a partial order on a semigroupoidG as follows: say thatb ≤ a if there existz,w ∈ Gsuch thata = zbw. By the existence of idempotents,a ≤ a. Transitivity is likewise readily verified.If a ≤ b and b ≤ a then a = zbw, b = z′aw′ and soa = (zz′)a(w′w). Then by the strongidempotent lawzz′ andw′w are idempotent. But then by the nonexistence of inverses,z, z′, w andw′ are idempotents and soa = b. Other partial orders are considered in Section 2.4.

By this definition, and the existence of idempotents, ifa = bc, then bothb andc are less than orequal toa. Also, by the nonexistence of inverses and uniqueness of idempotents, the idempotentscomprise the minimal elements ofG. We writeGe for the collection of idempotents.

We say that a setF ⊂ G is lower if a ∈ F andb ≤ a thenb ∈ F . Observe that for a lower setF ,Fe = F ∩Ge 6= ∅. Note too that ifH is a finite subset ofG, then there is a finite lower setF ⊃ H:simply letF = {a : there exists ab ∈ H such thata ≤ b}.

1.1.1. Examples.We list here several important examples of semigroupoids.

(1) Let G be a set, and assumeGe = G (so all elements are idempotent). We refer to suchsemigroupoids asPick semigroupoids.

(2) LetG = N = 0, 1, 2, . . . with the productab = a+ b. G is in fact a commutative cancellativesemigroup with idempotent0.

(3) The last example obviously generalizes toFn, the free (noncommutative) monoid onn gen-erators. This in turn is a special case of what we term theKribs-Power semigroupoids[29],

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which are defined as follows. LetΛ be a countable directed graph. The semigroupoidF+(Λ)determined byΛ comprises the vertices ofΛ, which act as idempotents, and all allowablefinite paths inΛ, with the natural concatenation of allowable paths inΛ defining the partialmultiplication. In particular,F+(Λ) = Fn whenΛ is a directed graph with one vertex andndistinct loops.

1.2. The convolution products. The product onG naturally leads to a product on functions overlower setsF ⊂ G in one or more variables.

1.2.1. The⋆-product for functions of one variable.Let F be a lower subset ofG. There is a naturalalgebra structure on the setP (F ) of functionsf : F → C which we call the semigroupoid algebraof F overC. Addition of f, g ∈ P (F ) is the usual pointwise addition of functions and the productisdefined by

(f ⋆ g)(a) =∑

rs=a

f(r)g(s),

which makes sense because of the artinian hypothesis onG and the assumption thatF is lower.The multiplicative unit ofP (F ) is given by

δ(x) =

{

1 x ∈ Fe,

0 otherwise.

The distributive and associative properties are readily checked, so we have an algebra. A functionfis invertible if and only iff(x) is invertible for allx ∈ Fe. The proof follows the same lines as in thematrix case given below, so we do not give it here.

If a ∈ F ′ ⊂ F andF ′ is itself lower, then

(f|F ′ ⋆ g|F ′)(a) = (f ⋆ g)(a).

Hence, we can be lax in specifying our lower set and usually act as if it is finite.Later we have need for powers of functions with respect to the⋆-product. To avoid confusion, for

a functionϕ onG, we letϕn⋆ denote then-fold ⋆-product ofϕ with itself.As it happens, it is unimportant that a function overF map intoC. For instance, the⋆-product

clearly generalizes to functionsf, g : F → C, whereC is aC∗-algebra.There will be times when we will want to interchanger ands in the definition of the convolution

product. OverC or, more generally, any commutativeC∗-algebraC this simply changesf ⋆ g intog ⋆ f . But in the noncommutative case this will not work. Hence we introduce the notation

(f ⋆ g)(a) =∑

rs=a

f(s)g(r).

For the⋆-product the multiplicative unit remainsδ, the associative and distributive laws continue tohold, andf is invertible with respect to this product if and only iff(x) is invertible for allx ∈Fe. We write f−1⋆ and f−1⋆ for the ⋆-inverse and⋆-inverse off , respectively. By consideringf−1⋆ ⋆ f ⋆ f−1⋆ , we see thatf−1⋆ = f−1⋆.

Another useful and easily checked property relating the twoproducts is that

(f ⋆ g)∗ = g∗ ⋆ f∗. (1.1)

Consequently(f ⋆ g)(g∗ ⋆ f∗) ≥ 0. We also have(f−1⋆)∗ = f−1⋆.In the examples listed above, the⋆-product is just pointwise multiplication for Pick semigroupoids.

For the second example, it is the usual convolution.5

1.2.2. The⋆-product for matrices.The following bivariate version of the convolution productis thecanonical generalization of Jury’s product [26] to semigroupoids.

For a lower setF , let M(F ) denote the set of functionsA : F × F → C. WhenF is finite,thinking of elements ofM(F ) as matrices (indexed byF ), the notationAa,b is used interchangeablywith A(a, b). The set of functions fromF × F toX will be denotedM(F,X).

Definition 1.1. LetF be a lower set and supposeA,B ∈M(F ). DefineA ⋆ B by

(A ⋆ B)(a, b) =∑

pq=a

∑

rs=b

A(p, r)B(q, s).

Once again, the artinian hypothesis onG guarantees the product is defined. Further,(A ⋆ B)(a, b)does not actually depend upon the lower setF which containsa andb. In particular, since there isalways a finite lower set containinga andb (just take the union of the set of elements less than orequal toa and those less than or equal tob), this product can and will be interpreted as a⋆-product ofmatrices.

The assumption that the entries ofA andB are inC is not important, and we will at times use the⋆-product when the entries are in other algebras. The⋆ notation should cause no confusion, since inessence the⋆-product is the bivariate analogue of the convolution product. Indeed, it is clear that the⋆-product could be defined for functions of three or more variables as well, though we have no needfor this here.

Unlike Jury’s⋆-product, ours will not necessarily be commutative (thoughthis will be the case ifG is commutative). In the special example of the Pick semigroupoids, the⋆-product is just the matrixSchur product.

As with functions we can also define the⋆-product of matrices:

(A ⋆ B)(a, b) =∑

pq=a

∑

rs=b

A(q, s)B(p, r).

OverC and any other commutative algebra,A ⋆ B = B ⋆ A. However we will need both productsin a noncommutative setting.

Define [1] (or [1]F if we want to make absolutely clear the lower set involved) tobe the matrixdefined by[1]a,b is 1 for a, b both elements ofFe and zero otherwise. It is easy to see that forA ∈M(F ), A ⋆ [1] = [1] ⋆ A = A. Note too that we can factor[1] = δδ∗.

1.3. The Toeplitz representation. Letϕ be a function on a (finite) lower setF . Define the associated(left) Toeplitz representationT(ϕ) by

[T(ϕ)]a,b =

{

∑

c ϕ(c), cb = a;

0 otherwise.

We drop the “left” hereafter, though we could also consider aright Toeplitz representation withbc = arather thancb = a. It is expedient to assumeF is finite to ensure thatT(ϕ) is bounded, though thedefinition makes sense formally whenF is not finite and in many interesting cases yields a boundedoperator.

As defined,T(ϕ) is a mapping ofF × F into C. Let CF denote the column vector space (of

dimension equal to the cardinality ofF ) with positions labelled by the entries ofF , which is naturallyisomorphic toP (F ) as a vector space. Then for a functionf ∈ P (F ), viewed as an element ofC

F ,

(T(ϕ)f)(a) =∑

b

[T(ϕ)]a,bf(b) =∑

cb=a

ϕ(c)f(b) = (ϕ ⋆ f)(a). (1.2)

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In this wayT(ϕ) is an operator inB(CF ) and the mappingP (F ) ∋ ϕ 7→ T(ϕ) ∈ B(CF ) is arepresentation. Indeed, it essentially acts as the left regular representation ofP (F ). The use of thenotationM(F ) to denote eitherB(CF ) or functions fromF × F → C should be clear from thecontext. Further, since the Toeplitz representation depends upon the lower setF in a consistent way,it should cause no serious harm that the notationT(ϕ) makes no reference toF . On occasions whenwe need to make the dependence onF explicit, we will write TF for T.

In the case thatG is the semigroupoidN, T(ϕ) is precisely the Toeplitz matrix associated with thesequence{ϕ(j)}. At the other extreme, whenG is a Pick semigroupoid,T(ϕ) is the diagonal matrixwith diagonal entriesϕ(a) for a ∈ G which, despite our terminology, seems very un-Toeplitz like!

WhenG is the Kribs-Power semigroupoidF+(Λ) determined by a countable directed graphΛ, alower setF of F+(Λ) is closed under taking left and right subpaths. The vector spaceP (F ) maybe regarded as a subspace of the generalized Fock spaceHΛ overΛ (see [27]). In this interpretationTF (χw) is the compression toP (F ) of the partial creation operators indexed byw ∈ F . Moregenerally,{χv : v ∈ F+(Λ)} may be thought as an orthonormal basis ofHΛ andT behaves as arepresentationF+(Λ) → B(HΛ). Hence, the weak operator topology closed subalgebra generatedby the family{T(χv)} is the free semigroupoid algebra of Kribs and Power [29], which includes,as particular case, the noncommutative Toeplitz algebra [10, 11, 20]. The set of generators can berestricted, as we see later.

Even whenF is not necessarily finite,T still behaves formally as a representation, but of course itneed not be the case thatT(ϕ) is bounded.

It is also possible to work with the⋆-product. Presumably, there is a distinction between a collectionof test functions, defined below with respect to the⋆-product, and those with respect to the⋆-product,though we do not develop this.

1.4. Test functions. For a functionϕ onG, recall thatϕn⋆ denotes then-fold ⋆-product ofϕ withitself, n = 1, 2, . . ..

Definition 1.2. A collectionΨ of functions onG into C is acollection of test functionsif

(i) for each finite lower setF ⊂ G andψ ∈ Ψ, ‖T(ψ)‖ ≤ 1;(ii) for eacha ∈ Ge,

limn→∞

ψn⋆(a) = limn→∞

ψn(a) = 0,

uniformly inψ; and(iii) for each finite lower setF , the algebra generated byΨ|F = {ψ|F : ψ ∈ Ψ} is all of P (F ).

The conditionΨ|F generatesP (F ) is not essential. It does however simplify statements of results.Givenx ∈ G, let f be the unique idempotent so thatxf = x. Since,

[T(ψ)]x,f =∑

cf=x

ψ(c) = ψ(x),

item (i) says that|ψ(x)| ≤ 1 for eachx ∈ G. By the same reasoning, ifψ1, ψ2 ∈ Ψ, then

|ψ1(x) − ψ2(x)| ≤ ‖T(ψ1) − T(ψ2)‖.Item (ii) says that for eacha ∈ Ge andǫ > 0 there is anN so that for alln ≥ N andψ ∈ Ψ,

|ψn(a)| < ǫ, and so for fixeda ∈ Ge, supψ∈Ψ |ψ(a)| < 1. Furthermore, for anya ∈ G, weautomatically obtainlimn→∞ ψn⋆(a) = 0. This follows from a straightforward counting argumentestimating the maximum number of ways of writinga ∈ G as a product ofn elements. Assumecard{b : b ≤ a} = r (which is finite sinceG is artinian) and thatn≫ r. Let c = maxb≤a |ψ(b)|, andce be the maximum of|ψ(b)| over all idempotents less than or equal toa. As noted above,ce < 1. In

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the product ofn terms, there are at most(nr

)

ways of choosing which of the at mostr terms are notidempotent, and then at mostrr ways of choosing these terms. The nonidempotents act as separatorsbetween at mostr + 1 blocks of idempotents. Within each block of idempotents, each term must bethe same idempotent (since the product of unequal idempotents is not defined). So there are at mostrr+1 ways of choosing which idempotent is in each block. Consequently

|ψn⋆(a)| ≤(

n

r

)

rrcrrr+1cn−re ≤ r2r+1(c/ce)rnrcne ,

which clearly goes to zero asn→ ∞.For a given semigroupoidG it is legitimate to wonder if there actually exists any family of test

functions. It so happens that the strong artinian conditionin the definition of a semigroupoid ensuresthis. Letκ = supa,c∈G card{b ∈ G : cb = a}, which we have assumed is finite. Letψ0 : G → D

with ψ0|Ge injective andψ0|G\Ge= 0. (This assumes the cardinality ofGe is less than or equal

to that of the continuum — it is only slightly more trouble to handle the more general case.) LetΨs = { 1

κχc : c ∈ G\Ge} ∪ {ψ0}. ThenΨs can be shown to be a collection of test functions (hereχc(x) equals1 if x = c and zero otherwise). In particular, the conditionκ < ∞ for all c ∈ G willhold if G is right cancellative (so in particular, for Kribs-Power semigroupoids). In the caseG = Ge,this choice of test functions will ultimately correspond toB(G), the normed algebra of all boundedfunctions onG.

1.5. Test functions and reproducing kernel Hilbert spaces.LetF ⊂ G be a lower set. A functionk : F ×F → C is a positive kernel if for each finite subsetA ⊂ F the matrix[k(a, b)]a,b∈A is positive(i.e., positive semidefinite).

More generally, it makes sense to speak of a kernel with values in the dual of aC∗ algebra. IfBis aC∗-algebra with Banach space dualB∗, then a functionΓ : F × F → B∗ is positive if for eachfinite subsetA ⊂ F and each functionf : A→ B,

∑

x,y∈A

Γ(x, y)(f(x)f(y)∗) ≥ 0.

In the sequel, unless indicated otherwise, kernels take their values inC.Given a set of test functionsΨ let KΨ denote the collection of positive (i.e., positive semidefinite)

kernelsk onG such that for eachψ ∈ Ψ, the kernel

G×G ∋ (x, y) 7→ kψ(x, y) = (([1] − ψψ∗) ⋆ k) (x, y) (1.3)

is positive. Here[1]−ψψ∗ is the function defined onG×G by ([1]−ψψ∗)(p, q) = [1]p,q−ψ(p)ψ(q)∗

so that the right hand side of equation (1.3) is the⋆-product of the functions (or matrices indexed byG) [1] − ψψ∗ andk, evaluated at(x, y) ∈ G×G.

The setKΨ is nonempty, since it at least containsk = 0. More importantly, from the hypothesis thatΨ is a family of test functions and the strong artinian law, it also contains the kernels : G ×G → C

given bys(x, y) = 1 if x = y and0 otherwise, which is strictly positive definite. We calls theToeplitzkernel.

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Let us verify thats ∈ KΨs for the collection of test functionsΨs constructed in the last subsection.For the test functionψ0,

(ψ0ψ∗0 ⋆ s)(a, b) =

∑

pq=art=b

ψ0(p)ψ∗0(r)sq,t =

∑

p,r∈Gepq=art=b

ψ0(p)ψ∗0(r)sq,t

=

{

ψ0(p)ψ∗0(p), p ∈ Ge, a = b, pa = a

0 otherwise.

Hence since[1] ⋆ s = s, ([1] − ψ0ψ∗0) ⋆ s is a diagonal matrix with entries of the form1 −

ψ0(p)ψ∗0(p) ≥ 0, and so is positive. On the other hand supposeψc = 1

κχc, c ∈ G\Ge. Then

(ψcψ∗c ⋆ s)(a, b) =

∑

pq=art=b

ψc(p)ψ∗c (r)sq,t = 1

κ2

∑

cq=a=b

1

2

∈ [0, 1],

and so([1] − ψcψ∗c ) ⋆ s is a positive diagonal matrix.

The kernels determined by a family of test functionsΨ in turn give rise to a normed algebra offunctions onG. LetH∞(KΨ) denote those functionsϕ : G→ C such that there exists aC > 0 suchthat for eachk ∈ KΨ, the kernel

G×G ∋ (x, y) 7→ ((C2[1] − ϕϕ∗) ⋆ k)(x, y)

is positive. The infimum of all suchC is the norm ofϕ. With this normH∞(KΨ) is a Banach algebraunder the convolution product. By constructionΨ is a subset of the unit ball ofH∞(KΨ).

There is a duality between kernels and test functions in Agler’s model theory [2, 4]. Roughly, theidea is, given a collectionK of positive kernels onG, to let Ψ = K⊥ denote those functionsψ ∈ Gsuch that for eachk ∈ K, the kernel

G×G ∋ (x, y) 7→ (([1] − ψψ∗) ⋆ k)(x, y)

is positive. In the case that Agler considers, where the semigroupoid consists solely of idempotents(i.e., a Pick semigroupoid), mild additional hypotheses onK guarantee thatΨ is a family of testfunctions, in which caseKΨ = K⊥⊥.

1.6. The evaluationE andC∗-algebra B. Let Ψ be a given collection of test functions andCb(Ψ)the continuous functions onΨ, whereΨ is compact in the bounded pointwise topology. DefineE ∈B(G,Cb(Ψ)) (the bounded functions fromG toCb(Ψ)) by

E(x)(ψ) = ψ(x), ψ ∈ Ψ,

with

‖E(x)‖ = supψ∈Ψ

{|E(x)(ψ)|}.

SoE(x) is the evaluation map onΨ, ‖E(x)‖ < 1 for eachx ∈ Ge and‖E(x)‖ ≤ 1 otherwise.Since evidently the collection{E(x) : x ∈ G} separates points and we include the identity, the

smallest unitalC∗-algebra containing all theE(x) isCb(Ψ). For convenience, we denote this algebraasB.

9

1.7. Colligations. Following [8] we define aB-unitary colligationΣ to be a tripleΣ = (U, E , ρ)whereE is a Hilbert space,

U =

(

A BC D

)

:E⊕C

→E⊕C

is unitary, andρ : B → B(E) is a unital∗-representation. The transfer function associated toΣ is

WΣ(x) = (Dδ + C(ρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ))(x). (1.4)

Observe that this looks like the standard transfer functionover a Pick semigroupoid.

1.8. The main event. We now state the realization theorem for elements of the unitball ofH∞(KΨ)and a concomitant interpolation theorem.

Theorem 1.3(Realization). If Ψ is a collection of test functions for the semigroupoidG, then thefollowing are equivalent:

(i) ϕ ∈ H∞(KΨ) and‖ϕ‖H∞(KΨ) ≤ 1;(iiF) for each finite lower setF ⊂ G there exists a positive kernelΓ : F × F → B∗ so that for all

x, y ∈ F([1] − ϕϕ∗)(x, y) = (Γ ⋆ ([1] − EE∗))(x, y);

(iiG) there exists a positive kernelΓ : G×G→ B∗ so that for allx, y ∈ G

([1] − ϕϕ∗)(x, y) = (Γ ⋆ ([1] −EE∗))(x, y); and

(iii) there is aB-unitary colligationΣ so thatϕ = WΣ.

Theorem 1.4(Agler-Jury-Pick Interpolation ). LetF be a finite lower set and supposef ∈ P (F ).The following are equivalent:

(i) There existsϕ ∈ H∞(KΨ) so that‖ϕ‖H∞(KΨ) ≤ 1 andϕ|F = f ;(ii) for eachk ∈ KΨ, the kernel

F × F ∋ (x, y) 7→ (([1] − ff∗) ⋆ k)(x, y)

is positive;(iii) there is a positive kernelΓ : F × F → B∗ so that for allx, y ∈ F

([1] − ff∗)(x, y) = (Γ ⋆ ([1] − EE∗))(x, y).

Remark 1.5. Of course, item (i) of Theorem 1.4 combined with item (iii) ofTheorem 1.3 says thatϕin Theorem 1.4 has aB-unitary transfer function representation.

The hypothesis thatΨ|F generates all ofP (F ) means that the representationπ : H∞(KΨ) →P (F ) which sendsϕ toϕ|F is onto and identifiesP (F ) with the quotientH∞(KΨ)/ker(π). Theorem1.4 can be interpreted as identifying the quotient norm.

2. FURTHER PROPERTIES OF THE⋆-PRODUCTS

2.1. The convolution products. The convolution products over finite lower setsF can be related tothe tensor product of matrices as follows. TakeV : C

n → Cn ⊗ C

n, wheren = card(F ), such thatV ea =

∑

pq=a ep⊗ eq, {ek} the standard basis forCn labelled with the elements ofF , and extendingby linearity. Thenf ⋆ g = V ∗(f ⊗ g) andf ⋆ g = (g⊗ f)V . Note thatV is an isometry only in thecase thatF = Fe, in which case the convolution products become the pointwise product. In all othercases it still has zero kernel and in fact maps orthogonal basis vectorsea andeb to orthogonal vectors,though it generally acts expansively.

10

2.2. The matrix ⋆-product—basic properties and an alternate definition. Straightforward calcu-lations show that the various associative and distributivelaws hold for the bivariate⋆-product. Here,for example, is the proof thatC ⋆ (A ⋆ B) = (C ⋆ A) ⋆ B:

[C ⋆ (A ⋆ B)]µ,ν =∑

lm=µ

∑

jk=ν

Cl,j[A ⋆ B]m,k

=∑

lm=µ

∑

jk=ν

Cl,j∑

pq=m

∑

rs=k

Ap,rBq,s

=∑

l,p,ql(pq)=µ

∑

j,r,sj(rs)=ν

Cl,jAp,rBq,s,

while

[(C ⋆ A) ⋆ B)]µ,ν =∑

iq=µ

∑

ns=ν

[C ⋆ A]i,nBq,s

=∑

iq=µ

∑

ns=ν

∑

lp=i

∑

jr=n

Cl,jAp,rBq,s

=∑

l,p,q(lp)q=µ

∑

j,r,sj(rs)=ν

Cl,jAp,rBq,s,

and sincel(pq) = (lp)q, j(rs) = (jr)s, the two are equal.There is an alternate equivalent definition of the⋆-product, just as with the convolution products.

TakeV defined as in the last subsection. Then it is easy to check that

A ⋆ B = V ∗(A⊗B)V. (2.1)

The Schur product is the matrix analogue of the pointwise product of functions in which caseV isisometric, though otherwise it will not be. From this formulation it is clear that the⋆-product iscontinuous.

Another important property which the⋆-product shares with the Schur product is that ifA,B ∈M(F ) are positive, then so isA ⋆ B. This follows immediately from the fact thatA ⊗ B ≥ 0 ifA,B ≥ 0. Similarly, since the tensor product of selfadjoint matrices is selfadjoint, the⋆-product ofselfadjoint matrices is selfadjoint.

2.3. Positivity and the ⋆-product. It should be emphasized that unlike with ordinary matrix multi-plication, the inverse with respect to the⋆-product of a positive matrix need not be positive. This isalready clear when considering Schur products, but we illustrate with another simple example. Sup-

pose thate, a ∈ G with e idempotent andeae = a. Consider the matrixA =

(

1 00 c

)

wherec > 0

and the first row and column is labelled bye while the second is labelled bya. An easy calculation

shows thatA−1⋆ =

(

1 00 −c

)

.

The⋆-product behaves somewhat unexpectedly with respect to adjoints (at least if you forget itsconnection to the tensor product). Using the formulation ofthe⋆-product given in (2.1), we see that(A ⋆ B)∗ = V ∗(A ⊗ B)∗V = V ∗(A∗ ⊗ B∗)V = A∗ ⋆ B∗. However with regard to inverses andadjoints,[1] = [1]∗ = (A ⋆ A−1⋆)∗ = V ∗(A∗ ⊗ (A−1⋆)∗)V , and so by uniqueness of the inverse,A∗ is invertible ifA is and(A∗)−1⋆ = (A−1⋆)∗. Consequently we see that ifA is selfadjoint andinvertible, thenA−1⋆ is selfadjoint.

11

Let F be a finite lower set. AnA ∈ M(F ) gives rise to the⋆-product operatorSA : M(F ) →M(F ) given bySA(B) = A ⋆ B = V ∗(A ⊗ B)V . The argument in Paulsen’s book ([41], The-orem 3.7) which shows that Schur product with a positive matrix gives a completely positive mapcarries over with the obvious modifications to show thatSA is completely positive. In particular, thecb-norm ofSA is given by‖A ⋆ 1‖, where1 ∈M(F ) is the identity (not the⋆-product identity).

All of the above carries over in total to the⋆-product, with a small change in the definition in termsof the tensor product, where we have

A ⋆ B = V ∗(B ⊗A)V.

2.4. More on order on semigroupoids. The following lemmas give general properties of an artinianorder on a semigroupoidG; i.e., a partial order� such that for anya ∈ G, the set{b ∈ G : b � a} isfinite. Since� is a partial order, it is permissible to use the notationy ≺ x to meany � x, buty 6= x.

As before, a setF is lower if for all a ∈ F , {b : b � a} ⊂ F . Clearly the intersection of lowersets is again lower. Forz ∈ G, let Sz = {x � z}. This is a lower set. Furthermore, ifH is anysubset ofG, x is minimal inH is equivalent toSx ∩ H = {x}. By the artinian assumption,Sx isfinite. Note thatb � a is equivalent toSb ⊆ Sa. (In fact there is an equivalence between artinianpartial orders(G,�) and functionsλ : G→ SG, SG the set of all finite subsets ofG, λ injective andλ(λ(x)) = λ(x), wherea � b if and only if λ(a) ⊆ λ(b).)

Lemma 2.1. Each nonempty subsetH of G contains a minimal element with respect to an artinianpartial order�.

Proof. Clearly any finite subset has a least element. SupposeH is any nonempty set and choosez ∈ H. Now Sz ∩H is a nonempty finite set, so it has a minimal elementx ∈ H. SinceSx ⊆ Sz,Sx ∩H = Sx ∩ (Sz ∩H) = {x}; that is,x is minimal inH. �

For a semigroupoidG with artinian order�, we define astratificationof G as follows. SetG0 =Ge. For natural numbersn, define

Gn = {x ∈ G : y ≺ x⇒ y ∈ Gm for somem < n andy ≺ x for somey ∈ Gn−1}.

We callGn thenth stratumwith respect to the order� and{Gn} whereGn is nonemptya stratifica-tion ofG with respect to the order�.

Lemma 2.2. For everyg ∈ G there is a uniquen ∈ N such thatg ∈ Gn.

Proof. SupposeH = G \ ∪∞0 Gn is nonempty. From Lemma 2.1,H has a minimal elementz (with

respect to�). In particular,z ∈ Sz ⊂⋃∞

0 Gn, a contradiction. �

For a lower setF ⊂ G with respect to the order� we define thestratification{Fn} ofF with strataFn = F ∩Gn whereFn 6= ∅.

The order≤ which we originally introduced on semigroupoids (whereb ≤ a if and only ifa = zbwfor somez,w ∈ G) is artinian by definition of a semigroupoid. Hence the abovelemmas apply toG with this order. There is another artinian order which will be useful in proving the existence ofinverses with respect to the⋆-product.

Define theleft order≤ℓ onG by declaringy ≤ℓ x if there is ana so thatx = ay.

Lemma 2.3. The relation≤ℓ is a partial order onG which is more restrictive than the order≤ onG;that is, ify ≤ℓ x, theny ≤ x.

12

Proof. The existence of idempotents implies thatx ≤ℓ x. If z ≤ℓ y ≤ℓ x, then there exista, b sothat x = ay and y = bz. Hence,x = a(bz) = (ab)z by the associative law and thusz ≤ℓ x.Finally, choosingx = z above givesx = (ab)x. By the strong idempotent law, it follows thatab isidempotent; and then by nonexistence of inversesa = b = ewheree is the idempotent so thatex = x.Thusx = ey. But by what it means to be idempotent,ey = y. Hence ifx ≤ℓ y andy ≤ℓ x, thenx = y. This proves that≤ℓ is an order onG.

If y ≤ℓ x, thenx = ay for somea. There is always an idempotentf so thatxf = x. Thus,x = xf = ayf (by associativity) andy ≤ x. Hence{y : y ≤ℓ x} ⊆ {y : y ≤ x}. The latter set isfinite, so both are finite. �

We use the notation{F ℓn} for the stratification of a lower setF with respect to the left order, andfor z ∈ G, we writeSℓz for Sz with respect to the left order.

2.5. ⋆-inverses. We next prove the statement about inverses of matrices with respect to the⋆-productmade in the introduction. A similar (and in fact easier) proof works for inverses of functions withrespect to the⋆-product. The arguments in the proof also apply to matrices over anyC∗-algebra,though the theorem is stated for matrices overC.

Theorem 2.4. LetF be a lower set. A matrixA ∈M(F ) is ⋆-invertible if and only ifAab is invertiblefor all a, b ∈ Fe. Furthermore the inverse is unique.

Proof. For a, b ∈ Fe = F ∩ Ge, the term(B ⋆ A)(a, b) is essentially the Schur product (that is,(B ⋆ A)(a, b) = BabAab if a, b ∈ Fe) and ifB is a⋆-inverse ofA, then(B ⋆ A)(a, b) = 1 in thiscase. Thus,Aab is invertible.

The proof of the converse proceeds as follows. Under the hypotheses of the theorem, a left⋆-inverseB for A is constructed which itself satisfies the hypotheses of the theorem. By what has already beenproved,B then has a left⋆-inverseC. Associativity of the⋆-product guarantees thatC = A and thusB is also a right⋆-inverse forA. Uniqueness of the⋆-inverse similarly follows from the construction.

So assumeAab is invertible for alla, b ∈ F ℓ0 = F ∩ Ge. Let {F ℓn} be the left stratification ofF .DefinePjk = {(a, b) : a ∈ F ℓj , b ∈ F ℓk}, andQN =

⋃

j,k≤N Pjk. The proof proceeds by inductiononN .

We require

(B ⋆ A)(a, b) =∑

pq=a

∑

rs=b

BprAqs =

{

1 botha, b ∈ Ge

0 otherwise.(2.2)

In the caseN = 0 (so thata, b ∈ F ∩ Ge), the choiceBab = A−1ab is the unique solution to this

equation.Now suppose thatBab have been defined for(a, b) ∈ QN satisfying equation (2.2) and suppose

(a, b) ∈ QN+1\QN . Isolating the(a, b) term in equation (2.2) gives

0 = BabAef +∑

pq=ap 6=a

∑

rs=br 6=b

BprAqs,

wheree, f ∈ Ge with ae = a andbf = b. In the second term on the right hand side,p <ℓ a andr <ℓ b. In particular,p, r ∈ QN and the matrixBpr is already defined. SinceAef is invertible,Bab isuniquely determined. �

Lemma 2.5. LetL,F be lower sets inG withL ⊃ F . SupposeA ∈M(L) is ⋆-invertible. ThenA|Fis ⋆-invertible and(A|F )−1⋆ = A−1⋆ |F .

Proof. This follows by observing thatA|F ⋆ A−1⋆ |F = (A ⋆ A−1⋆)|F = [1]F . �

13

3. REPRODUCING KERNELS

3.1. Generalized Szego kernels. In this section we investigate those kernels which play the roleover semigroupoids of Szego kernels. Recall, for a function ϕ defined on a lower setF , then-fold⋆-product ofϕ with itself is denotedϕn⋆ . We useAn⋆ similarly whenA is a matrix.

Theorem 3.1. LetA ∈ M(F ) be positive, and suppose‖An⋆‖ → 0 asn → ∞. Then[1] − A isinvertible (with respect to the⋆-product) and ([1] − A)−1⋆ ≥ 0. In particular, the result holds if‖A‖ < 1.

Proof. Observe that under the hypotheses,[An⋆ ]e,e = Ane,e → 0 asn → ∞ for e ∈ Fe. Hence|Ae,e| < 1 for all e ∈ Fe. Positivity ofA then implies that|Ae,f | < 1 for all e, f ∈ Fe. Consequently[1] −A is invertible.

It is easily seen that

1 +A+A2⋆ + · · · +An⋆ = ([1] −A)−1⋆ ⋆ ([1] −A(n+1)⋆).

But [1]−A(n+1)⋆ → [1] asn→ ∞ and1+A+A2⋆ + · · ·+An⋆ is an increasing sequence of positiveoperators, and so converges strongly to([1] − A)−1⋆ . Thus([1] − A)−1⋆ ≥ 0. The last part of thetheorem follows from the submultiplicativity of the operator norm. �

It is not difficult to verify that the above arguments also work if we instead consider matrices overa unitalC∗-algebra.

Corollary 3.2. If C is a unitalC∗-algebra,F a finite lower set, andϕ ∈ P (F,C). Suppose that

limn→∞

ϕn(a) = 0 (3.1)

for eacha ∈ Fe. Then([1] − ϕϕ∗)−1⋆ ∈ M(F,C) is well defined and positive. In particular, if‖T(ϕ)‖ < 1, the result follows (and in this caseF need not be finite).

Proof. Let A(a, b) = ϕ(a)ϕ(b)∗. Since fore ∈ Fe, ‖An⋆(e, e)‖ = ‖An(e, e)‖ → 0 asn → ∞, itfollows that‖A(e, f)‖ < 1 for all e, f ∈ Fe. A counting argument in the same vein as that followingDefinition 1.2 then shows that‖An⋆(a, b)‖ → 0 asn → ∞, and so sinceF is assumed to be finite,limn→∞ ‖An⋆‖ = 0. The conditions for Theorem 3.1 hold and the result follows directly.

If ‖T(ϕ)‖ < 1 then since

(A ⋆ 1)a,b =∑

pq=a

∑

rs=b

ϕ(p)ϕ(r)∗〈1q, 1s〉

=∑

pq=a

∑

rq=b

ϕ(p)ϕ(r)∗

and

(T(ϕ)T(ϕ)∗)a,b =∑

q

[T(ϕ)]a,q[T(ϕ)∗]q,b

=∑

q

[T(ϕ)]a,q[T(ϕ)]∗b,q

=∑

pq=a

∑

rq=b

ϕ(p)ϕ(r)∗.

the result is then a consequence of the last statement of Theorem 3.1, since‖A ⋆ 1‖ = ‖A‖cbdominates the operator norm. �

14

In what follows the theorem will be applied to test functionsψ and more generally the evaluationE. ThatE satisfies the hypothesis of Corollary 3.2 is equivalent to item (ii) in the definition of testfunctions (Subsection 1.4).

Lemma 3.3. For eacha ∈ Ge, the sequenceEn⋆(a) fromB converges to0.

3.2. The multiplier algebra for a single kernel. Let k : G × G → C be a positive kernel. Forb ∈ G, the functionkb : G → C defined bykb = k(·, b) is point evaluation atb. In the usual way weform a sesquilinear form〈·, ·〉 on linear combinations of kernel functions by setting〈kb, ka〉 = k(a, b)and modding out by the kernel. We then complete to get a Hilbert space,H2(k).

OnH2(k) addition is defined term-wise. The multiplier algebraH∞(k) consists of the collectionof operatorsTϕ : f 7→ ϕ⋆f for functionsϕ : G→ C satisfyingϕ ⋆ f ∈ H2(k) for eachf ∈ H2(k).(The product is well defined by the assumption thatG is artinian.) Note thatH∞(k) is nonempty,since it containsTδ, δ the⋆-product identity for functions onG. The closed graph theorem impliesthat the elements ofH∞(k) are bounded.

Observe that forf ∈ H2(k),

〈Tϕf, ka〉 = (ϕ ⋆ f)(a)

=∑

bc=a

ϕ(b)f(c)

=∑

bc=a

ϕ(b)〈f, kc〉

=∑

bc=a

〈f, ϕ(b)∗kc〉

=

⟨

f,∑

bc=a

ϕ(b)∗kc

⟩

,

which gives the formulaT ∗ϕka =

∑

bc=a ϕ(b)∗kc.For a lower setF , if we setM(F ) to the closed linear span of kernel functionska, a ∈ F , then the

usual sort of argument givesM(F ) invariant for the adjoints of multipliersTϕ.The⋆-product is useful in characterizing multipliers. Indeed,‖T ∗

ϕ|M(F )‖ ≤ C is equivalent to0 ≤ (

⟨

(C2 − TϕT∗ϕ)ka, kb

⟩

) which by the previous calculation is

‖T ∗ϕ|M(F )‖ ≤ C ⇐⇒ C2

k− ϕ∗ ⋆ k ⋆ ϕ ≥ 0.

In the aboveϕ∗ ⋆ k ⋆ ϕ stands for(ϕ∗ ⋆ k)(k∗ ⋆ ϕ) wherek(x) = kx in the factorizationk(x, y) =kxk

∗y for x, y ∈ F .

3.3. The Toeplitz kernel. A special case of interest is the kernels : F×F → C given bys(x, y) = 1if x = y and0 if x 6= y. This kernel is evidently positive and, as noted earlier is referred to as theToeplitz kernel. It arises naturally by declaring〈x, y〉 = s(x, y) for x, y ∈ F and extending bylinearity. That is, the Hilbert spaceH2(s) is nothing more than the Hilbert space with orthonormalbasis indexed byF ; i.e., C

F . The Toeplitz representation ofϕ : F → C determined bys as in theprevious subsection is thus the Toeplitz representationT(ϕ) of ϕ.

Note that(

s(x, y))

x,y∈F= 1 ∈M(F ), the usual identity matrix.

3.4. Kernels and representations.The results of Subsection 3.2 have an alternate interpretation. Letk be a reproducing kernel onG. Recall that we useP (F ) to denote the complex valued functions onthe finite lower setF ⊂ G, which under the⋆-product is an algebra. If we now compressk to F , it is

15

still a positive kernel (onF ) which we continue to callk (or kF if it is not absolutely clear from thecontext). Furthermore, sinceF is lower

F × F ∋ (x, y) 7→ (([1] − ψψ∗) ⋆ k)(x, y) (3.2)

is positive for eachψ ∈ Ψ|F . In this case, anyϕ ∈ P (F ) is a multiplier ofH2(kF ) since the algebragenerated byΨ|F is all of P (F ) andπ : P (F ) → B(H2(kF )) defined byπ(ϕ) = TF (ϕ) is arepresentation ofP (F ). Further, the assumption that (3.2) is positive implies‖π(ψ)‖ ≤ 1 for eachψ ∈ Ψ|F .

Define the functions

χa(x) =

{

1 x = a,

0 otherwise.

Routine calculation verifies(χa ⋆ χb)(x) =∑

pq=x χa(p)χb(q) = χab(x), where for conveniencewe takeχab = 0 if the productab is not in our partial multiplication.

Clearly the set{π(χa)δ} forms a spanning set forH2(k), and, sinceπ(ϕ)δ = ϕ = 0 if and only ifϕ = 0, it is in fact a basis. Indeed it is a dual basis to{ka}, since

〈ka, π(χb)δ〉 =⟨

TF (χb)∗ka, δ

⟩

=∑

pq=a

〈χ∗b(p)kq, δ〉 =

{

1 if p = b = a, q ∈ Fe

0 otherwise.

In some cases it is possible to reverse the above, obtaining akernel from a representationµ :P (F ) → B(H). For instance, supposeF is a finite lower set and assume thatµ is cyclic withdimension equal to the cardinality ofF . Write γ for the cyclic vector forµ, so thatH is spanned by{ℓa = µ(χa)γ : a ∈ F}. Since by assumption the dimension ofµ is the cardinality ofF , this set is infact a basis forF .

If µ is to come from a kernelk, we require that for any functionϕ onF ,

µ(ϕ)∗ka =∑

pq=a

ϕ(p)∗kq.

It suffices to have this for the functionsχb, in which case we need

µ(χb)∗ka =

∑

pq=a

χb(p)kq =∑

bq=a

kq.

Choose{ka : a ∈ F} to be a dual basis toℓa. Then compute,

〈µ(χb)∗ka, ℓc〉 = 〈ka, µ(χb)ℓc〉

= 〈ka, µ(χb)µ(χc)γ〉= 〈ka, µ(χb ⋆ χc)γ〉= 〈ka, ℓbc〉

=

{

1 bc = a

0 bc 6= a

=

⟨

∑

bq=a

kq, ℓc

⟩

.

Since this is true for allc ∈ F , it follows that

µ(χb)∗ka =

∑

bq=c

kq,

16

as desired.It is worth considering the example whereµ(ϕ) = T(ϕ), the Toeplitz representation. The function

δ(x), which is1 if x ∈ Fe and zero otherwise is a cyclic vector forµ. Moreover,

µ(χa)(x, y) = T(χa)(x, y) =∑

py=x

χa(p) = χay(x).

Therefore

ℓa(x) =∑

y

µ(χa)(x, y)δ(y) =∑

y

χay(x)δ(y) = χa(x),

which is just the standard basis, and so the assumption that{ℓa : a ∈ F} is a basis is automaticallymet. In this case we chooseka = χa, and the kernel is the Toeplitz kernels.

If F is infinite, this construction fails, since it need not be thecase thatχa ∈ H∞(KΨ).

3.5. P(F) as a normed algebra.Given a finite lower setF , let πF : H∞(KΨ) → P (F ) denote themappingπF (ϕ) = ϕ|F . The hypothesis on the collection of test functionsΨ imply that this mappingis onto and so ker(πF ) = {ϕ ∈ H∞(KΨ) : ϕ|F = 0}. ThusP (F ) is naturally identified with thequotient ofH∞(KΨ) by ker(πF ) and this givesP (F ) a norm for whichπF is contractive. There is analternate candidate for a norm onP (F ) constructed in much the same way as the norm onH∞(KΨ),called theH∞(KF

Ψ)-norm. LetKFΨ denote the kernelsk defined onF for which

F × F ∋ (x, y) 7→ (([1] − ψ|Fψ|∗F ) ⋆ k)(x, y)

is a positive kernel and, forϕ ∈ P (F ), say that‖ϕ‖ ≤ C (hereC ≥ 0) provided for eachk ∈ KFΨ,

the kernel

F × F ∋ (x, y) 7→ ((C2[1] − ϕϕ∗) ⋆ k)(x, y)

is positive.The following lemma ultimately implies that the quotient norm dominates theH∞(KF

Ψ)-norm.Theorem 1.4 then says that these norms are the same.

Lemma 3.4. Supposeµ : P (F ) → B(H) is a cyclic unital representation of the finite lower setFand letϕ ∈ H∞(KΨ) be given. LetπF : H∞(KΨ) → P (F ) be the restriction map,µF = µ ◦ πF . If‖µF (ψ)‖ ≤ 1 for eachψ ∈ Ψ, but‖µF (ϕ)‖ > 1, then there exists ak ∈ KΨ so that the kernel

F × F ∋ (x, y) 7→ (([1] − ϕϕ∗) ⋆ k)(x, y)

is not positive. In particular,‖ϕ‖ > 1.

Proof. Let γ denote a cyclic vector for the representationµ. Choosef ∈ P (F ) so that‖µ(f)γ‖ = 1but ‖µF (ϕ)µ(f)γ‖ = 1 + η > 1, andǫ so that(1 + ǫ2‖f‖2

H2(s))(1 + η/2)2 = (1 + η)2, where‖f‖H2(s) is the norm off in the space with the Toeplitz kernels.

Recall that for a finite lower setL, TL denotes the Toeplitz representation with its cyclic vectorδL. If L ⊇ F , let πLF be the restriction ofP (L) to P (F ) and setµLF = µ ◦ πLF . As above, defineπL : H∞(KΨ) → P (L) to be the restriction map.

ForL ⊇ F lower, there is a finite dimensional Hilbert space given byHL = {µLF (h)γ⊕ǫTL(h)δL :h ∈ P (L)}. RecallTL(h)δL = h. Define a representationρL : P (L) → B(HL) by

ρL(g)(µLF (h)γ ⊕ ǫTL(h)δL) = µLF (g ⋆ h)γ ⊕ ǫ g ⋆ h17

Since forψ ∈ Ψ, h ∈ P (L),

‖ρL(πL(ψ))(µ(h)γ ⊕ ǫh)‖ = ‖µ(πF (ψ))µ(h)γ ⊕ TL(πLψ)ǫh‖≤ max{‖µ(πF (ψ))‖, ‖TL(πLψ)‖}‖µ(h)γ ⊕ ǫh‖≤ ‖µ(h)γ ⊕ ǫh‖,

‖ρL(ψ)‖ ≤ 1, and in particular, takingL = F we have‖ρF (ψ)‖ ≤ 1.From the discussion in Subsection 3.4, there is a kernelk

F onF which implements the represen-tationρF . In particular, since

∥

∥

∥

∥

∥

∥

ρF (ϕ)µ(f)γ ⊕ ǫf

√

1 + ǫ2‖f‖2H2(s)

∥

∥

∥

∥

∥

∥

2

=1

1 + ǫ2‖f‖2H2(s)

(

‖µ(ϕ)µ(f)γ‖2 + ǫ2‖ϕ ⋆ f‖2)

≥ 1

1 + ǫ2‖f‖2H2(s)

(1 + η)2

= (1 + η/2)2,

the kernel

F × F ∋ (x, y) 7→ (([1] − ϕϕ∗) ⋆ kF )(x, y)

is not positive.Definek : G×G→ C by

k(a, b) =

{

kF (a, b) if (a, b) ∈ F × F

1ǫ2s(a, b) if (a, b) /∈ F × F.

In particular, ifa ∈ F andb /∈ F (or vice-versa), thenk(a, b) = 0. We will complete the proof byshowingk ∈ KΨ.

The representationρL defined as above is non-degenerate for anyL ⊇ F , in the sense of thediscussion in Subsection 3.4. In particular, for any suchL there is a reproducing kernelk

L whichimplements this representation. Consequently, for eachψ ∈ Ψ,

L× L ∋ (x, y) 7→ (([1] − ψψ∗) ⋆ kL)(x, y)

is positive. Our goal now is to show thatkL(x, y) = k(x, y) for x, y ∈ L from which it will follow

thatk ∈ KΨ.For this, once again recall the construction ofk

L from ρL. LetℓLa = ρL(χa)h, whereh = γ⊕ǫδL =µLF (δL)γ ⊕ ǫTL(δL)δL is the cyclic vector for the representationρL. Next, letkLb denote a dual basisto the basisℓLa and definekL by k

L(a, b) =⟨

kLb , kLa

⟩

.We calculate

ℓLa = ρL(χa)h = µLF (χa ⋆ δL)γ ⊕ ǫ χa ⋆ δ

L = µLF (χa) ⊕ ǫ χa,

which reduces to{0} ⊕ ǫ χa if a /∈ F , and which equalsℓFa ⊕ {0} if a ∈ F . Hence the dual basis is

kLa =

{

kFa ⊕ {0} a ∈ F,{0} ⊕ ǫ χa otherwise,

via which we immediately verify thatkL = k. �

18

3.6. Toeplitz representation for C∗-algebra-valued functions. The notion of the Toeplitz repre-sentation naturally generalizes to functionsf : F → C, whereF is a lower set andC is aC∗-algebrawith [T(ϕ)]a,b ∈ C andT(ϕ) ∈M(F,C), theC-valued matrices labelled by elements ofF .

Lemma 3.5. Suppose thatC is anotherC∗-algebra. Ifρ : C → C is a unital∗-representation, then

(1 ⊗ ρ)(T(f)) = T(ρ ◦ f)

and moreover,‖T(f)‖ ≥ ‖T(ρ ◦ f)‖.Proof. Simply compute

(1 ⊗ ρ)(T(f)) =[ρ([T(f)]a,b)]a,b

=

[

ρ

(

∑

cb=a

f(c)

)]

a,b

=

[

∑

cb=a

ρ(f(c))

]

a,b

= [[T(ρ ◦ f)]a,b]a,b.

The norm estimate follows sinceρ is completely contractive. �

In our applications of this lemmaf will be the functionE : F → B andρ : B → B(E) will bethe representation arising in aB-unitary colligation.

4. FACTORIZATION

Proposition 4.1. If Γ : G × G → B∗ is positive, then there exists a Hilbert spaceE and a functionL : G→ B(B, E) such that

Γ(x, y)(fg∗) = 〈L(x)f, L(y)g〉for all f, g ∈ B.

Further, there exists a unital∗-representationρ : B → B(E) such thatL(x)ab = ρ(a)L(x)b forall x ∈ G, a, b ∈ B.

Proof. The proof is a variant on a usual proof of the factorization ofpositive semidefinite kernels. Seethe book [6] Theorem 2.53, Proof 1. The statement should be compared with a similar result in [8].

LetW denote a vector space with basis labelled byG. On the vector spaceW ⊗ B introduce thepositive semidefinite sesquilinear form induced from

〈x⊗ f, y ⊗ g〉 = Γ(x, y)(fg∗),

wherex, y ∈ G andf, g ∈ B, makingW ⊗B into a pre-Hilbert space which is made into the HilbertspaceE by the standard modding out and completion.

One verifies that this is indeed positive as a consequence of the hypothesis thatΓ is positive. DefineL(x)a = x⊗ a. Since fora ∈ B,

‖L(x)a‖2 =〈L(x)a,L(x)a〉= Γ(x, x)(a∗a)

≤‖Γ(x, x)‖‖a∗a‖L(x) does indeed define a bounded operator onB with ‖L(x)‖2 ≤ ‖Γ(x, x)‖.

19

As for the∗-representation, it is induced by the left regular representation of B. That is, defineρ : B → B(E) by ρ(a)(x ⊗ f) = x ⊗ af . To see that this is indeed bounded, first note that‖a‖2 − a∗a is positive semidefinite inB and hence there exists ab so that‖a‖2 − a∗a = b∗b. Thus,

‖a‖2∥

∥

∥

∑

xj ⊗ fj

∥

∥

∥

2−∥

∥

∥

∑

xj ⊗ afj

∥

∥

∥

2

= ‖a‖2∑

Γ(xj , xℓ)(f∗ℓ fj) −

∑

Γ(xj, xℓ)(f∗ℓ a

∗afj)

=∑

Γ(xj , xℓ)(f∗ℓ b

∗bfj) ≥ 0

where the inequality is a result of the assumption thatΓ is positive. This shows at the same time thatρ is well defined.

We also have thatρ is unital, sinceρ(1)(x ⊗ f) = x⊗ 1f = x⊗ f .Finally,

〈ρ(a∗)(x⊗ f), y ⊗ g〉 = 〈x⊗ a∗f, y ⊗ g〉= Γ(x, y)(g∗a∗f)

= 〈x⊗ f, y ⊗ ag〉= 〈x⊗ f, ρ(a)(y ⊗ g)〉= 〈ρ(a)∗(x⊗ f), y ⊗ g〉

so thatρ(a∗) = ρ(a)∗. �

5. THE CONECF AND COMPACT CONVEX SETΦF

Given a finite subsetF ⊂ G, letM(F,B∗)+ denote the collection of positive kernelsΓ : F ×F →B∗ and define the cone

CF = {(

(Γ ⋆ ([1] − EE∗))(x, y))

x,y∈F: Γ ∈M(F,B∗)+}.

5.1. The cone is closed.

Theorem 5.1. LetF be a finite lower set. The coneCF is closed inM(F ).

Proof. LetM = Γ ⋆ ([1] − EE∗) ∈ CF , whereΓ : F × F → B∗ is positive. Positivity ofΓ meansin particular that ifF is a subset ofF , and{fq}q∈F is any collection of elements ofB, then

∑

p,q∈F

Γ(p, q)fpf∗q ≥ 0. (5.1)

For convenience we defineb ≤r a to meanbc = a for somec. As in Lemma 2.3, this can be shownto be an order onG andb ≤r a impliesb ≤ a.

Fix x ∈ F , and supposee is idempotent withxe = x. TakingFx = {y : y ≤r x}, we get a finitesubset ofF . Forq ∈ Fx, set

E(q) =∑

pqp=x

E(p).

20

Observe thatE(x) = E(e). With this notation, we have

Mx,x = Γ(x, x)(1 − E(x)E(x)∗) −∑

q<rxs<rx

Γ(q, s)(E(q)E(s)∗)

−∑

q<rx

Γ(q, x)(E(q)E(x)∗) −∑

s<rx

Γ(x, s)(E(x)E(s)∗).

(5.2)

Chooseǫ > 0 small enough that1 − (1 + ǫ2)E(x)E(x)∗ > 0. This can be done since1 −E(x)E(x)∗ = 1 − E(e)E(e)∗ > 0 by property (ii) of Definition 1.2. Letfx = −ǫE(x), fq =

(e−iθq/ǫ)E(q) for q ∈ Fx, q 6= x, andθq = arg(∑

q<rxΓ(q, x)E(q)E(x)∗). With this choice, (5.1)

gives

2|∑

q<rx

Γ(q, x)(E(q)E(x)∗)| ≤ (1/ǫ)2∑

q<rx

∑

s<rx

Γ(q, s)(E(q)E(s)∗) + ǫ2Γ(x, x)(E(x)E(x)∗).

(5.3)Combining the inequality in (5.3) with (5.2) we have

Γ(x, x)(1 − (1 + ǫ2)E(x)E(x)∗) ≤Mx,x +(

1 + 1ǫ2

)

∑

q<rx

∑

s<rx

Γ(q, s)(E(q)E(s)∗). (5.4)

Furthermore, positivity ofΓ and a calculation as for (5.3) yields forg ∈ B

2|Γ(x, y)g| ≤ Γ(x, x) 1 + Γ(y, y) gg∗ ≤ ‖Γ(x, x)‖ + ‖Γ(y, y)‖ ‖g‖2 ,

and so‖Γ(x, y)‖ ≤ 1

2 (‖Γ(x, x)‖ + ‖Γ(y, y)‖) . (5.5)

We show by induction on (right) strata that for eachp, q ∈ F , there is a constantcp,q, independentof Γ, such that‖Γ(p, q)‖ ≤ cp,q‖M‖. By (5.5), it suffices to prove this forp = q. SinceF is assumedfinite, it will then follow that‖Γ‖ ≤ c‖M‖ for somec ≥ 0 and independent ofΓ.

To begin with, if e ∈ F is idempotent, thenMe,e = Γ(e, e)(1 − E(e)E(e)∗), and since1 −E(e)E(e)∗ > 0, we have thatce,e exists. Now suppose that we havecp,q for all p, q in the(n − 1)stand lower strata. Letx be in thenth stratum. Then by the induction hypothesis and (5.4), we find cx,x.

Let {Mj} be a bounded sequence fromCF ,Mj = Γj ⋆ ([1] − EE∗), so that

Mj(x, y) = Γj(x, y)(1) −∑

pq=x

∑

rs=y

Γj(q, s)(E(p)E(r)∗), x, y ∈ F.

Then{Γj} is a bounded sequence inM(F,B∗)+; i.e., there is a uniform bound on the norm of thelinear functionalΓj(x, y) independent ofx, y, j. It follows from weak-∗ compactness, that there existsΓ ∈M(F,B∗) and a subsequence{Γjℓ} of {Γj} so that for eachx, y ∈ F , the sequence{Γjℓ(x, y)}converges toΓ(x, y) weak-∗. In particular,{Γjℓ(p, r)(E(q)E(s)∗)} converges toΓ(p, r)(E(q)E(s)∗)for eachp, q, r, s (and also withE(q)E(s)∗ replaced by1). If now {Mj} converges to someM , then

M = limℓ→∞

(

Γjℓ ⋆ ([1] − EE∗)(x, y))

x,y∈F=(

Γ ⋆ ([1] − EE∗)(x, y))

x,y∈F.

If f : F → B, then

0 ≤∑

x,y∈F

Γjℓ(x, y)(f(x)f(y)∗) →∑

x,y∈F

Γ(x, y)(f(x)f(y)∗),

which shows thatΓ is positive and completes the proof. �

21

5.2. The cone is big.

Lemma 5.2. LetΨ be a set of test functions forG. For eachψ ∈ Ψ the functionΓψ : G×G → B∗

given byΓψ(x, y)(f) = ([1] − ψψ∗)−1⋆(x, y)f(ψ), f ∈ B = C(Ψ),

is a positive kernel.

Proof. For eachx, y ∈ G, the functionalΓψ(x, y) is a multiple of evaluation atψ and hence doesindeed define an element ofB∗.

For a finite lower setF ⊂ G and a functionf : F → B,∑

x,y∈F

Γψ(x, y)(f(x)f(y)∗) =∑

x,y∈F

([1] − ψψ∗)−1⋆(x, y)(f(x)(ψ)f(y)(ψ)∗)

=∑

x,y∈F

([1] − ψψ∗)−1⋆(x, y)(g(x)g(y)∗)

whereg : F → C is given byg(x) = f(x)(ψ) andg is the vector withx entryg(x). By Corollary 3.2,

F × F ∋ (x, y) 7→ ([1] − ψψ∗)−1⋆(x, y)

is a positive matrix inM(F ). The conclusion follows. �

Lemma 5.3. SupposeF ⊂ G is a finite lower set. The coneCF contains all positive matrices. Inparticular, it contains[1] and so has non-trivial interior.

Proof. Let Γψ denote the positive kernel from the previous lemma. Then

[1](x, y) = Γψ ⋆ ([1] − EE∗)(x, y) ∈ CF .On the other hand, ifP ∈ M(F ) with P ≥ 0, thenP ⋆ Γψ ≥ 0 andP = P ⋆ [1] = P ⋆ (Γψ ⋆([1] −EE∗)). ThusCF contains all positiveP ∈M(F ). �

Lemma 5.4. The coneCF is closed under conjugation; i.e., ifM = (M(x, y)) ∈ CF andc : F → C,thenc ⋆ M ⋆ c∗ ∈ CF , where(c ⋆ M ⋆ c∗)(x, y) =

∑

pq=x

∑

rs=y c(p)M(q, s)c∗(r).

Proof. If M = Γ ⋆ ([1]−EE∗) ∈ CF , thenc ⋆ M ⋆ c∗ = Γ ⋆ ([1]−EE∗), whereΓ = c ⋆ Γ ⋆ c∗ ≥0. �

5.3. Separation.

Lemma 5.5. LetF be a finite lower set and supposeϕ ∈ H∞(KΨ). If

Mϕ =(

([1] − ϕϕ∗)(x, y))

x,y∈F/∈ CF ,

then there exists a cyclic unital representationµ : P (F ) → B(H) such that‖µ(ψ)‖ ≤ 1 for allψ ∈ Ψ|F , but‖µ(ϕ)‖ > 1.

Proof. By Theorem 5.1 the coneCF is closed (in the set ofF ×F matricesM(F )). As a consequenceof the Hahn-Banach Theorem (see, for example,§12.F of [25]), there is a linear functionalλ onM(F )such thatλ is nonnegative onCF andλ(Mϕ) < 0. As ‖Mφ‖ + Mφ ∈ CF by Lemma 5.3, we haveλ(1) > 0, where1 is the identity inM(F ). So in particular,λ is not identically zero onCF .

Next define a scalar product onP (F ) by

〈f, g〉 = λ(fg∗). (5.6)

For ease of notation, we will simply write “f ” for the restrictionf |F of f to the lower setF . We thenview f, g ∈ C

F as vectors so thatfg∗ ∈ M(F ) is the matrix with entriesfg∗(x, y) = f(x)g(y)∗.22

Since, by Lemma 5.3, the coneCF contains all positive matrices andλ is non-negative onCF , theform in equation (5.6) is positive semi-definite.

Mod out by the kernel and letq(f) denote the image off in the quotient. (Since the space is finitedimensional there is no need to complete to get a Hilbert space.) The resulting Hilbert space, whichwe callH, is nontrivial. In particular,q(δF ) 6= 0. To see this, first note that[1] ∈ CF , soλ([1]) ≥ 0.By assumptionλ(([1] − ϕϕ∗)) < 0, which impliesλ(ϕϕ∗) > 0. Since finite products of the testfunctions restricted toF spanP (F ), which is finite dimensional, we can writeϕ =

∑nk=0 ckξk, for

some finite collection of finite products of test functions{ξk}. Repeated use of the equality

[1] − ξkξjξ∗j ξ

∗k = ([1] − ξkξ

∗k) + ξk([1] − ξjξ

∗j )ξ

∗k

and Lemma 5.4 shows that[1] − ξξ∗ is in CF , and soλ([1] − ξξ∗) ≥ 0, for any finite product oftest functionsξ. By the Cauchy-Schwarz inequality, for anyj, k, |λ(ξjξ

∗k)| ≤ λ(ξjξ

∗j )λ(ξkξ

∗k), and

so if for all k, λ(ξkξ∗k) were zero, we would haveλ(ϕϕ∗) = 0. Hence there is some product of test

functionsξ such thatλ(ξξ∗) > 0. Consequentlyλ([1]) > 0, and so‖q(δF )‖ > 0.Letµ be the right regular representation ofP (F ) onH. That is,µ(g)q(f) = q(f ⋆ g) — provided

of course that it is well defined. Ifψ ∈ Ψ, then because of the definition ofCF ,

‖q(f)‖2 − ‖q(f ⋆ ψ)‖2 = λ(f∗ ⋆ ([1] − ψψ∗) ⋆ f) ≥ 0,

where the inequality follows from Lemma 5.4. Thus,µ(ψ) is well defined and sinceΨ|F generatesP (F ), µ is well defined.

Clearlyµ is cyclic with cyclic vectorq(δF ). Finally,

‖q(δF )‖2 − ‖µ(ϕ)q(δF )‖2 = λ([1] − ϕϕ∗) < 0

so that‖µ(ϕ)‖ > 1. �

5.4. A compact set. Fix ϕ : G → C and a collection of test functionsΨ. ForF ⊂ G a finite lowerset, let

ΦF = {Γ ∈M(F,B∗)+ : ([1] − ϕϕ∗)(x, y) = (Γ ⋆ ([1] − EE∗))(x, y) for x, y ∈ F}.The setΦF is naturally identified with a subset of the product ofB∗ with itself |F |2 times.

Lemma 5.6. The setΦF is compact.

Proof. Let Γα be a net inΦF . Arguing as in the proof of Theorem 5.1, we find eachΓα(x, x) is abounded net and thus eachΓα(x, y) is also a bounded net. By weak-∗ compactness of the unit ballin B∗ there exists aΓ and subnetΓβ of Γα so that for eachx, y ∈ F , the netΓβ(x, y) converges toΓ(x, y). �

6. PROOF OF THE REALIZATION THEOREM, THEOREM 1.3

6.1. Proof of (i) implies (iiF). Suppose that (iiF) does not hold. In this case there exists a finite lowersetF ⊂ G so that the matrix

Mϕ =(

([1] − ϕϕ∗)(x, y))

x,y∈F

is not in the coneCF = {

(

Γ ⋆ ([1] − EE∗))

x,y∈F: Γ ∈M(F,B∗)+}.

Lemma 5.5 produces a representationµ : P (F ) → B(H) so that‖µ(ψ)‖ ≤ 1 for all ψ ∈ Ψ|F , but‖µ(πF (ϕ))‖ > 1. Lemma 3.4 now implies‖ϕ‖ > 1.

23

6.2. Proof of (iiF) implies (iiG). The proof here uses Kurosh’s Theorem and in much the same wayas in [6].

The hypothesis is that for every finite lower setF ⊂ G, ΦF , as defined in Subsection 5.4 is notempty. The result in that section is thatΦF is compact. For a finite lower setF contained in a lowersetH, defineπHF : ΦH → ΦF by

πHF (Γ) = Γ|F×F .

Thus, withF equal to the collection of all finite lower subsets ofG partially ordered by inclusion,the triple(ΦG, π

GF ,F) is an inverse limit of nonempty compact spaces. Consequently, by Kurosh’s

Theorem ([6], p. 30), for eachF ∈ F there is aΓF ∈ ΦF so that wheneverF,H ∈ F andF ⊂ H,

πHF (ΓH) = ΓF . (6.1)

DefineΓ : G ×G → B∗ by Γ(x, y) = ΓF (x, y) whereF ∈ F is any lower set so thatx, y ∈ F .This is well defined by the relation in equation (6.1). IfF is any finite lower set andf : F → B isany function, then

∑

x,y∈F

Γ(x, y)(f(x)f(y)∗) =∑

x,y∈F

ΓF (x, y)(f(x)f(y)∗) ≥ 0

sinceΓF ∈ M(F,B∗)+. Any finite subset ofG is contained in a finite lower set, and so it followsthatΓ is positive.

6.3. Proof of (iiG) implies (iii). Let Γ denote the positive kernel in (iiG). Apply Proposition 4.1 tofind E , L : G→ B(B, E), andρ : B → B(E) as in the conclusion of the proposition.

Rewrite condition (iiG) as

[1](x, y) + (Γ ⋆ (EE∗))(x, y) = (ϕϕ∗)(x, y) + (Γ ⋆ [1])(x, y)

m[1](x, y) +

∑

pq=x

∑

rs=y

Γ(q, s)(E(p)E(r)∗) =ϕ(x)ϕ(y)∗ +∑

pq=x

∑

rs=y

Γ(q, s)(δ(p)δ(r)∗)

mδ(x)δ(y)∗ +

∑

pq=x

∑

rs=y

〈L(q)E(p)1, L(s)E(r)1〉 =ϕ(x)ϕ(y)∗ +∑

pq=x

∑

rs=y

〈L(q)δ(p)1, L(s)δ(r)1〉

mδ(x)δ(y)∗ + 〈(ρ(E) ⋆ L)(x)1, (ρ(E) ⋆ L)(y)1〉 =ϕ(x)ϕ(y)∗ + 〈(ρ(δ) ⋆ L)(x)1, (ρ(δ) ⋆ L)(y)1〉,

(6.2)

where1 is the identity inB. We have used the intertwining relation betweenL andρ from Proposi-tion 4.1. Notice that in doing so the⋆-product is replaced by the⋆-product.

From here the remainder of the proof is the standard lurking isometry argument.Let Ed denote finite linear combinations of

(

(ρ(E) ⋆ L)(x)1δ(x)

)

∈E⊕C

and letEr denote finite linear combinations of(

(ρ(δ) ⋆ L)(x)1ϕ(x)

)

∈E⊕C

.

24

DefineV : Ed → Er by

V

(

(ρ(E) ⋆ L)(x)1δ(x)

)

=

(

(ρ(δ) ⋆ L)(x)1ϕ(x)

)

and extending by linearity. Equation (6.2) implies∥

∥

∥

∥

∑

cj

(

(ρ(E) ⋆ L)(xj)1δ(xj)

)∥

∥

∥

∥

2

=∑

j,ℓ

cjc∗ℓ ([1]xj ,xℓ

+ 〈(ρ(E) ⋆ L)(xj)1), (ρ(E) ⋆ L)(xℓ)1)〉)

=∑

j,ℓ

cjc∗ℓ (ϕ(xj)ϕ(xℓ)

∗ + 〈(ρ(δ) ⋆ L)(xj)1, ρ(δ) ⋆ L)(xℓ)1〉)

=

∥

∥

∥

∥

∑

cj

(

ρ(δ) ⋆ L)(xj)1ϕ(xj)

)∥

∥

∥

∥

2

which shows simultaneously thatV is well defined and an isometry. ThusV (the lurking isometry)extends to an isometry from the closure ofEd to the closure ofEr. There exists a Hilbert spaceHcontainingE and a unitary map

U :H⊕C

→H⊕C

so thatU restricted toEd is V ; i.e.,Uγ = V γ for γ ∈ Ed.Write

U =

(

A BC D

)

(6.3)

with respect to the decompositionH⊕ C. In particular,(

A BC D

)(

(ρ(E) ⋆ L)(x)1δ(x)

)

=

(

(ρ(δ) ⋆ L)(x)1ϕ(x)

)

which gives the system of equations

A(ρ(E) ⋆ L)(x)1 +Bδ(x) = (ρ(δ) ⋆ L)(x)1 (6.4)

C(ρ(E) ⋆ L)(x)1 +Dδ(x) = ϕ(x).

From the first equation in (6.4) we have

L(x)1 = ((ρ(δ) −Aρ(E))−1⋆ ⋆ (Bδ))(x), (6.5)

where the inverse is with respect to the⋆-product. Plugging this into the second equation of (6.4)gives

ϕ(x) = Dδ(x) + C(ρ(E) ⋆ (ρ(δ) −Aρ(E))−1⋆ ⋆ (Bδ))(x), (6.6)

which, using the fact thatρ is unital, can be written

ϕ(x) = Dδ(x) + C(ρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ))(x), (6.7)

as desired.25

6.4. Proof of (iii) implies (i). Supposeϕ = WΣ as in equation (1.4) (equivalently, equation (6.7)above). We want to show thatk − ϕ ⋆ k ⋆ ϕ∗ ≥ 0 for all k ∈ KΨ. First, factork(x, y) = kxk

∗y.

To make the notation consistent with that used in the calculations below, we writek(x) for kx. Wecompute the (four) terms in(k−ϕ ⋆ k ⋆ ϕ∗)(x, y), using the identities implied byU being unitary andthe equality(δ ⋆ k)(k∗ ⋆ δ) = δ ⋆ k ⋆ δ = k. Recall that for functionsf andg, (f ⋆ g)∗ = g∗ ⋆ f∗.

To begin with,CC∗ = 1 −DD∗ and so we haveDk(x, y)D∗ = k(x, y) − Ck(x, y)C∗. Hence

k(x, y) −Dk(x, y)D∗ = (C(δ ⋆ k))(x)((k∗ ⋆ δ)C∗)(y)

=(

C(δ − ρ(E)A)−1⋆ ⋆ (δ − ρ(E)A) ⋆ k)

(x)(

k∗ ⋆ (δ − ρ(E)A)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y).

For the next few terms it is useful to observe that

ρ(E) ⋆ (δ −Aρ(E))−1⋆ = (δ − ρ(E)A)−1⋆ ⋆ ρ(E),

or equivalently,(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (δ −Aρ(E)) = ρ(E),

which follows from

ρ(E) ⋆ (δ −Aρ(E)) = ρ(E) ⋆ δ − ρ(E) ⋆ Aρ(E) = (δ − ρ(E)A) ⋆ ρ(E).

The second term we consider is(

Cρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ) ⋆ k)

(x)(k∗ ⋆ δ∗D∗)(y)

=(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (Bk))

(x)(k∗D∗)(y)

=∑

pqr=x

C(δ − ρ(E)A)−1⋆(p)ρ(E(q))Bk(r)k(y)∗D∗

=∑

pqr=x

C(δ − ρ(E)A)−1⋆(p)ρ(E(q))k(r, y)BD∗

= −∑

pqr=x

C(δ − ρ(E)A)−1⋆(p)ρ(E(q))Ak(r, y)C∗

= −∑

pqr=x

C(δ − ρ(E)A)−1⋆(p)ρ(E(q))Ak(r)k(y)∗C∗

= −(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ Ak)

(x)(

k∗ ⋆ (δ − ρ(E)A)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y).

For the third term,

(Dδ ⋆ k)(x)(

Cρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ) ⋆ k)

(y)∗

= (Dk)(x)(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (Bδ) ⋆ k)

(y)∗

= (Dk)(x)(

(k∗B∗) ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y)

= − (Ck)(x)(

A∗k∗ ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y)

= −(

C(δ − ρ(E)A)−1⋆ ⋆ (δ − ρ(E)A) ⋆ k)

(x)(

A∗k∗ ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y).

Finally, the last term is(

Cρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ) ⋆ k)

(x)(

Cρ(E) ⋆ (δ −Aρ(E))−1⋆ ⋆ (Bδ) ⋆ k)

(y)∗

=(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (Bk))

(x)(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (Bk))

(y)∗

=(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ (Bk))

(x)(

(k∗B∗) ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y)

=(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ k)

(x)(1 −AA∗)(

k∗ ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y).26

Putting them together we have

k− ϕ ⋆ k ⋆ ϕ∗

=(

C(δ − ρ(E)A)−1⋆ ⋆ (δ − ρ(E)A) ⋆ k) (

k∗ ⋆ (δ − ρ(E)A)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

−(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ Ak) (

k∗ ⋆ (δ − ρ(E)A)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

−(

C(δ − ρ(E)A)−1⋆ ⋆ (δ − ρ(E)A) ⋆ k) (

A∗k∗ ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

+(

C(δ − ρ(E)A)−1⋆ ⋆ ρ(E) ⋆ k)

(1 −AA∗)(

k∗ ⋆ ρ(E)∗ ⋆ (δ − ρ(E)A)∗−1⋆C∗)

(y),

which after a bit of algebra is seen to simplify to

k− ϕ ⋆ k ⋆ ϕ∗ = C(δ − ρ(E)A)−1⋆ ⋆ (k − ρ(E) ⋆ k ⋆ ρ(E)∗) ⋆ (δ − ρ(E)A)∗−1⋆C∗.

Given a finite lower setF ⊂ G, the matrix

P =(

(k −E ⋆ k ⋆ E∗)(x, y))

x,y∈F∈M(F,B)

is a positive since its value atψ ∈ Ψ is

P (ψ) =(

(k − ψ ⋆ k ⋆ ψ∗)(x, y))

.

Consequentlyk− ϕ ⋆ k ⋆ ϕ∗ ≥ 0 overF , which completes the proof.

7. AGLER-JURY-PICK INTERPOLATION

We now turn to the proof of Theorem 1.4. Condition (i) impliescondition (ii) simply by the defini-tion of the norm onH∞(KΨ).

If condition (iii) does not hold, then an argument just as in the proof of Theorem 1.3 produces akernelk ∈ KΨ so that the relevant kernel onF is not positive. Hence (ii) implies (iii).

To prove that (iii) implies (i), first argue along the lines ofthe proof of (iiG) implies (iii) in The-orem 1.3, but work with the finite setF in place ofG. Next verify that the transfer functionWΣ soconstructed and defined on all ofG satisfiesWΣ(x) = f(x) for x ∈ F (since we worked withF ).The implication (iii) implies (i) in Theorem 1.3 now says that ‖WΣ‖ ≤ 1. �

This leads to the following, which is reminiscent of resultson left tangential Nevanlinna-Pickinterpolation.

Theorem 7.1. LetF be a finite lower set in a semigroupoidG. Supposew(a), z(a) ∈ C, a ∈ F aregiven. Then there is a functionϕ ∈ H∞(KΨ) with ‖ϕ‖H∞(KΨ) ≤ 1 such that

(ϕ ⋆ z)(a) = w(a), for all a ∈ F,

if and only if(z∗z − w∗w) ⋆ k ≥ 0, for all k ∈ KΨ.

Proof. If w = ϕ ⋆ z with ϕ ∈ H∞(KΨ) and‖ϕ‖H∞(KΨ) ≤ 1, then for allk ∈ KΨ,

(z∗z − w∗w) ⋆ k = (z∗ ⋆ ([1] − ϕ∗ϕ) ⋆ z) ⋆ k

= ([1] − ϕ∗ϕ) ⋆ (z∗z) ⋆ k

= ([1] − ϕ∗ϕ) ⋆ (z∗ ⋆ k ⋆ z)

≥ 0,

since by Lemma 5.4,z∗ ⋆ k ⋆ z ∈ KΨ.27

Now suppose(zz∗ − ww∗) ⋆ k ≥ 0 for all k ∈ KΨ. Begin by assuming thatz is an invertiblefunction with respect to the⋆ product (i.e.,z(a) is invertible for alla ∈ Fe). Setf = w ⋆ z−1⋆ onF .Then restricting toF ,

0 ≤ (z∗z − w∗w) ⋆ k = (z∗ ⋆ ([1] − f∗f) ⋆ z) ⋆ k = ([1] − f∗f) ⋆ (z∗ ⋆ k ⋆ z)

for all k ∈ KΨ. Again by Lemma 5.4,z−1⋆ ∗ ⋆ k ⋆ z−1⋆ is inKΨ if k ∈ KΨ. Hence([1]−f∗f) ⋆ k ≥0 onF ×F for all k ∈ KΨ, and so by Theorem 1.4f extends toϕ ∈ H∞(KΨ) with ‖ϕ‖H∞(KΨ) ≤ 1

such that([1] − ϕ∗ϕ) ⋆ k ≥ 0 onG × G. Padz with zeros to make it a function inH2(k) for allk ∈ Kψ, and setw = ϕ ⋆ z (which agrees with the original definition ofw on the lower setF ).

If z is not⋆-invertible, thenz(a) = 0 for somea ∈ Fe. This means that{z(a) : a ∈ F} (wherez(a) is identified with the vector with this value in theath position and zero elsewhere) is not a basisfor C

F . Choose a vectorg with g(a) = 1 for eacha in Fe wherez(a) = 0 and zero otherwise. Fixǫ > 0. Let z′ beg normalized so thatRe 〈z, z′〉 ≥ 0 and‖z′‖ < ǫ. Then forzǫ = z + z′,

(z∗ǫ zǫ − w∗w) ⋆ k ≥ (z∗z + z′∗z′ − w∗w) ⋆ k ≥ (z∗z − w∗w) ⋆ k ≥ 0,

andzǫ is invertible, so we obtain by the last paragraph a correspondingfǫ for which([1]−f∗ǫ fǫ) ⋆ k ≥0 for all k ∈ KΨ and fǫ ⋆ zǫ = w. Since we are on a finite dimensional space, the sequence{f1/n}n=1,2,... converges to somef ∈ P (F ) with ([1] − f∗f) ⋆ k ≥ 0 for all k ∈ KΨ. Alsoz1/n −→ z. Consequently,f ⋆ z = w. �

A right tangential problem could very easily be formulated and solved. One way to do this wouldbe to replace “⋆” with “ ⋆ ” at appropriate points in the left interpolation theorem and proof, and thentake adjoints. The details are left to the interested reader.

Finally note that takingz = δF andw = f in the last theorem recovers the first two equivalencesin Theorem 1.4.

8. EXAMPLES

8.1. The classical examples.View D as a Pick semigroupoid. The partial multiplication is trivial andso eachz ∈ D is idempotent. TakeΨ = {z} (z meaning here the identity function) as the collectionof test functions. The Agler-Jury-Pick interpolation theorem in this case is Pick interpolation.

ChooseG = N with the usual semigroup(oid) structure. LetΨ = {z}, where byz we mean thefunctionz : N → C given byz(j) = 0 if j 6= 1 andz(1) = 1 (we think ofz(j) as the derivatives ofzat0). In this case Agler-Jury-Pick interpolation is Caratheodory-Fejer interpolation.

For mixed Agler-Pick and Caratheodory-Fejer chooseG = D×N with the semigroupoid structure,

(z, n)(w,m) =

{

is not defined ifz 6= w

(z, n +m) if z = w

and letΨ = {z} denote the functionz(w, 0) = w, z(w, 1) = 1 andz(w,m) = 0 for m ≥ 2.

8.2. Agler-Pick interpolation on an annulus. Let A denote an annulus{q < |z| < 1}, viewed as aPick semigroupoid.

There is a family of analytic functionsψ : A → D which are unimodular on the boundary ofA andhave precisely two zeros inA (counting with multiplicity), normalized byψ(

√q) = 0 andψ(1) = 1.

If ϕ is any other analytic function onA which is unimodular on the boundary and has exactly twozeros (counting with multiplicity), then there is a Mobiusmapm from the disk onto the disk such thatm ◦ ϕ ∈ Ψ. There is a canonical parameterization ofΨ by the unit circle.

28

Theorem 8.1. The collectionΨ is a family of test functions forA and the norm inH∞(KΨ) is thesame as the norm onH∞(A). Moreover, no proper subset ofΨ is a set of test functions which givesthe norm ofH∞(A).

In the case of Agler-Pick interpolation(on a finite setF ⊂ A), the realization formula for a solutionis in terms of a single positive measure on the unit circle.

Look for the details of this example in the forthcoming paper[22].

8.3. Caratheodory interpolation kernels. Let N denote the natural numbers with the usual semi-group(oid) structure. A kernelk on N is a Caratheodory interpolation kernel [33] provided (by wayof normalization)k(0, 0) = 1, k(0, n) = 0 for n > 0, and

b = [1] − k−1⋆

is positive.For illustrative purposes, supposeb has finite rankd and so factors asb = B∗B, whereB : N →

(Cd)∗. AlthoughB is not scalar-valued,[1] −B(a)B(b)∗ = [1] − b is scalar and moreover,

([1] −BB∗) ⋆ k = ([1] − b) ⋆ k = k−1⋆ ⋆ k = [1] ≥ 0.

ChoosingΨ = {B}, it turns out thatϕ ∈ H∞(KΨ) and‖ϕ‖H∞(KΨ) ≤ 1 if and only if ([1]−ϕϕ∗) ⋆ k

is positive.

8.4. NP kernels and Arveson-Arias-Popescu space.The situation for Nevanlinna-Pick (NP) ker-nels is similar to that for Caratheodory kernels. In particular, it requires a version of our results forvector valued test functions.

As a particular example, consider the semigroupNg with the (single) vector valued test function

Z =(

z1 z2 · · · zg)T

. This pair(Ng, Z) gives rise to symmetric Fock space; i.e., the space ofmultipliers of the space of analytic functions on the unit ball in C

g with reproducing kernelk(z,w) =(1−〈z,w〉)−1 studied by Arveson ([12, 13], in the commutative case) and byArias and Popescu ([10,11], in both the commutative case and the noncommutative case discussed in the next subsection).

8.5. Noncommutative Toeplitz algebras.The following have been considered in the context ofNevanlinna-Pick and Caratheodory-Fejer interpolationby Davidson and Pitts [20] and Arias andPopescu [11], as well as by Popescu in [45, 46].

Let F = Fg denote the free monoid on theg letters{x1, . . . , xg}. Let ψj : F → C denote thefunctionψj(xj) = 1 andψ(w) = 0 if w is any word other thanxj . The matrixT(ψj) is a (truncated)shift on Fock space.

Given a wordw = xj1xj2 · · · xjn , let

ψw⋆ = ψj1 ⋆ ψj2 ⋆ · · · ⋆ ψjn .Sinceψw⋆(v) = 1 if w = v and0 otherwise, it follows that ifF is any finite subset ofF, thenP (F )contains all functions onF .

Letψ =(

ψ1 · · · ψg)T

and considerψ as a (single) test function. We calculate

s(x, y) = ([1] − ψ∗ψ)−1⋆

wheres is the Toeplitz kernel (s(x, y) = 1 of x = y ands(x, y) = 0 if x 6= y). Then ifk is any kernelfor which ([1] − ψ∗ψ) ⋆ k = Q is positive, we haves ⋆ Q = k. It follows that‖ϕ‖ ≤ 1 if and onlyif the kernel

F × F ∋ (x, y) 7→ ([1] − ϕϕ∗) ⋆ s(x, y)29

is positive, if and only if‖T(ϕ)‖ ≤ 1. The versions of the Nevanlinna-Pick theorem considered inthe papers cited above coincide with Theorem 7.1, while Caratheodory-Fejer interpolation is given byTheorem 1.4.

As a final remark, note that eachT(ψj) is an isometry and∑

T(ϕ)T(ϕ)∗ = P∅ ≥ 0.

HereP∅ is the projection onto the span of the vacuum vector∅ in the Fock space.

8.6. The Polydisk. The semigroupoidNg (theg-fold product of the nonnegative integers) with the setof test functionszj, the characteristic function ofej the vector with1 in thej-th entry and0 elsewhere,gives rise to the Schur-Agler class of the polydiskD

g returning us to the introduction and [3].

8.7. Semigroupoid algebras of Power and Kribs.Kribs and Power [29, 30] consider a generaliza-tion of the noncommutative Toeplitz algebras which they term a free semigroupoid algebra. Orderarises from the assumption of freeness, the resulting semigroupoid is cancellative, and there is a rep-resentation (related to our Toeplitz representation on characteristic functionsχa) in terms of partialisometries and projections.

A notion of a generalized Fock space is developed, which is simply the Hilbert space with or-thonormal basis labelled by the elements of the semigroupoid. The algebras of interest in these papersare obtained from the weak operator topology closure of the algebras generated by the left regularrepresentations (i.e., the projections and partial isometries mentioned above).

The algebras are closely related to those in the present paper whenG is a semigroupoid in thismore restrictive sense and the collection of test functionsconsists of the characteristic functions ofnon-idempotent elements from the first stratum (to use our terminology).

It is assumed that for every idempotente ∈ G, there is a non-idempotenta such thatae is defined.LetG1 be the first (left) stratum inG, and assume that this set is countable. ThenG is generated byG1, in the sense that ifx is in thenth stratum, thenx = ay, wherey is in the(n− 1)st stratum andais in the first stratum. LetP have the property that

P (x, y) =

{

1 x = y andx, y /∈ Ge,

0 otherwise,

ands = [1] + P . Clearlys is invertible. Now mimic the proof in Section 8.5 by lettingψj(xj) = 1 if

xj ∈ G1. It is not difficult to verify that forψ =(

ψ1 . . . ψg)T

,

s(x, y) = ([1] − ψ∗ψ)−1⋆ ,

and so just as in that subsection, ifk is any kernel for which([1] − ψ∗ψ) ⋆ k = Q is positive,s ⋆ Q = k. It follows that the statements‖ϕ‖ ≤ 1,

F × F ∋ (x, y) 7→ ([1] − ϕϕ∗) ⋆ s(x, y)

positive, and‖T(ϕ)‖ ≤ 1 are all equivalent.A number of interesting algebras can be generated in this manner, including the noncommutative

Toeplitz algebras above and the norm closed semicrossed product Cn ×σβ Z+ [30]. Indeed, the con-

dition of being freely generated can be replaced by our more general conditions for a semigroupoid(again assuming though that for every idempotente ∈ G, there is a non-idempotenta such thatae isdefined). Our results allow for interpolation in all of thesealgebras.

30

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