INTRODUCTION TO NONCOMMUTATIVE · PDF fileContents 1 TOPOLOGY 9 1.1 ... 1.1.4 Spectrum of a...

INTRODUCTION TO NONCOMMUTATIVEGEOMETRY

A.G.Sergeev

June 18, 2016

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Contents

1 TOPOLOGY 9

1.1 Commutative Banach algebras . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.2 Adding of unit and compactification . . . . . . . . . . . . . . 10

1.1.3 Characters and spectrum . . . . . . . . . . . . . . . . . . . . . 10

1.1.4 Spectrum of a commutative Banach algebra. Gelfand transform 12

1.1.5 Gelfand–Naimark theorem . . . . . . . . . . . . . . . . . . . . 14

1.1.6 Positive elements and states . . . . . . . . . . . . . . . . . . . 14

1.1.7 Gelfand–Naimark–Segal construction(GNS-construction) . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.8 Embedding of C∗-algebras into the algebra of bounded linearoperators in a Hilbert space . . . . . . . . . . . . . . . . . . . 16

1.1.9 Correspondence: compact spaces ↔ unital commutative Ba-nach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.1 Complex vector bundles . . . . . . . . . . . . . . . . . . . . . 18

1.2.2 Functor Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2.3 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.4 Serre–Swan theorem . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 Functional analysis over C∗-algebras . . . . . . . . . . . . . . . . . . 24

1.3.1 C∗-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.3.3 Operators of A-finite rank and A-compact operators . . . . . . 29

1.3.4 Projectors in C∗-modules . . . . . . . . . . . . . . . . . . . . . 30

1.3.5 Unitary operators. Adjoint operators . . . . . . . . . . . . . . 31

1.3.6 Projective C∗-modules . . . . . . . . . . . . . . . . . . . . . . 32

1.4 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4.1 K0-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4.2 Higher K-groups . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.5 Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.5.1 Topological theory . . . . . . . . . . . . . . . . . . . . . . . . 42

1.5.2 Fredholm operators in C∗-modules . . . . . . . . . . . . . . . 45

1.5.3 Index of A-Fredholm operators . . . . . . . . . . . . . . . . . 47

1.6 Morita-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.6.1 Morita-equivalence of algebras . . . . . . . . . . . . . . . . . . 50

1.6.2 Morita-equivalence of C∗-algebras . . . . . . . . . . . . . . . . 52

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4 CONTENTS

2 ANALYSIS 552.1 Noncommutative integral . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.1.1 Ideals in the algebra of compact operators . . . . . . . . . . . 552.1.2 Dixmier trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.1.3 Pseudodifferential operators . . . . . . . . . . . . . . . . . . . 612.1.4 Wodzicki residue . . . . . . . . . . . . . . . . . . . . . . . . . 642.1.5 Connes trace theorem . . . . . . . . . . . . . . . . . . . . . . 67

2.2 Noncommutative differential calculus . . . . . . . . . . . . . . . . . . 682.2.1 Universal differential algebra . . . . . . . . . . . . . . . . . . . 682.2.2 Cycles and Fredholm modules . . . . . . . . . . . . . . . . . . 722.2.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.2.4 Chern character . . . . . . . . . . . . . . . . . . . . . . . . . . 802.2.5 Hochschild homology and cohomology . . . . . . . . . . . . . 82

3 SPINOR GEOMETRY 873.1 Spinor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.1.1 Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . 873.1.2 Spinor groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.1.3 Relation to the exterior algebra . . . . . . . . . . . . . . . . . 923.1.4 The group Spinc . . . . . . . . . . . . . . . . . . . . . . . . . . 953.1.5 Spin representation . . . . . . . . . . . . . . . . . . . . . . . . 97

3.2 Spinor geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2.1 Spin structures . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2.2 Spinor connections . . . . . . . . . . . . . . . . . . . . . . . . 1043.2.3 Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.2.4 Spinc-structures . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4 NONCOMMUTATIVE SPINOR GEOMETRY 1214.1 Spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.2 Definition of the noncommutative spinor geometry . . . . . . . . . . . 122

4.2.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.2.2 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.3 Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.2.4 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.5 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.6 Poincare duality . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.7 Definition of noncommutative spinor geometry . . . . . . . . . 126

4.3 Dirac geometry as a noncommutative spinor geometry . . . . . . . . . 1264.3.1 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3.3 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.4 Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.5 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.6 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.7 Poincare duality . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.4 Noncommutative spinor geometry over the algebra A = C∞(M) . . . 1284.4.1 Construction of the volume form . . . . . . . . . . . . . . . . 129

CONTENTS 5

4.4.2 Construction of the spin structure and metric . . . . . . . . . 1294.4.3 Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 CONTENTS

FOREWORD

The roots of noncommutative geometry lie in the theory of commutative Banach al-gebras and its connections with topology established by I.M.Gelhand, M.A.Naimark,G.E.Shilov, B.Mazur and other mathematicians in the middle of XXth century.Their main idea was that basic notions of the topology of compact topological spacesmay be translated into the language of commutative Banach algebras of continuousfunctions on these compacta.

One of the goals of noncommutative geometry is to establish such correspondencebetween topology, analysis and differential geometry, on one hand and Banach al-gebras, on the other hand. Otherwise speaking, we would like to translate basicnotions of topology, analysis and geometry into the language of Banach algebras.In this case we cannot restrict any longer to the theory of commutative Banachalgebras but have to use also noncommutative Banach algebras, more precisely theC∗-algebras of operators in a Hilbert space.

At that point a natural question arises: why do we need such a translation?One of the motivations behind it comes from theoretical physics. It is well knownthat the quantum field theory remains to a large extent the physical theory withouta solid mathematical basis. In contrast with quantum mechanics, which may beconsidered with some caution as a mathematically rigorous discipline, many resultsof quantum field theory (and string theory, in particular) are established only on the”physical level of rigorousness” and do not have correct mathematical proofs. Oneof the reasons for such a situation, in our opinion, is the absence of the adequatemathematical language adjusted for the description of the arising physical problems.

We think that such language should incorporate as one of its important ingre-dients the well-developed differential geometry of smooth infinite-dimensional man-ifolds. However, the classical notions of differential geometry such as connection,curvature et al. do not survive after transition to infinite dimensions. For example,various equivalent definitions of connection, known in the finite-dimensional case,acquire completely different sense (or loose it at all) for infinite-dimensional mani-folds. It is even more true with respect to the curvature which in most cases cannotbe correctly defined by analogy with the finite-dimensional case.

In this situation it seems that the most adequate language for the descriptionof notions of topology, analysis and differential geometry on infinite-dimensionalmanifolds should be the most ”robust” among available mathematical languages,namely, the algebraic one. This language has the highest chances to ”survive” un-der transition to infinite dimensions. To make this language ready for use in theinfinite-dimensional situation we should have available a ”dictionary” translatingbasic notions of topology, analysis and differential geometry into the algebraic lan-guage in the usual finite-dimensional setting. The elaboration of such dictionary is

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8 CONTENTS

precisely the main goal of this lecture course. Having it in mind, we concentratehere mostly on the ideas rather than technical details. By the same reason we omitsome proofs providing instead the references to other books.

This text is based on the lecture course delivered by the author at the ScientificEducational Center (SEC) of Steklov Institute during the spring semesters of years2014 and 2015.

The lectures were accompanied by the seminar of SEC on the same topic. Manyproblems which are only touched upon in this text were considered in detail atthe seminar. I am grateful to Alexander Komlov, Innocenti Maresin and RomanPalvelev who carried out the main burden of organizing this seminar. I would alsolike to thank the listeners of the course who helped a lot to improve the originaltext of the lectures with their questions and remarks.

While preparing this text the author was partly supported by the RFBR grants16-01-00117, 13-02-91330 and scientific program of Presidium of RAS ”Nonlineardynamics”.

Moscow Armen Sergeev

Chapter 1

TOPOLOGY

1.1 Commutative Banach algebras

1.1.1 C∗-algebras

Definition 1. The Banach algebra is an associative algebra A over the field C whichis simultaneously a complete normed space with the norm satisfying the inequality:

‖ab‖ ≤ ‖a‖ · ‖b‖for all elements a, b ∈ A. The algebra A is called unital if it contains the unit 1 withthe norm ‖1‖ = 1.

Definition 2. The involution in A is an isometric antilinear map a 7→ a∗ with thefollowing properties:

a∗∗ = a, (ab)∗ = b∗a∗

for any a, b ∈ A. A Banach algebra with involution is called otherwise the Banach*-algebra. The C∗-algebra is a Banach *-algebra A with the following additionalproperty:

‖a2‖ = ‖a∗a‖for all a ∈ A.

The standard example of the C∗-algebra is the algebra of continuous functionson a compact.

Example 1. Let X be a compact topological space (assumed to be Hausdorff asall other topological spaces considered in this text). Denote by C(X) the algebra ofcontinuous functions f : X → C provided with the norm

‖f‖ = supx∈X

|f(x)|.

The unit in C(X) coincides with the function f ≡ 1 and involution f 7→ f ∗ is givenby the complex conjugation: f ∗(x) := f(x). The introduced norm has the property

‖f‖2 = ‖f ∗f‖so C(X) is a C∗-algebra. Summing up, C(X) is a commutative unital C∗-algebra.

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10 CHAPTER 1. TOPOLOGY

1.1.2 Adding of unit and compactification

Any non-unital Banach algebra A may be embedded into a unital one by addingformally the unit 1A. In other words, we can extend A to the algebra A+ := A×Cwith evident rules of summation, multiplication by complex numbers and involution.The product in A+ is defined by

(a, λ) · (b, µ) = (ab + µa + λb, λµ)

so that 1A is identified with the element (0, 1). The norm of an element (a, λ) isdefined by:

‖(a, λ)‖ := sup‖b‖≤1

‖ab + λb‖.

The constructed algebra A+ is a unital C∗-algebra if A is a C∗-algebra.

Consider this unitalization procedure in the case of the commutative Banachalgebra of continuous functions.

Let Y be a locally compact topological space. Denote by Y + := Y ∪ ∞ theone-point compactification of Y and consider the subalgebra of C(Y +) consisting offunctions vanishing at infinity. By restricting the functions from this subalgebra toY we can identify it with the subalgebra C0(Y ) in C(Y ), consisting of the functionsvanishing at infinity. By definition the elements of this subalgebra are the functionsfrom C(Y ) which have the following property: the set y ∈ Y : |f(y)| ≥ ε iscompact for any ε > 0. The unitalization of this subalgebra coincides with thealgebra C(Y +).

Conversely, if we exclude from a compact topological space X a non-isolatedpoint x0 ∈ X then the obtained space Y := X \ x0 will be locally compact withY + = X and C0(Y ) = f ∈ C(X) : f(x0) = 0.

So the described unitalization procedure corresponds in the language of topolog-ical spaces to the one-point compactification.

1.1.3 Characters and spectrum

Definition 3. A character of a Banach algebra A is an algebra homomorphismµ : A → C. In other words, it is a non-zero linear functional µ : A → C on thealgebra A with the multiplicativity property, i.e. µ(ab) = µ(a)µ(b) for all a, b ∈ A.If the algebra A is unital then µ(1A) = 1. The set of characters of the algebra A isdenoted by M(A) and called otherwise the spectrum of the algebra A.

Example 2. An example of a character for the algebra A = C(X) of continuousfunctions on a compact topological space X is given by the evaluation at x ∈ X:

εx : f 7−→ f(x), f ∈ A.

Remark 1. In the case when A is a non-unital algebra any character µ ∈ M(A) maybe extended to its unitalization A+ by setting: µ(0, 1) = 1. Consider the characteron A+ determined by the formula: (a, λ) 7→ λ. Its restriction to A yields the zerofunctional. So the space M(A+) may be identified with M(A) ∪ 0.

1.1. COMMUTATIVE BANACH ALGEBRAS 11

Definition 4. The spectrum sp(a) of an element a of a unital Banach algebra A isthe set of complex numbers λ such that the element a− λ1A is not invertible in A.If the algebra A is non-unital that the spectrum of a consists of complex numbersλ such that the element a− λ1A+ is not invertible in A+.

We formulate two important properties of the spectrum of an element of a unitalBanach algebra A which are proved in the same way as the analogous properties ofthe spectrum of a bounded linear operator in a Hilbert space.

• The spectrum sp(a) of an arbitrary element a ∈ A is a closed set. It is boundedand contained in the disk λ ∈ C : |λ| ≤ ‖a‖, in particular, sp(a) is a compactset.

• The spectrum sp(a) of an arbitrary element a ∈ A is not empty.

For a unital algebra A the value µ(a) of any character µ ∈ M(A) at an arbitrarypoint a ∈ A belongs to the spectrum: µ(a) ∈ sp(a), since otherwise the elementµ(a − µ(a)1A) = 0 would be invertible in C. Since sp(a) ⊂ λ ∈ C : |λ| ≤ ‖a‖, itimplies that |µ(a)| ≤ ‖a‖, i.e. ‖µ‖ ≤ 1 where

‖µ‖ = supa∈A

|µ(a)‖‖a‖ . (1.1)

It follows that ‖µ‖ = 1 since µ(1A) = 1.

Definition 5. An element a ∈ A of a Banach *-algebra A is called self-adjoint ifa∗ = a.

Note that an arbitrary element a of a Banach algebra A may be written as thesum a = α + iβ of elements of A given by

α =a + a∗

2, β =

a− a∗

2i(1.2)

which are self-adjoint. It follows that a∗ = α− iβ.

Lemma 1. Let a be a self-adjoint element of a C∗-algebra A. Then the value µ(a)of any character µ ∈ M(A) at a is a real number.

Proof. Consider the exponential u := exp(ia) of the element ia. It is given by theseries

u =∞∑

k=0

(ia)k

k!

which converges and satisfies the estimate: ‖u‖ ≤ e‖a‖. Moreover, u∗ = exp(−ia)so uu∗ = u∗u = 1A. It implies that the element u is convertible and u−1 = u∗. Therelation ‖u‖2 = ‖u∗u‖ = 1 implies that ‖u‖ = 1 and, analogously, ‖u−1‖ = 1. Since‖µ‖ ≤ 1 the two last equalities imply that |µ(u)| ≤ 1 and |µ(u)|−1 = |µ(u−1)| ≤ 1whence |µ(u)| = 1. But

µ(u) =∞∑

k=0

µ(ia)k

k!= eiµ(a).

Since |µ(u)| = 1, it is possible only if µ(a) ∈ R.


Introduce one more important notion related to the spectrum.

Definition 6. The spectral radius of an element a ∈ A is the number r(a) equal to

r(a) = max|λ| : λ ∈ sp(a).

In other words, r(a) coincides with the radius of the smallest closed disk with thecenter at zero containing sp(a). In particular, r(a) ≤ ‖a‖.

The spectral radius may be computed by the Cauchy–Hadamard formula

r(a) = limn→∞

‖an‖1/n.

It implies that for a self-adjoint element a = a∗ ∈ A its spectral radius coincideswith the norm: r(a) = ‖a‖ (why?).

1.1.4 Spectrum of a commutative Banach algebra. Gelfandtransform

Denote by A′ the Banach space of continuous linear functionals ϕ : A → C on thealgebra A with the norm (1.1). Provide it with the weak-∗ topology, i.e. topologyof the uniform convergence on elements from A. In the case when A = C(X) is thealgebra of continuous functions on a compact X the space A′ consists of complexmeasures on X with the usual topology. By the Banach–Alaoglu theorem the unitball A′

1 in the space A′ is compact in the weak-∗ topology. Since the spectrum M(A)of the algebra A is contained in A′

1 we can provide it with the induced topology.

Lemma 2. The spectrum of a commutative Banach algebra is a locally compacttopological space.

Proof. We show first that the space M(A)∪0 is closed in the weak-∗ topology, i.e.that the weak-∗ limit of elements from M(A) ∪ 0 belongs again to M(A) ∪ 0.Since the weak-∗ limit of continuous linear functionals from A′ belongs to A′ wehave to check only that the multiplicativity property is preserved under this limit.Indeed, for fixed elements a, b ∈ A the map

M(A) ∪ 0 3 µ 7−→ µ(a)µ(b)− µ(ab)

is weak-∗ continuous and vanishes on M(A) ∪ 0, hence also in the closure ofM(A) ∪ 0 in A′. This proves that the space M(A) ∪ 0 is closed. Hence, it iscompact being a closed subset of the compact set A′

1. It follows that the space M(A)is locally compact.

If the algebra A is unital then already the space M(A) is compact since the point0 (corresponding to the evaluation at infinity) is isolated from M(A) (note thatthe continuous functional M(A) 3 µ 7→ µ(1A) separates it from M(A)).

Definition 7. Let A be a commutative Banach algebra. Its Gelfand transform isthe map

G : A −→ C0(M(A))


given by the formula

A 3 a 7−→ a : M(A) → C where a(µ) := µ(a).

We denote by C0(M(A)) as above the space of continuous functions on M(A) van-ishing at infinity.

The Gelfand transform A → C0(M(A)) is evidently continuous. Let us considerits action on the involution in A. If A is a C∗-algebra and µ ∈ M(A) then, inaccordance with decomposition (1.2),

µ(a∗) = µ(α− iβ) = µ(α)− iµ(β) = µ(a).

The last equality holds since by Lemma 1 the values of the character µ on self-adjoint elements are real. So, a∗(µ) = a(µ), i.e. G(a∗) = G(a). In other words,Gelfand transform commutes with involutions in A and C0(M(A)), i.e. it is a *-homomorphism.

For the characters the following analogue of the Hahn–Banach theorem holds.

Lemma 3. Let A be a unital commutative C∗-algebra and a ∈ A. If λ ∈ sp(a) thenthere exists a character µ ∈ M(A) such that µ(a) = λ.

Proof. We prove that the kernel of any character of A is a maximal ideal and,conversely, any maximal ideal in A coincides with the kernel of some character.

Assume that this fact is already established and deduce the assertion of ourlemma from it. Consider the ideal of A of the form (a − λ1A)A. Then this idealis contained in some maximal ideal (this assertion follows from the Zorn lemma;why?) which, according to the assumed fact, coincides with the kernel Ker µ ofsome character µ. In other words, µ(a− λ1A) = 0, i.e. µ(a) = λ.

Return to the proof of the fact formulated above. If µ is a character of the algebraA then its kernel Ker µ is an ideal in A. Suppose that this ideal is not maximal,i.e. there exists a proper ideal I of the algebra A which contains Kerµ but does notcoincide with it. Take an element a0 ∈ I which does not belong to Ker µ. Then thecharacter µ does not vanish on this element. But the element a0 − µ(a0)1A belongsto the kernel of µ, hence to the ideal I. So the element a0−(a0−µ(a0)1A) = µ(a0)1A

also belongs to the ideal I which implies that I contains 1A, i.e. I = A, contradictingour assumption that it is proper.

Conversely, let I be a proper maximal ideal in A. Note first of all that I isclosed. Indeed, the closure of I is an ideal in A which, due to the maximality of Ishould coincide either with the ideal I itself, or with the whole algebra A. But aproper ideal cannot be dense in A since the set of invertible elements of the algebraA is, evidently, open and the ideal I does not contain any of them (otherwise, itwould coincide with A). Hence, the closure of I should coincide with I, i.e. ideal Iis closed.

Since I is closed the quotient A/I modulo this ideal is a commutative Banachalgebra with unit. Moreover, it is simple, i.e. does not contain any proper ideals.Indeed, if such an ideal would exist then its preimage under the natural projectionA → A/I would be a proper ideal in A, containing I, which is impossible due tothe maximality of I. It means that the quotient algebra A/I is a field, i.e. any


nonzero element of this algebra is invertible (since the principal ideal generated bysuch element should coincide with the whole algebra).

This field coincides necessarily with C. Indeed, take an arbitrary element x ∈A/I. Since its spectrum sp(x) is not empty there exists a λ ∈ sp(x) such thatthe element x − λ1A is not invertible, i.e. it should be equal to zero according tothe above argument, whence x = λ1A. Thus, we have established an isomorphismbetween A/I and C.

Having proved that A/I = C we deduce that the natural projection A → A/I isa character of the algebra A with the kernel equal to I.

1.1.5 Gelfand–Naimark theorem

Before we formulate this theorem recall the Stone–Weierstrass theorem which willbe used in the proof of Gelfand–Naimark theorem.

Suppose that X is a locally compact topological space and B is a subalgebra inC0(X). We say that B does not vanish at x ∈ X if there exists a function from Bwhich does not vanish at this point.

Theorem 1 (Stone–Weierstrass). Let X be a locally compact topological space andB is a closed subalgebra in C0(X) which separates the points of X. Suppose thatB does not vanish at any point of X and is closed under the complex conjugation.Then B = C0(X).

Theorem 2 (Gelfand–Naimark). Let A be a commutative C∗-algebra. Then theGelfand transform establishes an isometric *-isomorphism G : A → C0(M(A)).

Proof. We have already proved that Gelfand transform is a *-homomorphism. Itsisometricity follows from the following chain of equalities:

‖a‖2 = ‖a∗a‖ = ‖a∗a‖ = r(a∗a) = ‖a∗a‖ = ‖a‖2

where the equality ‖a∗a‖ = r(a∗a) (r is the spectral radius) follows from Lemma 3(why?).

In particular, Gelfand transform is injective. Hence, the image G(A) of the al-gebra A in C0(M(A)) is a subalgebra which is complete (since the algebra A iscomplete and the map G is isometric) hence, also closed. The evaluation map sep-arates multiplicative functionals on A and the algebra G(A) does not vanish at anypoint of M(A). Moreover, it is closed with respect to complex conjugation. Hence,by Stone–Weierstrass theorem the subalgebra G(A) coincides with C0(M(A)).

1.1.6 Positive elements and states

Definition 8. An element a ∈ A of a C∗-algebra A is called positive if it is self-adjoint and its spectrum sp(a) is non-negative, i.e. belongs to [0,∞).

It is easy to show that an element a ∈ A is positive if and only if it can berepresented in the form a = b∗b for some element b ∈ A. (To prove this assertionuse the fact that any positive element has the square root, i.e. an element

√a ∈ A

such that (√

a)2 = a.)Using Definition 8, we can introduce a partial ordering relation on self-adjoint

elements of the C∗-algebra A.


Definition 9. Let a and b be self-adjoint elements of a C∗-algebra A. We say thata 6 b if the element b− a is positive.

Any positive element a ∈ A satisfy the inequality

0 ≤ a ≤ ‖a‖ 1A

since sp(‖a‖ 1A − a) ⊂ [0, +∞).

Definition 10. A linear functional ϕ : A → C is called positive if ϕ(a) > 0 for anypositive element a ∈ A or, equivalently, ϕ(b∗b) ≥ 0 for any element b ∈ A.

If the algebra A is unital then any positive functional ϕ on this algebra is boundedand ‖ϕ‖ = ϕ(1A). Conversely, any linear bounded functional on such algebra withthe property: ‖ϕ‖ = ϕ(1A), is necessarily positive.

Definition 11. A positive linear functional ϕ on a C∗-algebra A is called the stateif its norm is equal to 1: ‖ϕ‖ = 1. The state is called the trace if ϕ(ab) = ϕ(ba) forany a, b ∈ A.

The set of states is convex, i.e. for any states ϕ, ψ on the algebra A their convexlinear combination tϕ + (1− t)ψ with t ∈ [0, 1] is also a state.

Definition 12. A state ϕ is called pure if it cannot be represented as a convexlinear combination of other states, i.e. if ϕ cannot be represented as the sum ϕ =tψ1 + (1− t)ψ2 where ψ1, ψ2 are the states on A different from ϕ and t ∈ (0, 1).

1.1.7 Gelfand–Naimark–Segal construction(GNS-construction)

The GNS-construction allows to construct from any state ϕ on a C∗-algebra A a *-representation πϕ of the algebra A in some Hilbert space Hϕ. This representation isgiven by a homomorphism of A into the algebra L(Hϕ) of bounded linear operatorsin Hϕ.

We turn to the detailed description of the GNS-construction.Define a positive semi-definite (perhaps, degenerate) sesquilinear form on A by

the formula(a, b)ϕ = ϕ(a∗b).

It is linear in the second argument and anti-linear in the first one. Moreover, thisform satisfies the Cauchy inequality

|(a, b)ϕ|2 6 (a, a)ϕ(b, b)ϕ.

The constructed form degenerates on elements of the left ideal

Nϕ := a ∈ A : (a, a)ϕ = 0 = b ∈ A : ϕ(a∗b) = 0 for all a ∈ A.Consider the quotient A/Nϕ consisting of the elements given by the classes [a] :=a + Nϕ. Define an inner product on this quotient by:

([a], [b])ϕ := (a, b)ϕ.


This inner product is correctly defined on A/Nϕ and non-degenerate (check it!).Introduce now the Hilbert space Hϕ given by the completion of the space A/Nϕ

with respect to the norm associated with this inner product.Define the representation πϕ of the algebra A in the Hilbert space Hϕ by the

formula:πϕ(a) : [b] 7−→ [ab].

The operator πϕ(a) is bounded on A/Nϕ and ‖πϕ(a)‖ 6 ‖a‖. Moreover, the rep-resentation πϕ is a *-homomorphism, i.e. πϕ(a∗) = (πϕ(a))† where (πϕ(a))† is theHermitian conjugate of πϕ(a).

Since the operator πϕ(a) is bounded on the subspace A/Nϕ, which is dense inHϕ , it can be extended to a bounded linear operator denoted by the same letterπϕ(a) which is defined on the whole Hilbert space Hϕ and satisfies the estimate:‖πϕ(a)‖ ≤ ‖a‖.

Note that if the algebra A is unital then the vector ξ := [1A] is cyclic for the rep-resentation πϕ, i.e. the set of elements πϕ(a)ξ : a ∈ A is dense in Hϕ. Moreover,for any a ∈ A we have the equality

(ξ, πϕ(a)ξ)ϕ = ϕ(a). (1.3)

It can be shown that the constructed representation πϕ is irreducible if and onlyif the state ϕ is pure.

1.1.8 Embedding of C∗-algebras into the algebra of boundedlinear operators in a Hilbert space

An arbitrary C∗-algebra may be embedded into the algebra L(H) of bounded lin-ear operators in some Hilbert space according to the following Gelfand–Naimarktheorem.

Theorem 3 (Gelfand–Naimark). Any C∗-algebra A is isomorphic to a closed sub-algebra in the algebra L(H) of bounded linear operators acting in some Hilbert spaceH.

Idea of the proof. Using the Hahn–Banach theorem, it may be proved that for anyelement a ∈ A \ 0 there exists a state ϕa such that

ϕa(a∗a) = ‖a‖2.

Then the equality (1.3) implies that ‖πϕ(a)ξ‖ϕ = ‖a‖. It follows that πϕ(a) canvanish only for a = 0.

Consider now the representation π equal to the direct sum of GNS-representationscorresponding to all possible ϕa:

π :=⊕

ϕa:a∈A

πϕa .

This representation acts in the space

H :=⊕

ϕa:a∈A

Hϕa .

Since ‖πϕa‖ = ‖a‖ for any a ∈ A we have the equality: ‖π(a)‖ = ‖a‖, i.e. therepresentation π is isometric which proves the theorem.


1.1.9 Correspondence: compact spaces ↔ unital commuta-tive Banach algebras

A continuous map f : X → Y of compact topological spaces generates the homo-morphism of their algebras of continuous functions Cf : C(Y ) → C(X) given by theformula: ϕ 7→ ϕ f . It is a unital (i.e. preserving the units) *-homomorphism withthe following functorial property: if there is another continuous map g : Y → Z ofcompact topological spaces then C(g f) = Cf Cg.

Hence, the correspondence

F : X 7−→ C(X), f 7−→ Cf

determines a contravariant functor from the category of compact topological spaces(with morphisms given by continuous maps) into the category of unital commutativeC∗-algebras (with morphisms given by unital *-homomorphisms).

The inverse functor Φ is defined in the following way. Recall that the weak∗-topology on M(A) is the weakest topology for which all evaluation maps a : M(A) →C, a ∈ A, are continuous. In particular, the map f : X → M(A) is continuous ifand only if all maps a f : X → C are continuous.

Let ϕ : A → B be a unital *-homomorphism of unital commutative C∗-algebras.Denote by Mϕ : M(B) → M(A) the map given by the formula µ 7→ µ ϕ. It is

continuous since all functions of the form a Mϕ = ϕ(a) are continuous for a ∈ Aand has the functorial property: if ψ : B → C is another unital *-homomorphismof unital commutative C∗-algebras then M(ψ ϕ) = Mϕ Mψ.

The constructed functors establish an equivalence of the above categories.

Corollary 1. Two unital commutative C∗-algebras are isomorphic if and only iftheir spectra are homeomorphic.

Corollary 2. The group of automorphisms AutA of a unital commutative C∗-algebra A is isomorphic to the group of homeomorphisms Homeo(M(A)) of its spec-trum.

The constructed equivalence of categories establishes the following dictionary ofcorrespondence between the topology and algebra:

topology ←→ algebra

homeomorphism ←→ automorphism

compactness ←→ unitality

compactification ←→ adding of the unit

open subset ←→ ideal

closed subset ←→ quotient algebra

metrizability ←→ separability

connectedness ←→ absence of

nontrivial idempotents

We add comments only to the last two correspondences leaving the check of theothers to the reader.


Proposition 1. A compact topological space X is metrizable if and only if thealgebra C(X) is separable.

Proof. If the space X is metrizable then there exists a countable family of open ballsUn generating its topology. Consider the functions

fn(x) := dist(x,X \ Un).

They belong to the algebra C(X) and separate the points of X. Hence the sub-algebra, generated by the functions fn and constants, is dense in C(X) by theStone–Weierstrass theorem. Hence, the algebra C(X) is separable.

Conversely, if C(X) is separable then it contains a countable dense family ofcontinuous functions fn∞n=0. We may suppose that all of them have their normsless than 1 (if one the functions fn does not satisfy this condition we can replaceit by the function fn/(1 + |fn|)). The sequence fn should separate the pointsof X since otherwise it could not approximate all continuous functions separatingdifferent points of X. Introduce the function

dist(x, y) :=∞∑

n=0

|fn(x)− fn(y)|2n

.

It defines a metric on X with the balls being open subsets of X. Hence the identitymap of the original space X to the same space provided with the topology, inducedby the metric dist, is a homeomorphism which proves the metrizability of X.

Proposition 2. The connectedness of a compact topological space X is equivalentto the absence of nontrivial idempotents in the algebra C(X).

Proof. Recall that an idempotent in an algebra A is the element e such that e2 = e.If the algebra A = C(X) has a nontrivial idempotent e, i.e. the idempotent, differingfrom 1A and zero, then such idempotent is given by a function e ∈ C(X) which cantake only two values, namely 0 and 1. Its nontriviality means that e cannot be aconstant. So X decomposes into the disjoint union of two sets x : e(x) = 1 andx : e(x) = 0. Hence, X is not connected. To prove the converse statement it issufficient to reverse the given argument.

1.2 Vector bundles

1.2.1 Complex vector bundles

Definition 13. The complex vector bundle of rank r over a Hausdorff topologicalspace M is a continuous surjective map π : E → M of topological spaces with fiberEx = π−1(x) at any point x ∈ M given by a complex vector space of dimension rsuch that the following local triviality condition holds: for any x ∈ M there existsits open neighborhood U and a fiberwise homeomorphism

ϕU : U × Cr −→ π−1(U)

which, being restricted to the fiber, is a vector space isomorphism. If it is possibleto take for U in this definition the whole space M then such bundle is called trivial .

1.2. VECTOR BUNDLES 19

In analogous way one can define the C∞-smooth vector bundles: these are theC∞-smooth surjective maps of smooth manifolds π : E → M having the localtriviality property.

Complex vector bundles may be defined with the help of transition functions. LetE be a complex vector bundle of rank r and Uj is its trivializing covering , in otherwords, Uj is an open covering of the space M together with homeomorphisms

ϕj ≡ ϕUj: Uj × Cr −→ π−1(Uj)

having the properties listed in Definition 13. Then the functions ϕij := ϕ−1j ϕi,

defined on the intersections Uij := Ui ∩Uj, are called the transition functions of thebundle E. They take values in the group GL(r,C) of nondegenerate linear mapsCr → Cr and satisfy the following cocycle condition:

ϕii = id, ϕij ϕjk = ϕik on Uijk := Ui ∩ Uj ∩ Uk.

Denote by Γ(U,E) the set of sections of the bundle E over a subset U ⊂ M , i.e.continuous maps s : U → E such that π s = id.

Proposition 3. Let E be a complex vector bundle of rank r over a compact topo-logical manifold M . Then there exists another complex vector bundle E ′ → M suchthat for some n we shall have the following isomorphism of vector bundles

E ⊕ E ′ ∼= M × Cn.

Remark 2. In the real case if E coincides with the tangent bundle of the manifoldM embedded into a vector space, one can take for E ′ the normal bundle of thismanifold.

Proof. Using the compactness of M , we can choose a finite trivializing coveringUjm

j=1 of M . For every set Uj from this covering there exists a collection ofr linearly independent sections sj1, . . . , sjr : Uj → E of the bundle E. Denoteby ψjm

j=1 a continuous decomposition of unity subordinate to the covering Uj,consisting of continuous functions ψj with compact support in Uj, for which

∑ψj ≡

1. Introduce the maps

σjk : M → E, equal to

ψjsjk on Uj

0 outside Uj.

The vectors σj1(x), . . . , σjr(x) generate the fiber Ex for any x ∈ M .Set n := mr and consider the map β : M × Cn → E defined by the formula

β(x, t) =∑

j,k

tjkσjk(x).

It determines a surjective bundle morphism (i.e. a fiber-linear map) which can beincluded into the following exact sequence of morphisms

0 −→ E ′ := Kerβα−→ M × Cn β−→ E −→ 0.


We show that this sequence splits , i.e. there exists a right inverse morphism tomorphism β. This will imply that E ⊕ E ′ ∼= M × Cn.

Indeed, at any point x ∈ M the exact sequence of linear maps of vector space

0 −→ E ′x

αx−→ Cn βx−→ Ex −→ 0

splits since dim Kerβx + dim Im βx = n. In some neighborhood Ux of the point xthe morhism β is given by a matrix function b : Ux → Hom(Cn,Cr) having at thepoint x the rank equal to rk b(x) = r. By restricting, if necessary, the neighborhoodUx to a neighborhood Vx ⊂ Ux we can assert that rk b(y) = r for any y ∈ Vx. Theexact sequence

0 −→ E ′ −→ M × Cn −→ E −→ 0

over this neighborhood Vx still splits. We choose now from the collection Vxx∈M

a finite subcovering Vj of M and denote by γj the morphism which is the rightinverse to β over the neighborhood Vj. For a continuous decomposition of unityχj subordinate to the covering Vj we set

γ :=∑

j

χjγj.

Then γ is the right inverse to β over M .

Remark 3. The proved proposition remains true in the case when M is a paracom-pact manifold.

1.2.2 Functor Γ

The space of sections Γ(M,E) ≡ Γ(E) of a vector bundle π : E → M is a moduleover the commutative Banach algebra C(M) which we consider as a right moduleby setting

(sa)(x) := s(x)a(x) for s ∈ Γ(E), a ∈ C(M).

It is a covariant functor from the category of vector bundles over M into the categoryof right modules over the algebra C(M). Indeed, we can associate with any bundlemorphism τ : E → E ′ a homomorphism of C(M)-modules

Γτ : Γ(E) −→ Γ(E ′)

acting by the formula: (Γτ)s = τ s. This homomorphism is linear, i.e. (Γτ)(sa) =(Γτ)(s)a for a ∈ C(M) since the maps τx : Ex → E ′

x are linear for x ∈ M . Moreover,the functor Γ transforms the operations of duality, direct sum and tensor productfor the bundles into the analogous operations for C(M)-modules.

Apart from that, this functor has the following important properties:

1. Γ is faithful which means that the equality Γf = Γg for two bundle morphismf, g : E → E ′ implies the equality f = g.

2. Γ is full which means that the map

τ 7−→ Γτ : Hom(E, E ′) −→ HomC(M)(Γ(E), Γ(E ′))

is surjective.

3. Γ preserves the short exact sequences and transforms the splitting short exactsequences again into the splitting short exact sequences.


1.2.3 Projective modules

In this section A denotes a unital ring.

Definition 14. A right A-module E over a unital ring A is called free if it has an A-basis, i.e. a set of generators T such that every relation of the form t1a1+. . .+trar = 0with tj ∈ T , aj ∈ A, implies that a1 = . . . = ar = 0. A module E is called finitelygenerated if it has a finite set of generators or a finite A-basis.

Example 3. The standard free A-module of rank r has the form Ar =A⊕ . . .⊕A︸︷︷︸

r

and consists of vector-columns with entries from A. It has the standard

basis consisting of elements ej = t(0, . . . , 0, 1, 0, . . . , 0) (with 1 at jth place). It maybe also realized as a module rA = A⊕ . . .⊕A︸︷︷︸

r

consisting of vector-rows with entries

from A. Any finitely generated free A-module is isomorphic to Ar and rA for somer.

Definition 15. A right A-module P is called projective if it has the followinguniversal property: for any surjective A-linear map of right A-modules f : E → Gand any A-linear map of right A-modules ϕ : P → G there exists an A-linear mapψ : P → E such that the following diagram

E f // G // 0

Pψ

OOÂÂÂ ϕ

??ÄÄÄÄÄÄÄ

is commutative.

Properties of projective modules:

1. Any free right A-module is projective.

2. The direct sum of right A-modules is projective ⇐⇒ every summand in thissum is projective.

3. A right A-module P is projective ⇐⇒ any short exact sequence of A-linearmaps of right A-modules of the form

0 −→ E −→ G −→ P −→ 0 (1.4)

splits.

4. A right A-module P is projective ⇐⇒ P is a direct summand in a free A-module.

5. A right A-module P is projective ⇐⇒ it has the form P = eF where F is aright free A-module and e is an idempotent in the algebra EndAF of A-linearendomorphisms of F .


6. If a projective A-module is finitely generated, , i.e. it has a finite system ofgenerators, then the free module in properties 4 and 5 can be also chosenfinitely generated.

Proofs of properties of projective modules

Proof of property (1). Let F be a free right A-module with a system of generatorstαα∈I . Let ϕ : F → G be a A-linear map of right A-modules and f : E → G is asurjective A-linear map. Since this map is surjective we can find for any generatortα of the module F an element sα ∈ E such that f(sα) = ϕ(tα). Define the mapψ : F → E by setting it equal to ψ(tα) = sα on generators and extending further onby A-linearity. The constructed diagram

E f // G // 0

Fψ

OOÂÂÂ ϕ

??ÄÄÄÄÄÄÄÄ

is commutative since on generators we have: f(ψ(tα)) = f(sα) = ϕ(tα) and byA-linearity the equality f(ψ(t)) = ϕ(t) holds for all t ∈ F .

Proof of property (2). Suppose that the module P =⊕

α∈I Pα, which is the directsum of right A-modules, is projective. We show that in this case any A-module Pα

is also projective. Suppose that we are given an A-linear map of right A-modulesϕα : Pα → G and surjective A-linear map of right A-modules f : E → G. The mapϕα may be extended to an A-linear map ϕ : P → G by setting it equal to zero onall summands Pβ with β 6= α. Since the module P is projective it should exist anA-linear map ψ : P → E such that f ψ = ϕ. By restricting this map to Pα weobtain an A-linear map ψα : Pα → E such that f ψα = ϕα. This proves that themodule Pα is projective.

Conversely, suppose that all A-modules Pα are projective. Let ϕ : P → G bean A-linear map of right A-modules and f : E → G is a surjective A-linear map ofright A-modules. For every α ∈ I the restriction ϕα := ϕ|Pα is an A-linear mapPα → G so due to the projectivity of the A-module Pα there exists an A-linear mapψα : Pα → E such that f ψα = ϕα. Then the direct sum ψ :=

⊕α∈I ψα of the

maps ψα will give an A-linear map ψ : P → E with the property: f ψ = ϕ whichimplies that the module P is projective.

Proof of properties (3) and (4). We prove first that if a right A-module P is pro-jective then every short exact sequence of the form

0 −→ E f−→ G g−→ P −→ 0,

where G is a right A-module, splits. Indeed, due to the projectivity of the moduleP there exists an A-linear map ψ : P → G for which the diagram

G g // P // 0

Pψ

OOÂÂÂ idP

??ÄÄÄÄÄÄÄÄ


is commutative, i.e. g ψ = idP . Hence, the map ψ is the right inverse of g, i.e. theconsidered exact sequence splits.

We prove now that for any projective right A-module P there exists an exactsequence of the form (1.4) in which G ≡ F is a free right A-module. Choose inP some system of generators tαα∈I and consider the free right A-module F withan A-basis sαα∈I parameterized by the same set of indices I. In other words,the module F consists of all finite linear combinations

∑nk=1 sαk

ak where ak ∈ A,k = 1, . . . , n. Define the map g : F → P on generators by the formula: g(sα) = tαand extend it to the whole module F byA-linearity. Setting E := Ker g and denotingby f the embedding E → F , we shall obtain the exact short sequence

0 −→ E f−→ F g−→ P −→ 0. (1.5)

If for a right A-module P any exact sequence of the form (1.4) splits then thisproperty holds also for the above sequence (1.5) which implies that the module Pis a direct summand in the free A-module F . By property (1) the free module F isprojective and by property (2) the A-module P , which is a direct summand in themodule F , should be also projective.

To finish the proof of the property (4) it remains to note that if an A-moduleP is a direct summand in a free A-module F , which is projective by property (1),then it is projective by property (2).

Proof of property (5). If a right A-module P is projective then by property (4) itis a direct summand in a free right A-module F and we can take for the desiredidempotent the projector p : F → P .

Conversely, if e ∈ EndAF is an idempotent, i.e. e2 = e, acting in a free rightA-module F , and the right A-module P has the form P = eF , then

F = eF ⊕ (idF − e)Fwhere the map idF − e is also an idempotent. So the module P = eF is a directsummand in the free A-module F , hence a projective module.

Proof of property (6). If P is a finitely generated right A-module with a system ofgenerators tαr

α=1 then in the proof of property (4) the A-basis sαrα=1 of the

free A-module F will be also finite. Hence, P is a direct summand in the finite-dimensional free A-module F so coincides with the image of an idempotent e, actingin F .

1.2.4 Serre–Swan theorem

Proposition 4. Let M be a compact manifold and E → M is a complex vectorbundle of rank r over it. Then the C(M)-module Γ(E) is finitely generated andprojective.

Proof. According to Proposition 3 there exists a vector bundle E ′ → M such thatE ⊕ E ′ ∼= M × Cn. Since

Γ(E)⊕ Γ(E ′) = Γ(M × Cn) = C(M)n

the module Γ(E) is a direct summand in the free module C(M)n. It implies alsothat Γ(E) is finitely generated.


Theorem 4 (Serre–Swan). The functor Γ establishes an equivalence between thecategory of vector bundles over the compact manifold M and the category of finitelygenerated projective modules over the algebra C(M).

Proof. Due to Proposition 4 it remans to prove that any finitely generated projectiveC(M)-module E coincides with Γ(M,E) for some vector bundle π : E → M . Byproperty (5) of projective modules (cf. Sec. 1.2.3) the module E has the formE = eC(M)n for some idempotent e ∈ EndC(M)(C(M)n) ∼= Matn(C(M)). Theexact sequence

0 −→ Ker e −→ C(M)n −→ E −→ 0

splits by property (3) of projective modules (cf. Sec. 1.2.3). Since the functor Γ isfull it follows that the endomorphism e ∈ Matn(C(M)) is generated by some bundlemorphism τ : M × Cn → M × Cn so that eC(M)n coincides with Γ(M,E) whereE := Im τ .

It remains to show that E is a subbundle of M × Cn. For that it is sufficientto prove that rk τx is a locally constant function of x ∈ X. Note that the rank ofidempotent τx, as any other idempotent in Matn(C(M)), is upper semicontinuous.On the other hand, id− τ is also an idempotent in Matn(C(M)) and rk(idx− τx) =n − rk τx for any x ∈ M . It follows that the map x 7→ rk τx is not only upper butalso lower semicontinuous, so it is continuous, hence, locally constant.

Remark 4. The given proof admits the following interpretation. The idempotente ∈ Matn(C(M)) may be considered as a continuous map from M into the spaceof matrix idempotents. Denote by εx the evaluation map at the point x ∈ M . Asit was shown in the proof of the theorem, the map ε(e) : x 7→ εx(e) has constantrank m = rk εx(e) ≤ n, hence it generates a map M → Gm(Cn) into the Grassmannmanifold Gm(Cn). In other words, the space Ex = εx(e)Cn lies in the fibre of thetautological bundle T → Gm(Cn) over the point x. The desired bundle E → M ,corresponding to the projective module E , coincides with the inverse image of thetautological bundle T under the map ε(e):

E = ε(e)∗(T ) −−−→ Tyy

M −−−→ε(e)

Gm(Cn)

Remark 5. The C∞-smooth version of Serre–Swan theorem is also true. It assertsthat the category of C∞-smooth vector bundles over a C∞-smooth compact manifoldM is equivalent to the category of finitely generated projective modules over thealgebra C∞(M). The equivalence of these categories is established by the functorΓ∞ associating with a smooth bundle E → M the module Γ∞(M,E) of its C∞-smooth sections.

1.3 Functional analysis over C∗-algebras

1.3.1 C∗-modules

Definition 16. The (right) C∗-premodule over a C∗-algebra A is a complex vec-tor space E which is simultaneously a right A-module provided with a sesquilinear

1.3. FUNCTIONAL ANALYSIS OVER C∗-ALGEBRAS 25

pairing (· , ·) : E × E → A having the following properties:

(1) (r, s + t) = (r, s) + (r, t);

(2) (r, sa) = (r, s) a;

(3) (r, s) = (s, r)∗;

(4) (s, s) > 0 for s 6= 0,

satisfied for any r, s, t ∈ E , a ∈ A.

It follows that the pairing (· , ·) is A-linear in second variable, antilinear in thefirst variable and positively defined. In particular, (ra, s) = a∗(r, s) for a ∈ A,r, s ∈ E .

Using the pairing (· , ·) we can introduce on the C∗-premodule E the norm bysetting

‖s‖E :=√‖(s, s)‖

for s ∈ E . Here, ‖ · ‖ denotes the norm in the C∗-algebra A. (We shall omit in thesequel the lower index E in the notation of the norm ‖ · ‖E on the C∗-premodule Ewhen it is clear which norm is considered.)

The following lemma is an analogue of the Cauchy inequality for the introducednorm.

Lemma 4. Let E be a C∗-premodule over a C∗-algebra A. Then for any r, s ∈ Ethe following inequality holds

‖(r, s)‖ ≤√‖(r, r)‖

√‖(s, s)‖.

Proof. We shall give the proof of this inequality only in the case of a unital algebraA leaving the general case as an exercise. For an arbitrary element a ∈ A we havethe following relationn

0 ≤ (ra + s, ra + s) = a∗(r, r)a + a∗(r, s) + (s, r)a + (s, s). (1.6)

Since (r, r) and (s, s) are positive elements of A they satisfy the estimates: a∗(r, r)a ≤‖(r, r)‖a∗a and (s, s) ≤ ‖(s, s)‖ · 1A. So the inequality (1.6) implies that

0 ≤ (ra + s, ra + s) = ‖(r, r)‖a∗a + a∗(r, s) + (s, r)a + ‖(s, s)‖ · 1A.

Setting a = − (r,s)(r,r)

in the last relation we get

0 ≤ −(s, r)(s, r)∗

‖(r, r)‖ + ‖(s, s)‖ · 1A.

Hence,0 ≤ (s, r)(s, r)∗ ≤ ‖(r, r)‖ · ‖(s, s)‖ · 1A

which implies the desired inequality.

Using the proved lemma it is easy to check that ‖s‖E =√‖(s, s)‖ is indeed a

norm on E , in particular, it satisfies the triangle inequality.


Definition 17. A C∗-premodule E is called the C∗-module if it is complete withrespect to the norm ‖s‖E .

In particular, the completion of an arbitrary C∗-premodule with respect to thenorm ‖s‖E is a C∗-module.

Examples of C∗-modules

1. A complex Hilbert space is a C∗-module over the algebra C with the pairinggiven by the inner product.

2. Any C∗-algebra A is a C∗-module over itself with the pairing given by theformula: (a, b) := a∗b.

3. The free A-module An, consisting of vector-columns with entries from A, isa C∗-module over A with the pairing of elements a = t(a1, . . . , an) and b =t(b1, . . . , bn) given by the formula: (a, b) := (a∗1, . . . , a

∗n) t(b1, . . . , bn) = a∗1b1 +

. . . + a∗nbn.

4. The free A-module nA, consisting of vector-rows with entries from A, is a C∗-module over the algebra Matn(A) with the pairing of elements a = (a1, . . . , an)and b = (b1, . . . , bn) given by the formula: (a, b) := t(a∗1, . . . , a

∗n)(b1, . . . , bn) =

(a∗i bj)ni,j=1.

An important example of a C∗-module over the C∗-algebra A is given by thetensor product H ⊗ A where H is a Hilbert space. First of all we should preciselydefine the meaning of the tensor product used in this formula. Recall that thealgebraic tensor product H¯A of a Hilbert space H and a C∗-algebra A is the rightA-module, consisting of finite sums of pure tensors of the form

∑nk=1 ξk ⊗ ak with

ξk ∈ H, ak ∈ A, provided with an A-valued pairing given on pure tensors by theformula

(ξ ⊗ a, η ⊗ b) = (ξ, η)a∗b

where a, b ∈ A, ξ, η ∈ H. It can be shown that this pairing has all properties listed inDefinition 16, in particular, it is positively defined. Hence, H¯A is a C∗-premoduleover A. Its completion with respect to the norm, induced by the introduced pairing,is a C∗-module over A which is denoted by H⊗ A and called the tensor product ofa Hilbert space H and the algebra A .

We shall use also the C∗-moduleHA ≡ `2A over A consisting of sequences a = ak

of elements from A for which the series∞∑

k=1

a∗kak

converges in A. It can be provided with the sesquilinear pairing of the form

(a, b) :=∞∑

k=1

a∗kbk.

As a first example of linear operators, acting in HA, consider the operators Pn

of the formPn(a1, . . . , an, an+1, . . .) = (a1, . . . , an, 0, . . .)


which are, evidently, projectors in HA, i.e. P 2n = Pn and P ∗

n = Pn. Also ‖ξ−Pnξ‖ →0 for n →∞ for any ξ ∈ H.

We shall return to the study of linear operators, acting in C∗-modules, in Sec.1.3.3. Consider now in more detail the notion of the tensor product we have metalready in this subsection.

1.3.2 Tensor products

Functor Eϕ

Let A, B be unital rings (i.e. rings with units) and ϕ : A → B is a unital(i.e. sending 1A to 1B) homomorphism. We can associate with such homomorphismϕ the functor Eϕ from the category of right A-modules into the category of rightB-modules defined in the following way.

Let E be a right A-module. Then the ring B may be provided with the structureof left A-module by setting: a · b := ϕ(a)b. This allows us to define the tensorproduct

Eϕ(E) = E ⊗A Bwhich will be called the tensor product of the right A-module E and the ring B withrespect to the homomorphism ϕ.

Namely, E ⊗A B is an Abelian group with elements being finite sums of the form∑nk=1 sk ⊗ bk, sk ∈ E , bk ∈ B, provided with the unique relation

(sa)⊗ b = s⊗ ϕ(a)b for any a ∈ A.

Introduce the right action of B on E ⊗A B by setting: (s⊗ b)c := s⊗ (bc) for alls ∈ E , b, c ∈ B.

If F is another right A-module and τ : E → F is an A-linear map then Eϕ(τ) :=τ ⊗ idB is a B-linear map from E ⊗AB into F ⊗AB. Hence, Eϕ does define a functorfrom the category of right A-modules into the category of right B-modules.

Tensor product of C∗-algebras

Recall first the definition of the tensor product of Hilbert spaces H1 and H2.Consider the algebraic tensor product H1 ¯ H2 with elements given by the finitesums of pure tensors of the form

∑nk=1 ξk ¯ ηk. We can introduce on the space

H1¯H2 the structure of a pre-Hilbert space with the pairing given on pure tensorsby the formula

(ξ1 ¯ η1, ξ2 ¯ η2) := (ξ1, ξ2)(η1, η2)

where ξ1, ξ2 ∈ H1, η1, η2 ∈ H2.This pairing is positively defined and the tensor product H1⊗H2 is the completion

of H1 ¯ H2 with respect to the norm generated by this pairing. Note that theintroduced norm has the cross property which may be written for pure tensorsξ ⊗ η ∈ H1 ⊗H2 in the form

‖ξ ⊗ η‖ = ‖ξ‖ · ‖η‖.The norms, having this property, will be called the cross-norms.

In the case of Banach spaces E1,E2 the algebraic tensor product E1 ¯ E2 mayhave several different norms with the cross property leading to different completions.


We switch now to the case of C∗-algebras A, B and consider the cross-norms onthe algebraic tensor product A ¯ B which have also the C∗-property, i.e. ‖z∗z‖ =‖z‖2 for all z ∈ A¯B.

We call a C∗-algebra A nuclear if the algebraic tensor product A¯ B with anyother C∗-algebra B has a unique C∗-cross-norm. The completion of this productwith respect to this norm is called the tensor product A⊗ B of C∗-algebras A andB.

The examples of nuclear C∗-algebras are given by the finite-dimensional C∗-algebras Matn(C) for which Matn(C)⊗B ∼= Matn(B) and commutative C∗-algebrasC0(Y ) for which C0(Y ) ⊗ B ∼= C0(Y, B). The algebra K(H) of compact operatorsin a Hilbert space H yields one more important example of nuclear C∗-algebras.However, the C∗-algebra L(H) of bounded linear operators in an infinite-dimensionalHilbert space H is already not nuclear.

Tensor product of C∗-modules

Consider first the construction of the tensor product of a right C∗-module E overa C∗-algebra A with another C∗-algebra B extending the construction of the abovefunctor Eϕ to the case of C∗-modules and C∗-algebras.

Suppose that ϕ : A → B is a morphism of C∗-algebras. We want to define thetensor product Eϕ(E) = E ⊗A B.

Consider the algebraic tensor product E ¯B over C and introduce the B-valuedpairing on it given on pure tensors s¯ b by the formula

(s¯ b, t¯ c) := b∗ϕ((s, t))c (1.7)

where s, t ∈ E , b, c ∈ B.However, the introduced pairing has the kernel and in order to introduce a norm

on E ¯ B it is necessary first to factor this kernel out. It can be shown (cf. [3],p.73) that the kernel N := z ∈ E ¯ B : (z, z) = 0 is generated by the elementsof the form sa ¯ b − s ¯ ϕ(a)b. Restricting to the quotient (E ¯ B)/N , we obtaina positive definite pairing on it which is defined on pure tensors s⊗ t ∈ (E ¯B)/Nby the same formula

(s⊗ b, t⊗ c) = b∗ϕ((s, t))c.

It follows that the quotient (E ¯ B)/N is a C∗-premodule over the algebra B andits completion with respect to the norm, generated by the introduced pairing, is aC∗-module. We denote it by Eϕ(E) = E ⊗A B and call the tensor product of theC∗-module E and the C∗-algebra B.

The tensor product of C∗-modules is defined following the same scheme. Sup-pose that we have two right C∗-modules E and F over the C∗-algebras A and Brespectively. Suppose moreover that we are given a morphism ρ : A → EndBF .

Provide the algebraic tensor product E ¯ F , being a right B-module, with thepairing given on pure tensors by the formula

(s1 ¯ t1, s2 ¯ t2) = (t1, ρ ((s1, s2)) t2) = (ρ ((s2, s1)) t1, t2) (1.8)

where s1, s2 ∈ E , t1, t2 ∈ F . Consider again the B-submodule in E ¯ F consistingof the elements z ∈ E ¯ F such that (z, z) = 0. This submodule is generated, asbefore, by the elements of the form sa¯ − s¯ ρ(a)t.


The pairing (1.8) defines a positive definite inner product on the quotient (E ¯F)/N . The completion of (E ¯ F)/N with respect to the norm, generated by thisinner product, is a C∗-module which is denoted by E ⊗ρ F = E ⊗A F and called thetensor product of the C∗-module E and C∗-module F .

1.3.3 Operators of A-finite rank and A-compact operators

We start the study of linear operators acting in C∗-modules from the simplest oper-ators of rank 1. These operators, acting from a C∗-module E to a C∗-module F , arecalled the ketbra-operators and denoted by |r〉〈s| where s ∈ E , r ∈ F . The operator|r〉〈s| is defined by the formula

|r〉〈s| : t 7−→ r(s, t)

where t ∈ E . It is an A-linear operator since r(s, ta) = r(s, t)a for a ∈ A. Its adjointis again the ketbra-operator given by the formula (|r〉〈s|)∗ = |s〉〈r|.

In the case when E = F the composition of two ketbra-operators

|r〉〈s| |t〉〈u| = |r〉(s, t)〈u| = |r〉〈u(t, s)|

is again a ketbra-operator so that the finite sums of ketbra-operators form an algebraof operators acting in E . This algebra is a two-sided ideal FinAE in the algebraEndAE . Its completion with respect to the operator norm is denoted by KA(E).

More generally, let us denote by FinA(E ,F) the space of ketbra-operators of theform

n∑

k=1

|rk〉〈sk|

called otherwise the operators of A-finite rank . The completion of FinA(E ,F) withrespect to the operator norm is denoted by KA(E ,F) and its elements are called theA-compact operators .

Note that an A-compact operator need not be compact in the usual sense.

Example 4. For any unital C∗-algebra A we have an isomorphism

KA(A) ∼= A.

Indeed, the map T 7→ T (1A) determines a bijection of the algebra of bounded A-linear operators acting in A onto the algebra A. Moreover, any operator of the forma 7→ b∗a, which coincides in fact with the ketbra-operator |1A〉〈b|, is adjointable andhas finite A-rank. Hence KA(A) ∼= A.

In analogous way it can be shown that

KA(An) ∼= Matn(A).

Later we show that for the algebra A = C(M), where M is a compact manifold,we have an isomorphism

KA(Γ(M,E)) ∼= Γ(M, EndE)


for any Hermitian vector bundle E → M .We introduce one more important C∗-algebra associated with the C∗-algebra A.

Namely, consider the tensor product K ⊗ A of the algebra of compact operatorsK = K(H), acting in a Hilbert space H, and the C∗-algebra A. Recall that K ⊗ Ais the completion of the algebraic tensor product K ¯ A with respect to a uniqueC∗-norm having the cross-norm property :

‖ξ ⊗ η‖ = ‖ξ‖ · ‖η‖.

Definition 18. The tensor product AS := K ⊗ A is called the stabilization of theC∗-algebra A. A C∗-algebra A is called stable if AS

∼= A. Two C∗-algebras A andB are stably equivalent if AS

∼= BS.

1.3.4 Projectors in C∗-modules

We study now the projectors in C∗-modules. Let F be a closed submodule in aC∗-module E . It is well known that not any such submodule admits the orthogonalcomplement. Suppose however that F has the orthogonal complement denoted byF⊥ so that F ⊕ F⊥ ∼= E . Then any element s ∈ E can be written uniquely in theform

s = t + u

where t ∈ F , u ∈ F⊥ so that the map s 7→ t determines a projector p ∈ EndAE withthe image, equal to F .

Conversely, if p ∈ EndAE is a projector then the orthogonal complement to Im pexists and coincides with

(Im p)⊥ = Im(1E − p) = Ker p.

Thus we have established a bijective correspondence between complementableC∗-submodules in E and images of projectors from EndAE .

Definition 19. Let A be a unital C∗-algebra. Then the A-compact projectors inHA form a subset in the C∗-algebra AS denoted by P(AS). It is a closed subset inthe unit ball of AS.

As examples of projectors in P(AS) one can consider the operators of the form

Pn =n∑

j=1

|ej〉〈ej|

where ej = (0, . . . , 0, 1, 0, . . .) (with 1 at the jth place).In conclusion we give a description of A-compact operators in finitely generated

projective modules over the algebra A. Consider the modules of the form pAn wherep is a projector in Matn(A).

Proposition 5. Let p be a projector in Matn(A) generating the C∗-module pAn overthe algebra A. Then

KA(pAn) ∼= pMatn(A)p.


Proof. Consider the map

FinA(pAn) −→ p Matn(A)p

which sends the ketbra-operator of the form |pa〉〈pb| to the matrix with (i, j)-entriesequal to ∑

k,l

pikakb∗l plj.

This map is an isometric ∗-isomorphism which extends to an isomorphism ofKA(pAn)onto p Matn(A)p.

1.3.5 Unitary operators. Adjoint operators

We continue to study linear operators in C∗-modules over an algebra A. Recall thedefinition of the adjoint operator.

Definition 20. Let T : E → F be a bounded A-linear operator acting from aC∗-module E over the C∗-algebra A to a C∗-module F over the same algebra. Wesay that T is adjointable or admits an adjoint operator if there exists an A-linearoperator T † : F → E , called the adjoint operator of T , such that

(r, Ts) = (T †r, s)

for all s ∈ E , r ∈ F .

As in the case of bounded linear operators, acting in a Hilbert space, the adjointoperator is uniquely defined and T †† = T . However, in contrast with the boundedoperators in a Hilbert case, bounded A-linear operators, acting in C∗-modules, doesnot always admit the adjoint operators. The reason is that not every closed sub-module F in a C∗-module E over a C∗-algebra A admits the orthogonal complementG such that F ⊕ G = E . (The reader may try to find the corresponding counterex-ample.)

So we should assume as an extra condition the existence of the adjoint operator.Namely, we shall denote by HomA(E ,F) the vector space of A-linear operatorsT : E → F which admit the adjoint operator. For F = E we denote by EndA(E) thealgebra of A-linear adjointable endomorphisms of E .

Introduce one more class of operators which we shall constantly meet below.

Definition 21. A unitary operator U : E → F from a C∗-module E over a C∗-algebra A to a C∗-module F over the same algebra is a map U ∈ HomA(E ,F) suchthat

U †U = 1E , UU † = 1F .

If such an operator exists then the modules E and F are called unitary equivalent .

One of the most important properties of C∗-algebras is that the operator algebrasover them are again C∗-algebras.

Proposition 6. Let E be a (right) C∗-module over a C∗-algebra A. Then the algebraEndAE of bounded linear adjointable operators in E is also a C∗-algebra.


Proof. The norm in the algebra EndAE is defined by

‖T‖ = sup‖s‖≤1

‖Ts‖

where s ∈ E . By the Cauchy inequality

‖Ts‖2 = ‖(s, T †Ts)‖ ≤ ‖s‖ · ‖T †Ts‖ ≤ ‖T †T‖ · ‖s‖2

whence

‖T‖2 ≤ ‖T †T‖ ≤ ‖T †‖ · ‖T‖,i.e. ‖T‖ ≤ ‖T †‖, so ‖T‖ = ‖T †‖ since T †† = T . Hence,

‖T‖2 ≤ ‖T †T‖ ≤ ‖T‖2 =⇒ ‖T †T‖ = ‖T‖2,

i.e. ‖ · ‖ is a C∗-norm.To proof the completeness of the space EndAE we note first of all that a Cauchy

sequence of adjointable operators Tn should converge to some bounded linearoperator T . By the above argument the sequence of adjoint operators T †

n is alsoa Cauchy sequence and so converges to some bounded linear operator S. Since

(r, Ts) = limn

(r, Tns) = limn

(T †nr, s) = (Sr, s)

for all r, s ∈ E the operator T admits an adjoint operator which coincides with S.This proves the completeness of the space EndAE .

Corollary 3. The algebra KA(E) which consists of A-compact operators, acting ina C∗-module E, is a C∗-algebra and two-sided ideal in the algebra EndAE.

In particular, the algebra K ≡ K(H) which consists of compact operators, actingin a Hilbert space H, is a two-sided ideal in the algebra L(H) of all bounded linearoperators in H.

1.3.6 Projective C∗-modules

In this section we shall prove an analogue of the Serre–Swan theorem for arbitraryunital C∗-algebras, more precisely, we establish a bijective correspondence betweenA-compact projectors and finitely generated projective C∗-modules over A.

Theorem 5. Let A be a unital C∗-algebra. Then the right C∗-modules of the formpHA with p ∈ P(AS) are finitely generated projective modules over A. In otherwords, each of them is isomorphic to the direct summand in a free module An forsome n.

Idea of the proof: consists in associating with the module pHA an isomorphic mod-ule embedded into the free module PnHA

∼= An where Pn is the standard projectorin HA introduced in the beginning of Sec.1.3.2. In order to compare the originalprojector p with Pn we should first ”rotate” it with the help of a unitary elementun ∈ EndAHA to achieve the inequality unpu∗n ≤ Pn.


Turning to the construction of such operator note that for a given ε > 0 thereexists n for which

‖p− PnpPn‖ < ε/3.

The operator an := PnpPn is positive since

an ≥ a2n = a∗nan ≥ 0.

Indeed, the inequality an ≥ a2n may be written as the relation PnpPn ≥ PnpPnpPn

which follows from the inequality p ≥ pPnp implied by I ≥ Pn.Moreover, ‖an − a2

n‖ < ε since

‖an − a2n‖ ≤ ‖an − p‖+ ‖p(p− an)‖+ ‖(p− an)an‖ ≤

< ε/3 + ‖p‖ε/3 + ‖an‖ε/3 ≤ ε

because ‖p‖ ≤ 1 and ‖an‖ ≤ 1.It is easy to see that the spectrum of an is contained in the interval [0, 1] without

the point 1/2. One can also show that this spectrum is contained in the union ofintervals [0, 2ε] ∪ [1− 2ε, 1] (assuming that ε < 1/4).

We apply now the spectral theorem for selfadjoint elements of a Banach algebra(cf., e.g. [9]). Denote by pn the spectral projector associated with the interval[1 − 2ε, 1]. This projector is equal to f(an) for any continuous function f on theinterval [0, 1] such that 0 ≤ f ≤ 1 and

f(t) =

0 for 0 ≤ t ≤ 2ε

1 for 1− 2ε ≤ t ≤ 1.

We have pn ∈ P(AS) with ‖pn − an‖ < 2ε so that ‖pn − p‖ < 3ε < 3/4.We use now the following lemma.

Lemma 5. If p, q are two projectors in a unital C∗-algebra B satisfying the relation‖p− q‖ < 1 then there exists a unitary element u ∈ B such that q = upu†.

Proof of the lemma. Consider the elements 2p − 1 and 2q − 1 from the algebra B.They are unitary selfadjoint elements from B with spectra contained in the set−1, 1. Define

2r := (2p− 1)(2q − 1) + 1 =⇒ r = 2qp− p− q + 1.

Then qr = qp = rp and

r†rp = r†qr = (qr)†r = (rp)†r = pr†r,

i.e. p commutes with |r|2, hence with |r|. Moreover, the element r is invertible inB since

‖r − 1‖ = ‖2qp− q − p‖ = ‖(q − p)(2p− 1)‖ ≤ ‖q − p‖ < 1.

Set u = r|r|−1. It is a unitary element in B satisfying the condition

upu† = rp|r|−2r† = q.


By the above lemma pn = unpu†n for some unitary element un from the unital

C∗-algebra EndAHA. Moreover, pn ≤ Pn. Indeed, the evident relation

Pnan = anPn = an

implies (with the help of approximation of f by polynomials) that

Pnpn = pnPn = pn.

It follows that Pn − pn is a projector in P(AS) since

(Pn − pn)2 = Pn − pnPn − Pnpn + pn = Pn − pn.

The C∗-module pnHA is finitely generated and projective since it is a directsummand in the free module PnHA

∼= An:

pnHA ⊕ (Pn − pn)HA = PnHA.

The map s 7→ uns establishes a unitary equivalence of A-modules pHA and pnHA.So the C∗-module pHA is also isomorphic to a direct summand in An, i.e. it isfinitely generated and projective.

The converse result is also true.

Theorem 6. Let E be a (right) finitely generated projective module over a unitalC∗-algebra A. Then it can be provided with the structure of a C∗-module over A insuch a way that it will be isomorphic to pHA for some p ∈ P(AS).

Proof. The module E has the form E = eAn where e is an idempotent in Matn(A).Since the image of an idempotent is closed the structure of a C∗-module on E maybe defined by restricting the standard structure of C∗-module from An to E .

Note that the operator e admits an adjoint operator. Indeed, if ujnj=1 is the

standard basis in An so that

1An =n∑

j=1

|uj〉〈uj|

then the idempotent e may be rewritten in its terms in the form

e =n∑

j=1

|euj〉〈uj|,

whence

e† =n∑

j=1

|uj〉〈euj|.

As we have pointed out before,

(Im e)⊥ = Ker e†

since e†s = 0 ⇐⇒ (er, s) = (r, e†s) = 0 for all r ∈ An. It follows that (Ker e†)⊥ =Im e and, analogously, (Im e†)⊥ = Ker e, (Ker e)⊥ = Im e† since e† is also an idem-potent.

1.4. K-THEORY 35

Now,Ker e = Ker(e†e)

since the equality (e†e)s = 0 implies that (es, es) = (s, (e†e)s) = 0 =⇒ es = 0.Switching to the orthogonal complements, we obtain

Im e† = Im(e†e).

So if the element s ∈ An then e†s = (e†e)t for some t ∈ An and we can represent itas the sum

s = et + (s− et) ∈ Im e⊕Ker e† = Im e⊕ (Im e)⊥.

Thus, we have shown that An = E ⊕ E⊥, i.e. E admits the orthogonal complement.So there exists a projector p ∈ Matn(A) such that E = pAn. Since An ∼= Pn(HA)and p ≤ Pn under such identification we have E = pHA.

Remark 6 (Kaplansky formula). There is an explicit formula expressing the projectorp through the idempotent e. Namely, consider the operator

r = ee† = (1− e†)(1− e) = 1 + (e− e†)(e† − e) = 1− e† − e + ee† + e†e.

It is a positive and invertible element in Matn(A) (since these properties have allelements of the form 1 + a†a). Moreover, r commutes both with e and with e†.Indeed,

re = er = ee†e re† = e†r = e†ee†.

So r−1 also commutes with e and e†. We set now p := ee†r−1. Then p = p† and

p2 = ee†ee†r−2 = ere†r−2 = p,

i.e. p is a projector. We also have, ep = p and pe = e since

per = ee†e = er

so that the image of p coincides with the image of e.

The Theorem 6 implies

Corollary 4. For an arbitrary Hermitian vector bundle E → M over a compactmanifold M the following equality

KA(Γ(M,E)) = EndA(Γ(M, E)) ∼= Γ(M,EndE)

holds.

1.4 K-theory

1.4.1 K0-group

We introduce the following equivalence relation on the set of projectors in a C∗-algebra A.


Definition 22. Two projectors p, q ∈ P(AS) are equivalent if q = upu† = upu−1 forsome unitary element u ∈ EndAHA.

Denote byV top(A) := P(AS)/ ∼

the quotient of the space P(AS) with respect to the introduced equivalence relation.

Proposition 7. The set V top(A) is a unital commutative semigroup.

Proof. Define the direct sum of projectors p, q ∈ P(AS) by the formula

p⊕ q =

(p 00 q

).

Using this, we define the sum in V top(A) as

[p] + [q] = [p⊕ q].

It is correctly defined since

upu−1 ⊕ vqv−1 = (u⊕ v)(p⊕ q)(u−1 ⊕ v−1)

for unitary u, v ∈ EndAHA. Moreover,(

p 00 q

)=

(0 11 0

)(p 00 q

)(0 11 0

)=

(q 00 p

),

i.e. this semigroup is commutative. The role of the unit (or better say, zero) in thissemigroup is played by the zero class [0].

Grothendieck construction. We can associate in a canonical way with anyunital commutative semigroup S a group K called the Grothendieck group of thesemigroup S. It is a commutative group provided with a unital semigroup homo-morphism ϑ : S → K which has the following universal property: if G is any othergroup provided with a unital semigroup homomorphism γ : S → G then there existsa unique group homomorphism κ : K → G such that the following diagram

Kκ //___ G

S

ϑ

OO

γ

>>~~~~~~~~

commutes, i.e. γ = κ ϑ.The group K is uniquely determined up to an isomorphism. It can be constructed

in the following way. Consider on the set S × S the following equivalence relation:

(x, y) ∼ (x′, y′) ⇐⇒ if there exists z ∈ S such that x + y′ + z = x′ + y + z.

Then the group K is defined as K = S × S/ ∼ and the homomorphism ϑ is givenby the formula: ϑ(x) := [x, 0] so that [x, y] = ϑ(x)− ϑ(y) in the group K.

Definition 23. The topological K0-group of a unital C∗-algebra A is the Grothendieckgroup Ktop

0 (A) of the semigroup V top(A).

1.4. K-THEORY 37

We switch now to the construction of the algebraic K0-group. For that we provefirst the following lemma.

Lemma 6. Let e ∈ Matn(A) and f ∈ Matm(A) be two matrix idempotents overa unital ring A. Then the associated finitely generated modules eAn and fAm areisomorphic if and only if there exists an invertible matrix a ∈ MatN(A) with N >m,n such that

a

(e 00 0N−n

)a−1 =

(f 00 0N−m

).

Proof. Necessity. Suppose that there exists an isomorphism ϕ : eAn → fAm.Extending ϕ by zero to (1 − e)An, we obtain a morphism ψ : An → Am and,analogously, extending ϕ−1 by zero to (1− f)Am we get a morphism χ : Am → An.These morphisms may be written in the form

ψ(s) = gs and χ(t) = ht

for appropriate matrices g ∈ Matm,n(A) and h ∈ Matn,m(A). These matrices satisfythe following easily checked relations:

gh = f, hg = e and g = ge = fg, h = eh = hf.

Set now N = m + n and note that

(g 1− f

1− e h

)(e 00 0

)(h 1− e

1− f g

)=

(f 00 0

)

and also (g 1− f

1− e h

)(h 1− e

1− f g

)=

(1 00 1

).

This proves the necessity of conditions of the lemma.Sufficiency. If a(e⊕0)a−1 = f⊕0 then aeAN = faAN . Plugging AN = An⊕Am

into this relation, we obtain an explicit module isomorphism ϕ : eAn → fAm.

Denote by Qn(A) the set of idempotents in the algebra Matn(A) and by GLn(A)the group of invertible elements in Matn(A). There are natural embeddings

Matn(A) → Matn+1(A), m 7−→(

m 00 0

)

and

GLn(A) → GLn+1(A), a 7−→(

a 00 1

).

The first of them generates also the embedding Qn(A) → Qn+1(A). Using theseembeddings we can define the inductive limits

Mat∞(A) =∞⋃

n=1

Matn(A), Q∞(A) =∞⋃

n=1

Qn(A), GL∞(A) =∞⋃

n=1

GLn(A).


Definition 24. Two idempotents e, f ∈ Qm(A) are equivalent if they are conjugatein GL∞(A), i.e. if for some n there exists an element a ∈ GLn+m(A) such that

a

(e 00 0n

)a−1 =

(f 00 0n

).

Consider the setV alg(A) = Q∞(A)/ ∼

which is the quotient of Q∞(A) with respect to the introduced equivalence relation.Define the sum in V alg(A) by the rule

[e] + [f ] =

[(e 00 f

)]=

[(f 00 e

)].

This operation is correctly defined due to the relation e⊕f ∼ f ⊕e. Hence, V alg(A)is a commutative semigroup and we can define the algebraic K0-group Kalg

0 (A) asthe Grothendieck group of the semigroup V alg(A).

Theorem 7. For an arbitrary unital C∗-algebra A both K0-groups coincide, i.e.

K top0 (A) = Kalg

0 (A).

Proof. Suppose first that idempotents e, f ∈ Q∞(A) are equivalent. By Lemma 6there exists a sufficiently large n for which the right A-modules eAn and fAn areisomorphic. Then by Theorem 6 there exist projectors p, q ∈ Matn(A) such that

eAn = pHA fAn = qHA.

It follows that p ∼ e ∼ f ∼ q in Q∞(A), i.e. there exists an element z ∈ GL∞(A)such that

q = zpz−1.

In order to prove the equivalence of projectors p and q in the space P(AS) we haveto find a unitary operator u in HA such that q = upu−1. For that represent z in thepolar form z = u|z| where u is a unitary operator in the space EndAHA. Then

|z|p|z|−1 = u†qu = (u†qu)† = |z|−1p|z|,

i.e. p commutes with |z|2, hence also with |z| (why?). So, q = upu†.Conversely, suppose that projectors p, q ∈ P(AS) are equivalent to each other,

i.e. there exists a unitary operator u ∈ EndAHA such that q = upu†. Then, asin the proof of Theorem 5, we can find for some sufficiently large n projectorspn, qn ∈ Matn(A) such that pn ∼ p ∼ q ∼ qn by unitary conjugations which impliesthat

qn = vpnv† (1.9)

for some unitary operator v ∈ EndAHA. If v ∈ GLm(A) with some m ≥ n thenthe projectors pn and qn belong to the same class in Q∞(A), i.e. the modules pnHA

and qnHA are isomorphic. But in this case the modules pHA and qHA are alsoisomorphic, i.e. by Lemma 6 they belong to the same class in Kalg

0 (A).

1.4. K-THEORY 39

In the case when v is a general unitary operator in HA we can however supposethat v ∈ A+

S where A+S denotes the unitalization of the algebra AS. Indeed, as in

the proof of Lemma 5 we can construct unitary operators un and vn, belonging tothe group GLn(A) ⊂ A+

S , such that

pn = unpu†n and qn = vnqv†n.

The operator v in these terms is written in the form v = vnuu†n (one can check it byplugging this expression for v into (1.9)). This formula implies that v ∈ A+

S .Using this fact we can find for a sufficiently large m ≥ n a unitary operator

w ∈ Matm(A) approximating v in Matm(A) with the given accuracy ε < 1/4.Denote by qn ∈ Matm(A) the projector of the form qn = wpnw†. Then

qn − qn = Pm(qn − qn)Pm =

= Pm(v − w)pn(v − w)†Pm + Pmwpn(v − w)†Pm + Pm(v − w)pnw†Pm =

= Pm(v − w)PmpnPm(v − w)†Pm + PmwPmpnPm(v − w)†Pm+

+ Pm(v − w)PmpnPmw†Pm

where the latter equality follows from the evident relation pn = PmpnPm. It impliesthat the operator qn − qn admits the estimate

‖qn − qn‖ = ‖Pm(v − w)PmpnPm(v − w)†Pm+

+ PmwPmpnPm(v − w)†Pm + Pm(v − w)PmpnPmw†Pm‖ <

< ε2 + 2ε < 1.

So by Lemma 5 there exists a unitary operator wm ∈ Matm(A) such that qn =wmqnw†

m. Then the operator zm := wmw ∈ GLm(A) will satisfy the relation qn =zmpnz

−1m which implies that the projectors pn and qn belong to the same class in

Q∞(A), i.e. the modules pnHA and qnHA are isomorphic. But in this case themodules pHA and qHA are also isomorphic so by Lemma 6 they belong to the sameclass in Kalg

0 (A).

Due to the proved theorem we shall omit further on the indices “top” and “alg”in the notations of V (A) and K0(A).

Any element from K0(A) is represented in the form [p] − [q] where projectorsp, q ∈ P(AS). Let ϕ : A → B be a unitary morphism of C∗-algebras. Denote byK0ϕ : K0(A) → K0(B) the map given by the formula

K0ϕ : [p]− [q] −→ [ϕ(p)]− [ϕ(q)].

Proposition 8. The correspondence

(A,ϕ) 7−→ (K0(A), K0ϕ)

determines the covariant functor from the category of unital C∗-algebras into thecategory of Abelian groups.


Examples of K0-groups

1. K0(C) = Z. Indeed, V (C) = N since all projectors in P(CS) have finite rankwhich is their only invariant.

2. Analogously, K0(Matn(C)) = Z.

3. The group K0(L(H)) for the algebra L(H) of bounded linear operators in aHilbert space is equal to zero.

Properties of K0-functor

1. stability : K0(AS) = K0(A).

2. half-exactness : K0 transforms short exact sequences of the form 0 → J →A → B → 0 to the sequences

K0(J) → K0(A) → K0(B)

which are exact in the middle term.

3. K0 commutes with inductive limits.

1.4.2 Higher K-groups

In order to define the higher K-groups we introduce the notion of suspension of aC∗-algebra A. It is a C∗-algebra of the form

ΣA := A⊗ C0(R) ∼= C0(R, A)

where C0(X) (resp. C0(X,A)) denotes the space of continuous functions (resp. withvalues in A) on a locally compact topological space X vanishing at infinity. Usingthis notion we define the K-group of order n for the C∗-algebra A as

Kn(A) := K0(ΣnA).

Theorem 8 (Bott periodicity theorem). For any C∗-algebra A and any natural nwe have the following isomorphisms

K2n(A) ∼= K0(A), K2n+1(A) ∼= K1(A).

In view of this theorem it is sufficient to study, apart from the group K0(A),only the group K1(A) for which we shall give another, equivalent definition.

Namely, we introduce the group

Ktop1 (A) = [C0(R), AS]

identified with the set of homotopy classes of homomorphisms C0(R) → AS. Thisdefinition may be rewritten in the form

[C0(R), AS] ∼= [C(T), A+S ]+

1.4. K-THEORY 41

where A+S denotes the unitalization of the algebra AS and the index “+” in the

notation [X,Y ]+ indicates that we are considering the set of homotopy classes ofcontinuous maps X → Y of the pointed topological spaces X,Y .

Note that the C∗-algebra C(T) is generated by the unique unitary element t 7→eit. So a homomorphism from Ktop

1 (A) ∼= [C(T), A+S ]+ is determined by the choice

of a unitary element in A+S = (K⊗A)+. Hence, we can identify Ktop

1 (A) with groupπ0(U(A+

S )) of connected components of the unitary group U(A+S ).

Using the fact that K(H) = lim−→Matn(C), we can rewrite the latter definition of

Ktop1 (A) in the form

Ktop1 (A) = lim−→Un(A)/Un(A)0 = lim−→GLn(A)/GLn(A)0

where Un(A) denotes the subgroup in Matn(A) consisting of unitary elements, andUn(A)0 is the connected component of identity in Un(A).

Examples of topological K1-groups

1. Ktop1 (C) = 0. This fact follows from the connectedness of the group U(K+)

(where K = K(H)) which you may check by yourself.

2. Analogously, Ktop1 (Matn(C)) = 0.

The multiplication in the group Ktop1 (A) is defined by the formula:

[u] · [v] = [uv] =

[(u 00 v

)]

where the second equality and commutativity of multiplication follow from the chainof homotopies: (

uv 00 1

)∼

(u 00 v

)∼

(v 00 u

)∼

(vu 00 1

)

which the reader may check by himself.We switch to the definition of the algebraic K1-group. Recall first of all that the

commutant of an arbitrary group G is its normal subgroup G′ := [G,G] generatedby the elements of the form [g, h] := ghg−1h−1. The quotient

Gab := G/G′

is an Abelian group called the abelianization of the group G.Using this notion, we can define the K1-group of an arbitrary ring A as

Kalg1 (A) = GL∞(A)ab = GL∞(A)/GL∞(A)′.

Examples of algebraic K1-groups

1. If A = Z then Kalg1 (Z) = Z2.

2. If A = F is a field then Kalg1 (F ) = F× (the group of invertible elements in F ).


In particular, if a ring A is a unital C∗-algebra A then

Kalg1 (A) = GL∞(A)ab = GL∞(A)/GL∞(A)′.

On the other handKtop

1 (A) = GL∞(A)/GL∞(A)0.

Since the commutant is contained in the connected component of identity (why?)there is a natural surjective map

Kalg1 (A) −→ Ktop

1 (A) (1.10)

which is, however, in contrast with the case of K0-groups, not always injective.Indeed, in the case of the C∗-algebra A = C we have:

Kalg1 (C) = C× while Ktop

1 (C) = 0.

(The map (1.10) is bijective in the case of the so called stable C∗-algebras.)In the sequel we shall denote by K1(A) the group Ktop

1 (A).

1.5 Fredholm operators

1.5.1 Topological theory

Consider the algebra L(H) of bounded linear operators acting in a Hilbert space H.The ideal K ≡ K(H) of compact operators in the algebra L(H) will play later therole of the set of infinitesimal elements in this algebra. So it is wotrhwhile to studythe so called Kalkin algebra

Q(H) = L(H)/K(H).

Note that two operators S, T ∈ L(H) have the same image in the algebra Q(H) ifand only if S = T + K for some compact operator K.

Proposition 9. A bounded linear operator F ∈ L(H) has an invertible image inthe algebra Q(H) if and only if there exists an operator G ∈ L(H) such that theoperators 1 − GF and 1 − FG are compact. The latter condition is equivalent tothe fact that the image ImF is closed and the kernel KerF and cokernel CokerF ofoperator F are finite-dimensional.

Proof. The first equivalence is evident. To prove the second one suppose that thereexists an operator G ∈ L(H) such that the operators 1−GF and 1−FG are compact.Assume that we have already proved that Im F is closed and show that the kernelKerF is finite-dimensional. Note that Ker F is invariant under the operator 1−GFsince

(1−GF )ξ = ξ −GFξ = ξ

for ξ ∈ KerF . The same is true also for the unit ball in the space Ker F whichcoincides therefore with the image of the compact operator 1−GF . The compactnessof this ball implies that Ker F is finite-dimensional.

1.5. FREDHOLM OPERATORS 43

We prove now that Im F is closed. For that we choose an operator R of finiterank so that

‖(1−GF )−R‖ < 1/2.

For ξ ∈ KerR we shall have

‖ξ‖ − ‖ξ −GFξ‖ ≤ ‖GFξ‖ ≤ ‖G‖ · ‖Fξ‖.On the other hand,

‖ξ‖ − ‖ξ −GFξ‖ ≥ ‖ξ‖ − 1

2‖ξ‖ =

1

2‖ξ‖,

i.e.1

2‖ξ‖ ≤ ‖G‖ · ‖Fξ‖.

It follows that

‖Fξ‖ ≥ ‖ξ‖2‖G‖

hence, the restriction of the operator F to KerR has closed image. But the subspace(KerR)⊥ = Im R† is finite-dimensional since the operator R has finite rank. So thespace

ImF = F (KerR) + F ((KerR)⊥)

is closed.To prove that the cokernel CokerF is finite-dimensional we note, as above, that

the space KerF † is invariant under the operator (1−FG)† = 1−G†F † which impliesits finite-dimensionality. But

CokerF = H/ImF ∼= KerF †

and so is also finite-dimensional.Conversely, if the image Im F is closed and the subspaces Ker F and CokerF are

finite-dimensional then we can construct the desired operator G by setting

G(Fξ) = ξ ξ ∈ (KerF )⊥,

G(ξ) = 0 ξ ∈ (ImF )⊥ ∼= KerF †.

Indeed, this operator is correctly defined since the map F : (KerF )⊥ → ImF isbijective. Moreover, the operators 1−GF and 1− FG have finite rank, hence theyare compact.

Definition 25. A bounded linear operator F : H1 → H2 from a Hilbert space H1

into another Hilbert space H2 is called Fredholm if its image ImF is closed andthe spaces KerF and CokerF are finite-dimensional. The index of the Fredholmoperator F is equal to

indF = dim KerF − dim CokerF.

For H1 = H2 = H the space of Fredholm operators F : H → H is denoted byFred = Fred(H) and provided with the topology of uniform convergence from L(H).It is a multiplicative semigroup (cf. property 2 below).


Properties of the index:

1. The map ind : Fred → Z is continuous.

2. The map ind is a semigroup homomorphism, i.e.

ind (F1F2) = indF1 + indF2.

3. The value of the index does not change under compact perturbations, i.e.

ind (F + K) = indF

for any K ∈ K.

4. indF = 0 ⇐⇒ F is a compact perturbation of an invertible operator.

5. The standard right shift operator in the space `2 is Fredholm and its index isequal to −1.

6. indF = dim Ker (F †F )− dim Ker (FF †).

7. indF † = −indF .

Theorem 9 (Atiyah–Janich theorem). For any compact Hausdorff topological spaceX there exists a group isomorphism

ind : [X,Fred] −→ K0(X)

where K0(X) is the Grothendieck group K(Vect(X)) of the semigroup Vect(X) ofvirtual vector bundles over X. This isomorphism is functorial in the sense that forany continuous map ϕ : Y → X of compact topological spaces the following relation

ind ϕ∗ = K0ϕ ind

holds where ϕ∗ : [X,Fred] → [Y,Fred] is the homomorphism generated by the mapF 7→ F ϕ from the space C(X,Fred) into the space C(Y,Fred) and K0ϕ : K0(X) →K0(Y ).

Remark 7. We explain roughly the idea of the proof of this theorem. Let F :X → Fred, x 7→ Fx, be a continuous map from the space X into the space ofFredholm operators. We should associate with it an element of K0(X). The firstidea is to identify the desired element with the field of virtual vector spaces of theform x 7→ [KerFx] − [CokerFx]. But this map in general is not defined since thedimensions of the spaces Ker Fx and CokerFx may change from point to point.

In order to circumvent this difficulty the following method is used. The field ofspaces x 7→ KerFx is replaced by the trivial bundle over X with fibre H/V whereV is a closed subspace in H of finite codimension such that

V ∩KerFx = 0for all x ∈ X. Such V is chosen using the compactness of X as the intersection

V =m⋂

i=1

(KerFxi)⊥


where x1, . . . , xm ∈ X is an appropriate finite set of points in X. The class [H/V ]of the trivial vector bundle with fibre H/V in K0(X) is the required replacement ofthe field x 7→ KerFx. It is proved next that the union

⋃x∈X

H/Fx(V )

is the total space of some locally trivial vector bundle W → X over X and theclass [W ] of this bundle in K0(X) may be taken as the replacement of the fieldx 7→ CokerFx. Finally it is proved that

indF = [H/V ]− [W ] ∈ K0(X).

1.5.2 Fredholm operators in C∗-modules

Definition 26. Let E ,F be right C∗-modules over a C∗-algebra A and F ∈ HomA(E ,F)is a bounded A-linear operator. The operator F is called A-Fredholm if there existsan operator G ∈ HomA(F , E) such that 1F − FG ∈ KA(F) and 1E −GF ∈ KA(E).In the case when E = F this condition is equivalent to the invertibility of the imageof F in the quotient algebra EndA(E)/KA(E). The set of A-Fredholm operators isdenoted by FredA(E ,F).

In this definition one can replace the A-compact operators by the operators ofA-finite rank according to the following lemma.

Lemma 7. Let A be a unital C∗-algebra and J is an ideal in A which closure isdenoted by J . Then any element a ∈ A which is invertible modulo J is also invertiblemodulo J .

Proof. Denote by b the element of the algebra A such that 1− ab ∈ J . Then thereexists an element c ∈ J for which ‖1 − ab − c‖ < 1 and, consequently, ab + c isinvertible. Denote by b1 the element b1 := b(ab + c)−1. Then

1− ab1 = 1− ab(ab + c)−1 = c(ab + c)−1,

i.e. belongs to J . An analogous argument shows that there exists an element b2

such that 1 − b2a ∈ J . In other words, the element [a] := a + J of the quotientalgebra A/J is invertible both from the right and from the left which means that itis invertible in A/J .

Choosing for J = FinA(E ,F), we obtain that for an A-Fredholm operator F :E → F it always exists an operator G ∈ HomA(F , E) such that 1F −FG ∈ FinA(F)and 1E −GF ∈ FinA(E).

Remark 8. Note that the image Im F of an A-Fredholm operator F should notbe closed. Take, for example, for A the algebra A = C(I) of functions which arecontinuous on the unit interval I = [0, 1] and for E the algebra A itself. Define theoperator F by the formula: Fa(t) := ta(t) for t ∈ I. Since the algebra A is unital wehave KA(A) ∼= A so that any operator from EndAA is A-compact and A-Fredholm.But the image Im F is not closed since the function b(t) =

√t evidently does not

belong to Im F but it belongs to its closure since it can be uniformly approximatedby polynomials vanishing at t = 0, hence belonging to Im F . (For example, one canapproximate it by the Bernstein polynomials

∑nk=1 Ck

n

√k/ntk(n− t)n−k.)


Regular Fredholm operators

In order to avoid this problem we introduce the notion of a pseudoinverse opera-tor. In the proof of Proposition 9 we have constructed an operator G such that theoperators 1 − FG and 1 − GF are projectors onto KerF and KerF † respectively.The operators F and G satisfy the relations: FGF = F and GFG = G.

This observation motivates the following definition.

Definition 27. For a given operator T ∈ HomA(E ,F) its pseudoinverse is theoperator S ∈ HomA(F , E) such that

TST = T and STS = S.

In this case the operators TS and ST are idempotents and have closed images.Moreover, they have the following properties:

1. KerST = KerT ;

2. Im(1− ST ) = KerT ;

3. Im(TS) = ImT .

Since the notion of pseudoinvertibility is symmetric with respect to S and T wehave similar properties obtained by replacing T by S:

4. KerTS = KerS;

5. Im(1− TS) = KerS;

6. Im(ST ) = ImS.

The proof of these properties we leave as an exercise. For instance, for the firstof them we have, on one side, that Ker T ⊂ KerST and, on the other side, thatSTu = 0 =⇒ TSTu = Tu = 0.

The operators, having pseudoinverses, are called regular . It is clear that theusual Fredholm operators are regular.

Suppose that F is a regular A-Fredholm operator with pseudoinverse S. Denoteby G the operator existing by Definition 26. Note that

(1−GF )(1− SF ) = 1− SF

since Im(1 − SF ) = KerF . Hence, the operator 1 − SF ∈ KA(E). Analogously,1−FS ∈ KA(F). So for a regular A-Fredholm operator F we can use as the operatorG from the Definition 26 the pseudoinverse of F .

Moreover, the operators 1−FS and 1−SF have closed images. Indeed, denotethe idempotent 1− SF by e. We have the following lemma.

Lemma 8. Let p ∈ KA(E) be an A-compact projector in a C∗-module E over aunital C∗-algebra A. Then p ∈ FinAE. Moreover, any idempotent e ∈ KA(E) is infact an operator of A-finite rank.


Proof. The algebra pKA(E)p is a unital C∗-algebra in which the role of unit is playedby the projector p itself. This algebra contains p FinAEp as a dense ideal. Since anysuch ideal, being dense, cannot be proper it should contain the unit 1E which canbe represented in the form

1E =n∑

k=1

|rk〉〈sk|

for some elements rk, sk ∈ E . Then the projector p will be written in the form

p =n∑

k=1

p|rk〉〈sk|p =n∑

k=1

|prk〉〈psk|

which implies that p has A-finite rank. More generally, if e is an A-compact idempo-tent then the Kaplansky formula (cf. Remark 6) will give an A-compact projectorp such that pe = p. It follows from the first part of the proof that e ∈ FinAE .

Return to the operator 1− SF which we have identified with the idempotent e.If p is the projector corresponding to e by the Kaplansky formula then p ∈ FinAE aswell as e. It implies that the image of the operator e = 1− SF , coinciding with theimage of the projector p, is closed. In analogous way one can prove that the imageof the operator 1− FS is closed.

1.5.3 Index of A-Fredholm operators

In order to introduce the index of regular Fredholm operators we shall use thefollowing important theorem.

Theorem 10 (Kasparov absorption theorem). If E is an arbitrary countably gen-erated C∗-module over an algebra A then

E ⊕HA∼= HA

as A-modules.

Proof. We shall construct an intertwining operator T between HA and E ⊕ HA forwhich the operator T and the adjoint operator T † have dense images. Suppose forsimplicity the the algebra A is unital (the case of a non-unital algebra is treatedin [3], Theorem 4.6). Let uk be a countable family of unit vectors generating E .Denote by (sk) the sequence obtained by the ”reproduction” of the sequence (uk) sothat every uk in it is repeated infinite number of times. Moreover, denote by ξkthe canonical system of generators of the A-module HA (cf. Sec. 1.3.1). Define anA-compact operator T by the formula

T =∞∑

n=1

2−n|sn〉〈ξn| ⊕ 4−n|ξn〉〈ξn|.

Every time when sn coincides with uk we have T (2nξn) = (uk ⊕ 2−nξn). Since suchcoincidence occurs for an infinite number of values of n it follows that we can find asubsequence of the sequence uk ⊕ 2−nξn converging to an element (uk ⊕ 0) which


belongs to the closure of the image of T . It follows that any element (0 ⊕ ξn) alsobelongs to this closure. Since T †(0⊕ ξn) = 4−nξn the image of the adjoint operatorT † is dense in HA.

To finish the proof of the theorem we use the following lemma.

Lemma 9. Let T ∈ HomA(E ,F) be an A-linear operator acting from a C∗-moduleE over a C∗-algebra A into a C∗-module F over the same algebra A. Suppose thatthe operator T , as well as the adjoint operator T †, have dense images. Then theC∗-modules E and F are unitary equivalent.

Proof. Note, first of all, that in this case the image Im(T †T ) is dense in E (why?).

It implies that also the image of the operator |T | :=√

T †T is dense in E .Define an A-linear map

U : ImT −→ Im |T |

by setting U(Ts) := |T |s for s ∈ E . The constructed operator is isometric since

‖Ts‖2 = (T †Ts, s) = ‖|T |s‖2.

Since the subspaces Im T and Im |T | are dense in E the constructed isometric oper-ator extends in a unique way to a unitary operator in HomA(E ,F).

The assertion of the theorem follows from the proved lemma.

Index of A-Fredholm operators

We show now how, using the above theorem, one can define the index of regularA-Fredholm operators. The absorption theorem implies that any C∗-module E of A-finite rank may be considered as a submodule inHA of the form pHA with p ∈ P(AS).This projector determines a class [p] in K0(A) which is associated with the C∗-module E and denoted by [E ].

Let now F be a regular A-Fredholm operator. Then the C∗-modules KerF =Im(1− SF ) and Ker F † = Im(1− FS) have, as we have pointed out above, the A-finite rank and so determine the elements [Ker F ] and [KerF †] of the group K0(A).Taking this into account, we can give the following definition.

Definition 28. Let F ∈ FredA(E ,F) be a regular A-Fredholm operator. Define theindex of F by the formula

indF := [KerF ]− [KerF †] ∈ K0(A).

The index of a regular A-Fredholm operator has the following properties theproof of which we leave to the reader.

1. If F ∈ FredA(E ,F) is a regular operator with pseudoinverse S then S is alsoa regular A-Fredholm operator and ind S = −indF .

2. If F ∈ FredA(E ,F) is a regular operator with pseudoinverse S then the adjointoperator F † is also a regular A-Fredholm operator and ind F † = −indF .


3. If F1 ∈ FredA(E1,F1) and F2 ∈ FredA(E2,F2) are regular operators than theirdirect sum F1 ⊕ F2 is also a regular operator in FredA(E1 ⊕ E2,F1 ⊕ F2) andind (F1 ⊕ F2) = indF1 + indF2.

4. If F ∈ FredA(E ,F) is a regular operator and operators U ∈ EndA(E) andV ∈ EndA(F) are invertible then the operators FU and V F are regular A-Fredholm operators and ind FU = indV F = indF .

5. If F1, F2 ∈ FredA(E ,F) are regular operators and F1 − F2 ∈ KA(E ,F) thenindF1 = indF2.

Is it possible to define the index of an arbitrary, possibly not regular A-Fredholmoperator? The answer to this question is positive and we describe briefly how suchindex can be defined.

Let us represent the operators T ∈ HomA(E1⊕E2,F1⊕F2) in the block form sothat

T =

(T11 T12

T21 T22

)

where Tij ∈ HomA(Ei,Fj), i, j = 1, 2.To define the index of a general A-Fredholm operator we use the following lemma.

Lemma 10. Let E ,F be C∗-modules over a unital C∗-algebra A and F ∈ FredA(E ,F).Then there exist a natural number n ∈ N and a regular operator F ∈ FredA(E ⊕An,F ⊕ An) such that F11 = F .

The proof of this lemma may be found in [3], Lemma 4.10.We introduce now the index of an arbitrary A-Fredholm operator F setting by

definition

indF = ind F .

It is necessary to check only the correctness of this definition depending on the choiceof a regular extension F which is, in its turn, determined by the operator G. Butany other choice of an operator G′, for which the operators 1 − G′F and 1 − FG′

are operators of A-finite rank, will lead to an operator F ′ differing from F by anoperator of A-finite rank. Hence the operator F ′ will have the same index as F dueto the property 5 above.

The so defined index of general A-Fredholm operators has the properties similarto the index of regular A-Fredholm operators.

1. If F1, F2 ∈ FredA(E ,F) and F1−F2 ∈ KA(E ,F) then indF1 = indF2, i.e. doesnot change under A-compact perturbations.

2. The set FredA(E ,F) is open in HomA(E ,F) and the index map ind : FredA(E ,F) →K0(A) is locally constant.

3. If F ∈ FredA(E ,F) and G ∈ FredA(F ,G) then GF ∈ FredA(E ,G) and

ind (GF ) = indG + indF.


Theorem 11 (noncommutative Atiyah–Janich theorem). If A is a unital C∗-algebrathen the map

ind : π0(FredA) −→ K0(A)

where FredA := FredA(HA) is a group isomorphism.

The proof of this theorem may be found in [3], Theorem 4.19.

Remark 9. Theorem 11 is an extension of Atiyah–Janich Theorem 9, correspondingto the case A = C(X) where X is a compact topological space, since in this case wehave a group isomorphism

π0(FredC(X)) ∼= [X, Fred].

1.6 Morita-equivalence

1.6.1 Morita-equivalence of algebras

We shall study the notion of Morita-equivalence first in the case of algebras. Let Aand B be two algebras. Denote by MA, AM and AMB the categories of respec-tively right A-modules, left A-modules and (AB)-bimodules, i.e. bimodules over thealgebras (A,B).

Definition 29. The algebras A and B are called Morita-equivalent if the categoriesMA and MB are equivalent. It means that there exist an (AB)-bimodule E and(BA)-bimodule F with the following bimodule isomorphisms

E ⊗B F ∼= A, F ⊗A E ∼= B. (1.11)

Here A is considered as an (AA)-bimodule over A with bimodule structure definedby the equality: a(b)c := abc for all a, b, c ∈ A. Analogously, B is considered as a(BB)-bimodule with bimodule structure defined in the same way. The bimodules Eand F are called the equivalence bimodules .

Remark 10. If the algebras A and B are Morita-equivalent then the categories AMand BM and also the categories AMA and BMB are also equivalent.

Example 5. Any unital algebra A is Morita-equivalent to the matrix algebraB := Matn(A). In this case the equivalence bimodules are given by the bimodulesconsisting of row- and column-vectors. Indeed, denote by E = nA the space of row-vectors with n entries provided with left Matn(A)-action and right A-multiplication.Denote next by F = An the space of columns with n entries provided with the leftA-multiplication and right Matn(A)-action. Then the isomorphisms from (1.11) willbe given by the following formulas:

(a1, . . . , an)⊗ t(b1, . . . , bn) 7−→n∑

k=1

akbk,

t(b1, . . . , bn)⊗ (a1, . . . , an) 7−→ (biaj)ni,j=1.

1.6. MORITA-EQUIVALENCE 51

In order to give another definition of equivalence bimodules consider the followingalgebra homomorphisms generated by an arbitrary (AB)-bimodule E :

A −→ EndBE , a 7−→ La,

Bo −→ EndAE , b 7−→ Rb

where La is the operator of left multiplication by a and Rb is the operator of rightmultiplication by b. In this formula we have denoted by Bo the algebra opposite tothe algebra B. By definition it is the algebra

Bo = bo : b ∈ Bwith multiplication law: boco := (cb)o for b, c ∈ B.

Theorem 12 (Morita theorem). An (AB)-bimodule E is an equivalence bimodule ifand only if it is finitely generated and projective both as a left A-module and rightB-module, and the above homomorphisms

A −→ EndBE , Bo −→ EndAEare in fact isomorphisms.

The proof of this theorem may be found in [1].

Example 6. Let E be a finitely generated projective left A-module and B =(EndAE)o. Then the algebras A and B are Morita-equivalent and the equivalence(AB)-bimodule coincides with E . It implies, in particular, that if E is a complexvector bundle over a manifold M then the algebras A = C(M) and B = Γ(End E)are Morita-equivalent. This case reduces to the one just considered after settingE = Γ(E).

Before we switch to the Morita-equivalence of C∗-algebras consider as an inter-mediate step the construction of equivalence bimodules for involutive algebras.

Recall that the involutive or ∗-algebra is an algebra A provided by an anti-linearmap ∗ : A → A such that

(ab)∗ = b∗a∗, (a∗)∗ = a

for all a, b ∈ A.Let A be a unital ∗-algebra and E be an arbitrary unital (i.e. multiplication by

1A is the identity map on E) right A-module. Assume that we have on E an A-valuedinner product (· , ·)A which is full , i.e. for any a ∈ A there exist elements sk, tk ∈ E ,k = 1, . . . , n, such that

a =n∑

k=1

(sk, tk)A.

Suppose that A and B are two ∗-algebras and E is an (AB)-bimodule providedwith a full A-valued inner product A(· , ·) and full B-valued inner product (· , ·)B.Assume that these inner products are related by the following associativity condition:for any s, t, u ∈ E

A(s, t)u = s(t, u)B. (1.12)


We assert that such bimodule is necessarily finitely generated and projective bothas a left A-module and right B-module. Indeed, it follows from the fullness of E asa right B-module that there exist elements sk, tk ∈ E , k = 1, . . . , n, such that

1B =n∑

k=1

(sk, tk)B.

Denote by eknk=1 the standard basis of the module of row-vectors An and consider

the map P : An → E sending ek 7→ tk, k = 1, . . . , n. We assert that this maphas a right inverse which will imply that E is a finitely generated projective leftA-module. Indeed, Consider the A-linear map Q : E → An given by the formula:Q(s) =

∑nk=1 A(s, sk)ek. Then

PQ(s) =n∑

k=1

A(s, sk)tk =n∑

k=1

s(sk, tk)B = s.

An analogous proof shows that E is a finitely generated projective right B-module.In fact one can assert that E is an equivalence bimodule. For the proof denote

by E the complex vector space which is complex dual to E with elements denotedby s where s ∈ E . Then E is a (BA)-bimodule with multiplication defined by theequality: bsa := a∗sb∗ for s ∈ E , a ∈ A, b ∈ B. This bimodule may be provided alsowith A-valued and B-valued inner products by the formula

B(s, t) := (s, t)B, (s, t)A := A(s, t)

for s, t ∈ E .Consider the bimodule maps

f : E ⊗ E∗ −→ A, s⊗ t 7−→ A(s, t),

g : E∗ ⊗ E −→ B, s⊗ t 7−→ (s, t)B.

Both maps are surjective due to the fullness of inner products. It may be shown alsothat they are isomorphisms (cf. [1], proposition 4.4). Hence, E is an equivalencebimodule and the algebras A and B are Morita-equivalent.

1.6.2 Morita-equivalence of C∗-algebras

Suppose that A and B are two C∗-algebras and E is a right C∗-module over thealgebra A and F is the right C∗-module over the algebra B. We shall assume that itis given a representation of the C∗-algebra A in the module F , i.e. a homomorphismρ : A → EndBF . In Sec. 1.3.5 it was shown that in this case one can define theC∗-module over B which is the tensor product E ⊗ρ F = E ⊗A F of C∗-modules Eand F .

Definition 30. A C∗-module E over a C∗-algebra A is called full if the ideal

I =: (E , E) = span(s, t) : s, t,∈ E

is dense in A.

1.6. MORITA-EQUIVALENCE 53

With this definition we can introduce the notion of the equivalence bimodule forC∗-algebras.

Definition 31. Let A and B are two C∗-algebras. Then the (AB)-equivalencebimodule is an (AB)-bimodule E with the following properties:

1. E is a left full C∗-module over A and simultaneously a right full C∗-moduleover B;

2. for all s, t, u ∈ E the following associativity condition is satisfied:

A(s, t)u = s(t, u)B

where A(· , ·) and (· , ·)B are the pairings in E considered as a left C∗-moduleover A and right C∗-module over B respectively.

Having the notion of equivalence bimodule we can define the Morita equivalenceof C∗-algebras.

Definition 32. We call two C∗-algebras A and B Morita-equivalent if for themthere exists an equivalence (AB)-bimodule E .

It follows from the conditions of associativity and fullness that in this case theoperators La of left multiplication by elements a ∈ A and Rb of right multiplicationby elements b ∈ B, defined on the C∗-module E , admit adjoint operators. Indeed,let us check it for the operator La. For all s, t, u ∈ E , a ∈ A, b ∈ B we have:

u(as, t)B = A(u, as)t = A(u, s)a∗t = u(s, a∗t)B

which implies, due to the fullness of E , that (as, t)B = (s, a∗t)B. Hence, the operatorLa has the adjoint operator L†a = La∗ . In the same way one can prove that R†

b = Rb∗ .Thus, we have correctly defined the representations of C∗-algebras given by op-

erators La and Rb:

L : A −→ EndBE , R : Bo −→ EndAE .

We take this property as the definition of the following notion generalizing thatof equivalence bimodule.

Definition 33. Let A and B be two C∗-algebras. Then an (AB)-correspondencebetween the algebras A and B is a homomorphism ϕ : A → EndBE for some rightC∗-module E over B.

Every equivalence (AB)-bimodule, according to the above argument, determinesalso an (AB)-correspondence between the C∗-algebras A and B.

If E is a right C∗-module over the algebra A, F is a right C∗-module over thealgebra B and ϕ : A → EndBE is an arbitrary ∗-homomorphism then F inherits thestructure of (AB)-bimodule and we can define the tensor product E ⊗ϕF = E ⊗AFwhich is a C∗-module over B. This tensor product operation is associative. In fact,one can introduce an additive category with C∗-algebras as objects and morphismsgiven by the correspondences between them determined up to isomorphisms.


Using the notion of correspondence it is possible to give an equivalent definitionof Morita-equivalence which is close to that used in the algebraic case.

Note, first of all, that any C∗-algebra A is Morita-equivalent to itself since inthis case one can take for the equivalence (AA)-bimodule the algebra A consideredas a bimodule provided with the pairings

A(a, b) := ab∗, (a, b)A := a∗b

for a, b ∈ A.Using this, one can show that two C∗-algebras A and B are Morita-equivalent if

and only if there exist an (AB)-correspondence E and (BA)-correspondence F suchthat

E ⊗B F ∼= A, F ⊗A E ∼= B.

Remark 11. If E is an equivalence (AB)-bimodule then one can take for F theadjoint bimodule E consisting of elements s with s ∈ E . Then, as it was pointed outbefore, E is a (BA)-bimodule with multiplication given by the equality: bsa := a∗sb∗

for any s ∈ E , a ∈ A, b ∈ B, and pairings given by

B(s, t) := (t, s)B, (st)A := A(s, t).

Then E becomes an equivalence (BA)-bimodule and

E ⊗B E ∼= A, E ⊗A E ∼= B.

Remark 12. Morita-equivalence is the equivalence relation. The proof of this asser-tion we leave as an exercise.

Remark 13. If two C∗-algebras A and B are Morita-equivalent and E and F aretheir equivalence bimodules then we can identify the category MA of right C∗-modules over A with the category MB of right C∗-modules over B. Indeed, themap S 7→ S ⊗A F associates with a right A-module S the right B-module S ⊗A Fand, conversely, the map T 7→ T ⊗B E associates with a right B-module T the rightA-module T ⊗B E .

We formulate without proof several criterions of Morita-equivalence (their proofsmay be found in the book [3], Sec.4.5).

Proposition 10. Two C∗-algebras A and B are Morita-equivalent if and only ifthere exist a full right C∗-module E over A such that KA(E) ∼= B.

Proposition 11. Any C∗-algebra A is Morita-equivalent to its stabilization, i.e.AS

∼= K ⊗ A is Morita-equivalent to A.

Corollary 5. If A and B are two stable equivalent C∗-algebras then the algebra Ais Morita-equivalent to the algebra B.

Theorem 13 (Exel). If C∗-algebras A and B are Morita-equivalent then K0(A) ∼=K0(B).

Remark 14. If C∗-algebras A and B are Morita-equivalent then using the GNS-construction one can establish a bijective correspondence between the irreduciblerepresentations of algebras A and B.

Chapter 2

ANALYSIS

2.1 Noncommutative integral

2.1.1 Ideals in the algebra of compact operators

For the construction of the noncommutative version of the analysis on C∗-algebraswe have to introduce, first of all, the notion of ”infinitesimal” elements. In thealgebra of bounded linear operators in a Hilbert space their role is played by compactoperators for which the degree of ”smallness” is measured by the rate of decreasingof their singular values. Let us consider these notions in more detail.

Let T be a compact operator in a Hilbert space H and |T | =√

T †T . Denote byµn(T ) the sequence of the singular values (s-values) of operator T given by theeigenvalues of operator |T | in the decreasing order:

µ0(T ) ≥ µ1(T ) ≥ . . .

so that µn(T ) → 0 for n →∞.The singular values of operator T may be found by the minimax principle, more

precisely, they are given by the formula

µn(T ) = infE‖T |E⊥‖ : dim E = n

so that µn(T ) coincides with the infimum of the norms of restrictions of T to orthog-onal complements of various n-dimensional subspaces E ⊂ H. In fact, this infimumis attained on the subspace En generated by the first n eigenvectors of operator |T |corresponding to the eigenvalues µ0, . . . , µn−1.

In a different way, one can define µn(T ) as the distance from the operator T tothe subspace Finn of operators of rank≤ n. Namely,

µn(T ) = infR‖T −R‖ : R ∈ Finn.

The singular values of operator T have the following properties (cf. [10]).

55

56 CHAPTER 2. ANALYSIS

Properties of s-values:

1. |µn(T1) − µn(T2)| ≤ ‖T1 − T2‖, in particular, the functional µn(T ) dependscontinuously on T in the uniform topology for any n.

2. µn+m(T1 +T2) ≤ µn(T1)+µm(T2) where we use the embedding Finn +Finm ⊂Finn+m.

3. µn+m(T1T2) ≤ µn(T1) · µm(T2).

4. Since µ0(T ) = ‖T‖ we have

µn(T1T2) ≤ µn(T1)‖T2‖, µn(T1T2) ≤ ‖T1‖µn(T2).

Definition 34. Let T be a compact operator in a Hilbert space H. We say that Tbelongs to the space Lp = Lp(H), 1 ≤ p < ∞, if

∞∑n=0

µn(T )p < ∞.

The space Lp is an ideal in the algebra K of compact operators and in the algebraL(H) of bounded linear operators in H. We are especially interested in the class L1

of nuclear operators provided with the norm

‖T‖1 := Tr|T | =∞∑

n=0

µn(T ) =∞∑

n=0

(un, Tun)

where un∞n=0 is an orthonormal basis in H. (Note that this definition does notdepend on the choice of an orthonormal basis in H.)

We introduce now a quantity playing an important role in the sequel:

σN(T ) :=N−1∑n=0

µn(T ).

In a different way it may be defined as

σN(T ) = supE‖TPE‖1 : dim E = N

where PE is the orthogonal projector to the subspace E, and the supremum isattained again on the subspace EN generated by the first N eigenvectors of operatorT .

It follows from the latter definition that σN satisfies the triangle inequality

σN(T1 + T2) ≤ σN(T1) + σN(T2)

hence, it determines a norm on K.We shall give one more definition of this quantity which is used below.

σN(T ) = inf‖R‖1 + N‖S‖ : R, S ∈ K, R + S = T.

2.1. NONCOMMUTATIVE INTEGRAL 57

It allows to extend the definition of the function σN as a function of natural param-eter N to arbitrary nonnegative values λ ∈ [0,∞) by setting

σλ(T ) = inf‖R‖1 + λ‖S‖ : R, S ∈ K, R + S = T.

It can be shown that the function σλ(T ) has the following properties:

1. the function σλ(T ) is piecewise linear and convex; moreover, if λ = N + t with0 ≤ t < 1, so that [λ] = N , then

σλ(T ) = (1− t)σN(T ) + tσN+1(T );

2. σλ(S + T ) ≤ σλ(S) + σλ(T );

3. if operators S, T are positive then

σλ+µ(S + T ) ≥ σλ(S) + σµ(T ).

The last two properties imply that the inequality

σλ(S + T ) ≤ σλ(S) + σλ(T ) ≤ σ2λ(S + T ) (2.1)

holds for arbitrary compact positive operators S, T . This subadditivity propertyof the functional σλ(T ) on the cone of positive compact operators will play animportant role in the definition of Dixmier trace in the next section.

Apart from ideals Lp we introduce also the interpolation ideals Lp,q.

Definition 35. An operator T ∈ Lp,q if

∞∑N=1

N (α−1)q−1σN(T )q < ∞

where α = 1/p. Extend this definition to q = ∞ by stating that T ∈ Lp,∞ if thesequence of numbers Nα−1σN(T )∞N=1 is bounded.

Proposition 12. Each of the introduced spaces Lp,q is a two-sided ideal in thealgebra K of compact operators. For p1 < p2 and for p1 = p2, q1 < q2 there are thefollowing inclusions

Lp1,q1 ⊂ Lp2,q2 .

The proof is left as an exercise.Let us consider in more detail some particular examples of the spaces Lp,q.The space Lp,p, 1 ≤ p < ∞, coincides with the space Lp, introduced above, with

the norm given by the formula

‖T‖p = (Tr|T |p)1/p =

[ ∞∑n=0

µn(T )p

]1/p

.


The space Lp,∞, 1 < p < ∞, consists of compact operators T for which σN(T ) =O(N1−α), i.e. µn(T ) = O(n−α). There is a natural norm on this space given by

‖T‖p,∞ = supN

1

N1−ασN(T ).

The space Lp,1 consists of compact operators T for which the series

∞∑N=1

Nα−2σN(T )

is converging which is equivalent to the convergence of the series∑∞

n=1 nα−1µn−1(T ).Between these spaces there are the following embeddings:

Lp− ≡ Lp,1 ⊂ Lp ≡ Lp,p ⊂ Lp,∞ ≡ Lp+.

We extend the definition of the spaces Lp,q, 1 < p < ∞, 1 ≤ q < ∞, to p = 1,q = ∞ by setting

L1,∞ = T ∈ K : σN(T ) = O(log N)and providing this space with the norm

‖T‖1,∞ = supN≥2

σN(T )

log N.

If this norm is finite it implies the estimate of the form µn = O(1/n) on the s-valuesof operator T . The space L1,∞ is an ideal dual to the ideal

L∞,1 = T ∈ K :∞∑

n=1

µn(T )

n< ∞.

As we have pointed out before, for the nuclear operators T ∈ L1 we can definethe trace given by the sum of s-values of this operator. In the sequel the trace willplay the role of the noncommutative integral but the trace defined on the class ofnuclear operators does not suit this goal unlike the Dixmier trace, introduced in thenext section, which is defined on a larger class of operators T ∈ L1,∞.

2.1.2 Dixmier trace

Let T be a positive operator belonging to the ideal L1,∞. We would like to defineits trace by the formula

limN→∞

1

log N

N−1∑n=0

µn(T ) = limN→∞

σN(T )

log N. (2.2)

Then we are met with two questions:

1. If the limit in Formula (2.2) does exist?

2. If the functional given by the Formula (2.2) is linear?


Note that the problem of linearity of this functional is closely related to theexistence of the limit in Formula (2.2). Indeed, to prove the linearity we shouldcompare the quantity

γN =σN(T1 + T2)

log N

with the sum of quantities

αN =σN(T1)

log Nand βN =

σN(T2)

log N.

The triangle inequality for σN(T ) implies that γN ≤ αN +βN and from the inequalityσN(T1) + σN(T2) ≤ σ2N(T1 + T2), mentioned in Sec. 2.1.1, we deduce that

αN + βN ≤ log(2N)

log NγN .

Since log(2N)/ log N → 1 for N → ∞ we see that the existence of the limit inFormula (2.2) will imply the linearity of the functional (2.2).

Turning to the question of existence of the limit in Formula (2.2) note that forany T ∈ L1,∞ the sequence of numbers

σN(T )

log N

is bounded.It allows to treat the problem of existence of the limit in Formula (2.2) in the

following, more general setting. Namely, we look for a linear form on the space`∞(N) of bounded sequences a = an∞n=1 denoted by

` ≡ Limω

which satisfies the following conditions:

1. Limωa ≥ 0 if all an ≥ 0;

2. Limωa = limn→∞ an if the limit in the right hand side exists;

3. Limω(a1, a1, a2, a2, a3, a3, . . .) = Limωan.

The only non-evident condition is the last one which is treated as the asymptoticscale invariance. In order to explain the origin of this term let us switch from thesequences an to the functions of a real parameter as we have already done inthe case of the function σN . Namely, let us associate with the sequence an∞n=1 abounded function fa(λ) on the real line defined in the following way: if λ = N + twith 0 ≤ t < 1, i.e. [λ] = N , then we set fa(λ) = (1 − t)aN + taN+1. Thus, theintroduced function fa(λ) is piecewise linear.

Replace now the function f(λ) ≡ fa(λ) with its Cesaro average

(Mf)(λ) =1

log λ

∫ λ

3

f(t)

tdt.


This average on bounded functions f has the following property of asymptotic scaleinvariance:

|M(Sµf)(λ)−MF (λ| −→ 0 for λ → +∞where (Sµf)(λ) := f(λµ) for any µ > 0. Returning to the property (3), we can in-terpret it as the asymptotic scale invariance of the function fa = S1/2(fa) associatedwith the sequence a = (a1, a1, a2, a2, a3, a3, . . .).

We shall formulate now more precisely what kind of the limit we want to haveon the considered space Cb(R+) of bounded continuous functions on the halflineR+ := [1,∞). Since we are interested only in the limits of such functions at infinitywe replace the space Cb(R+) by the quotient B∞ := Cb(R+)/C0(R+) modulo thesubspace C0(R+) of functions vanishing at infinity.

Fix a positive linear form ω on the space Cb(R+) such that ω = 0 on the subspaceC0(R+) and ω(1) = 1. In other words, ω is a state on the C∗-algebra B∞. We canconsider ω(f) as a ”generalized limit” of function f ∈ Cb(R+) at infinity. Using theform ω, we can define the limit Limω(a) of a sequence a ∈ `∞(N) by the formula:

Limω(a) := ω(Mfa).

Hence, we come to the following definition.

Definition 36. For any state ω on the C∗-algebra B∞ = Cb(R+)/C0(R+) theDixmier trace of a positive operator T ∈ L1,∞ is defined by the formula

Trω(T ) = Limωσλ(T )

log λ.

Properties of Dixmier trace:

1. Additivity: Trω(T1 + T2) = Trω(T1) + Trω(T2).

2. Positivity: the Dixmier trace may be extended to the whole ideal L1,∞ so thatthe following property Trω(T ) ≥ 0 will hold on positive operators T ∈ L1,∞.

3. Unitary equivalence: Trω(UTU∗) = Trω(T ) for any unitary operator U .

4. Commutativity: for any bounded operator S ∈ L(H) and any T ∈ L1,∞ thefollowing inequality Trω(ST ) = Trω(TS) holds.

5. The trace Trω(T ) vanishes on the subspace L1,∞0 coinciding with the closure

of the space Fin of finite rank operators with respect to the norm ‖ · ‖1,∞. Inparticular, this trace vanishes on all nuclear operators from the space L1.

We have already explained why the introduced trace should be additive (a de-tailed proof of this fact cf. in [3], Lemma 7.14). We leave the proof of other propertiesto the reader as an exercise.

In general, the trace Trω depends on the choice of the state ω, however thefollowing proposition is true.


Proposition 13. The subspace

N = T ∈ L1,∞ : Trω(T ) does not depend on ω

is a closed linear subspace in L1,∞. This subspace contains the subspace L1,∞0 and is

closed under conjugation by invertible operators from L(H).

Definition 37. We call an operator T ∈ L1,∞ measurable if there exists the limit

limλ→∞

σλ(T )

log λ= lim

n→∞σn(T )

log n.

In this case the Dixmier trace Trω(T ), of course, does not depend on ω and wedenote it by

Tr+ T = limλ→∞

σλ(T )

log λ

in order to distinguish from the usual trace of nuclear operators.

Example 7. Consider as an example the formula for the trace of Laplace operatoron the sphere Sn provided with the standard metric. The eigenvalues of this operatorare equal to l(l + n− 1), where l is a nonnegative integer, with multiplicities ml =(

l+nn

)− (l+n−2

n

). The operator ∆−n/2 is measurable and its Dixmier trace is equal to

Tr+ ∆−n/2 =2

n!.

2.1.3 Pseudodifferential operators

Before we compute the Dixmier trace of concrete pseudodifferential operators recallbriefly their basic properties (more about pseudodifferential operators cf. in thebooks [4],[11]).

The pseudodifferential operators generalize the notion of the usual differentialoperators. Recall that a differential operator of degree d in a domain U ⊂ Rn isgiven by the formula

P (x,D) =∑

|α|≤d

aα(x)Dα

where the coefficients aα(x) are smooth functions in U . Here, α = (α1, . . . , αn) isthe multiindex α ∈ Z+ with nonnegative integer components and Dα = Dα1

1 . . . Dαnn

where Dj = −i∂/∂xj. Using the Fourier transform we can rewrite this operator inthe form

P (x,D) =1

(2π)n

∫ei(x−y)·ξp(x, ξ)f(y) dy dξ (2.3)

wherep(x, ξ) =

∑

|α|≤d

aα(x)ξα

is the symbol of operator P (x,D).In order to extend this definition to pseudodifferential operators we enlarge the

class of admissible symbols. Namely, introduce the class of symbols Sd(U) of degree


d which consists of functions p(x, ξ) ∈ C∞(U × Rn) satisfying on every compactsubset K ⊂ U the estimate: for any multiindices α, β ∈ Z+ we have

∣∣DβxDα

ξ p(x, ξ)∣∣ ≤ C

(1 + |ξ|2)

d−|α|2

for all x ∈ K, ξ ∈ Rn with the constant C, depending on α, β and K.

Definition 38. A pseudodifferential operator of degree d in a domain U ⊂ Rn isthe operator P given by Formula (2.3) with symbol p ∈ Sd(U). The space of allsuch operators is denoted by Ψd(U).

This definition implies that the operator P is correctly defined as a linear opera-tor acting continuously from the space D(U) of C∞-smooth functions with compactsupports in U to the space E(U) ≡ C∞(U) of C∞-smooth functions in the domainU . By duality it can be extended to a linear continuous operator P : E ′(U) → D′(U)acting on the distributions with compact support in the domain U . If, in particular,U = Rn, then such operator extends to a linear continuous operator acting in theSchwartz space S ′(Rn) of tempered distributions.

The kernel of operator P is a generalized function k ∈ D′(U × U) given by theintegral

k(x, y) =1

(2π)n

∫ei(x−y)·ξp(x, ξ)f(y) dξ

treated in the sense of distributions. If the kernel k ∈ C∞(U×U) then the associatedpseudodifferential operator is called smoothing and its degree is set to −∞.

It is convenient to represent the symbols of pseudodifferential operators byasymptotic series. Namely, for any sequence of symbols pk∞k=0, pk ∈ Sdk(U),where dk is a decreasing sequence of real numbers with dk → −∞ for k → ∞,there exists a symbol p ∈ Sd0(U) such that

p−n∑

k=0

pdk∈ Sdn(U) for all n = 0, 1, . . . ,

and this symbol is uniquely determined modulo the space S−∞(U) of smoothingsymbols. We use the notation: p ∼ ∑∞

k=0 pdk.

A standard class of symbols is formed by the so called classical symbols for whichthe exponents dk = d− k, and pk(x, ξ) are homogeneous functions in ξ of order dk.The asymptotic decomposition of the symbol in this case takes the form

p(x, ξ) ∼∞∑

k=0

pd−k(x, ξ).

The leading term pd(x, ξ) in this decomposition is called the principal symbol .Pseudodifferential operators form an algebra which properties are described in

the already mentioned books [4],[11]. We are especially interested in elliptic pseu-dodifferential operators defined in the following way.

Definition 39. A pseudodifferential operator P ∈ Ψd(U) is called elliptic if thereexist positive continuous functions c and C in the domain U for which the symbolof operator P satisfies the estimate

|p(x, ξ)| ≥ c(x)|ξ|d for |ξ| ≥ C(x), x ∈ U.


Elliptic pseudodifferential operators are invertible modulo smoothing operators,more precisely, we have the following

Proposition 14. A pseudodifferential operator P ∈ Ψd(U) is elliptic if and only ifthere exists a symbol q ∈ S−d(U) such that the corresponding operator Q ∈ Ψ−d(U)satisfies the relation

P Q = Q P ≡ 1 mod Ψ−∞(U).

In order to extend the definition of pseudodifferential operators to the operatorson manifolds it is necessary to study their behavior under the changes of variablesgenerated by smooth diffeomorphisms. Let ϕ : U → V be a diffeomorphism of adomain U ⊂ Rn onto another domain V ⊂ Rn. If P ∈ Ψd(U) is a pseudodifferentialoperator of degree d in the domain U then the fomula

ϕ∗P (f) := P (ϕ∗f) ϕ−1

determines a pseudodifferential operator in the domain V . In fact, we have thefollowing

Proposition 15. Suppose that an operator P ∈ Ψd(U) has the following pseudolo-cality property: both operator P and its adjoint operator P † map the space D(U)into itself. Then for a given diffeomorphism ϕ : U → V the operator Pϕ := ϕ∗Pbelongs to Ψd(V ) and has the same pseudolocality property. Moreover, the kernel pϕ

of this operator has the asymptotic decomposition of the form

pϕ(x, ξ) ∼∑

|α|≥0

1

α!qα(x, ξ)Dα

ξ (ψ(x), t(ψ′(x)−1)ξ)

where ψ := ϕ−1, q0(x, ξ) = 1 and qα(x, ξ) is a polynomial in ξ of degree≤ 12|α|.

Explicit expressions for the coefficients qα(x, ξ) may be found in the book [4](Volume III, Theorem 18.1.17), we note only that for the principal symbol pϕ,d(x, ξ)the change of variables formula has the form

pϕ,d(ϕ(x), ξ) = pd(x, tϕ′(x)ξ).

With this proposition we can define pseudodifferential operators on a compactmanifold in such a way that they will satisfy automatically the pseudolocality prop-erty from the last proposition.

Definition 40. Let M be a compact manifold. A linear operator P : D(M) →C∞(M) is called the pseudodifferential operator of degree d if its kernel is smoothaway from the diagonal in M ×M and for any coordinate chart (U,ϕ) the operatorϕ∗P , acting from D(ϕ(U)) to C∞(ϕ(U)), is a pseudodifferential operator from thespace Ψd(ϕ(U)). Such operator is called the classical pseudodifferential operator ifall its local expressions are pseudodifferential operators with classical symbols.

The above change of variables formula for the principal symbol implies thatthis symbol is correctly defined as a function on the cotangent bundle T ∗M → M .Elliptic pseudodifferential operators are defined as the operators for which all theirlocal expressions are elliptic operators.


2.1.4 Wodzicki residue

Prior the definition of Wodzicki residue we recall several auxiliary facts on homoge-neous functions and forms on the cotangent bundles of smooth manifolds.

Let M be a smooth compact manifold of dimension n > 1 and T ∗M is itscotangent bundle with local coordinates given by (x, ξ) where x = (x1, . . . , xn) arelocal coordinates on M and ξ = (ξ1, . . . , ξn) ∈ Rn are coordinates in the fibre T ∗

xM .Denote by σξ the differential (n− 1)-form on Rn \ 0 defined by

σξ :=n∑

j=1

(−1)j−1ξjdξ1 ∧ . . . ∧ dξj ∧ . . . ∧ dξn

where the ”hat” over dξj means that this term should be omitted. The form σξ

coincides with the inner product of the volume form dnξ = dξ1 ∧ . . .∧ dξn and Eulervector field E =

∑nj=1 ξj∂/∂ξj: σξ = Ey dnξ.

Lemma 11. For any homogeneous function p−n(ξ) of homogeneity degree −n theform p−nσξ on the space Rn \ 0 is closed.

Proof. Indeed,

dp−n ∧ σξ = dp−n ∧ (Ey dnξ) = Ey dp−n ∧ dnξ = −np−ndnξ

which implies that

d(p−nσξ) = dp−n ∧ σξ + p−ndσξ = −np−ndnξ + np−ndnξ = 0.

We shall use the above lemma to compute the integrals of the form∫

Sn−1 p−n|σξ|.According to this lemma, we can replace the integration space given by the unitsphere Sn−1 by any section of the bundle Rn \0 → Sn−1, the result will be the same.

Recall that according to Euler theorem, the homogeneous functions f of homo-geneity degree λ satisfy the identity

1

n + λ

n∑j=1

∂(ξjf)

∂ξj

=1

n + λ(nf + Ef) = f.

For λ = −n this identity has no sense, however the following assertion holds.

Lemma 12. The integral ∫

Sn−1

p−n|σξ| = 0

if and only if the function p−n is the sum of derivatives.

Proof. It is well known that the kernel of Laplace operator acting on the smoothfunctions on a compact manifolds, consists only of the constants. If the consideredintegral vanishes then the right hand side of the equation

∆Sn−1f = p−n|Sn−1


is orthogonal to the kernel of ∆Sn−1 so this equation should have a solution. Weextend this solution f to a function f , defined on the whole space Rn \0, by setting:f(ξ) := |ξ|−(n−2)f(ξ/|ξ|). The Laplace equation for this function will rewrite as

∆Rn f = |ξ|−np−n(ξ/|ξ|) = p−n(ξ).

Conversely, assume first that p−n is the derivative of some function, say, p−n =∂q1−n/∂ξ1 where q1−n is some homogeneous function of degree 1−n. The cycle Sn−1

may be replaced by the cycle of the form Sn−2 × R since the function q1−n, takinginto account its homogeneity degree, should tend to zero for |ξ1| → ∞. Denoting(ξ2, . . . , ξn) by η, we get

∫

Sn−1

p−n|σξ| = ±∫

Sn−2

∫ ∞

−∞

∂q1−n

∂ξ1

|dξ1||ση| = 0.

The same argument goes on in the case when p−n is the sum of derivatives withrespect to different variables.

We shall also need the notion of density on a manifold. Let us start from thecase of the real vector space V of dimension n.

Definition 41. A density on the vector space V is the continuous map λ : V n → Rhaving the following property

λ(Av1, . . . , Avn) = | det A|λ(v1, . . . , vn)

for all v1, . . . , vn ∈ V , A ∈ EndV .

If ω is the volume form on V then it determines a density |ω| on V by theformula: |ω|(v1, . . . , vn) := |ω(v1, . . . , vn)|.

The definition of densities is easily extended to arbitrary Riemannian manifoldsby the following

Lemma 13. Let (M, g) be a Riemannian manifold with Riemannian metric g. Thenthere exists a unique density |νg| on M which takes values 1 on all orthonormal basesof tangent spaces TxM , x ∈ M . If the vectors v1, . . . , vn ∈ TxM then

|νg|(v1, . . . , vn) = | det (gx(vi, vj)) |1/2.

It is natural to call this density Riemannian.

The existence of the Riemannian density follows from the fact that the bundleof densities on any smooth manifold M (with the transition functions given by themodulus of Jacobians) is trivial since there are no objections to the construction of anon-zero section (in contrast with the bundle of volume forms which is trivial only inthe case of oriented manifolds). In Riemannian case we can, using the Riemanniandensity, identify the space of all densities on the manifold M with the direct productM × R. The local expression for the Riemannian density is written in the form

√det g|dx1 ∧ . . . ∧ dxn| =

√det g|dnx|.


It follows from the change of variables formula on M that there exists a linearform

∫M

on the space of sections of the bundle of densities, defined in a unique way,which invariant under diffeomorphisms and coincides with the Lebesgue integral inlocal charts.

We can give now the definition of the Wodzicki residue. Recall that the classicalpseudodifferential operator on a compact manifold M is a linear operator P withthe symbol having local expressions of the form

p(x, ξ) ∼∞∑

k=0

pd−k(x, ξ)

where the function pd−k(x, ξ) is homogeneous in ξ of degree d− k.

Theorem 14 (Wodzicki). Suppose that P is a classical pseudodifferential operatordefined on a smooth compact manifold M of dimension n. Then there exists a densityresxP on M having local expressions of the form

resxP =

(∫

|ξ|=1

p−n(x, ξ)|σξ|)|dnx|. (2.4)

The integral determined by this density is called the Wodzicki residue of operator P :

ResP :=

∫

M

resxP. (2.5)

Proof. The change of variables formula for pseudodifferential operators gives a for-mula for the transformation of a pseudodifferential operator P ∈ Ψd(U) to thepseudodifferential operator ϕ∗P ∈ Ψd(V ) under the action of a diffeomorphismϕ : U → V from a domain U ⊂ Rn onto a domain V ⊂ Rn. The symbol p(x, ξ)of operator P under this map is transformed to the symbol pϕ(x, ξ) =: p(x, ξ) ofoperator ϕ∗P given by the formula (where ξ = tψ′(x)η):

p(x, tψ′(x)η) =∑

α

cα(x, η)∂αη p(ψ(x), η).

Here c0(x, η) = 1 and the other coefficients cα(x, η) are polynomials in η. It means, inparticular, that the coefficient p−n(x, tψ′(x)η) differs from the coefficient p−n(ψ(x), η)by the sum of the terms which are derivatives in ξ.

Let us see how the integral∫|ξ|=1

p−n(x, ξ)|σξ| changes under linear changes of

the variable ξ for a fixed x. Suppose that such change is given by a map h. It iseasy to see that h∗σξ = (det h)σhξ.

Note that the integral over the sphere S = ξ : |ξ| = 1 of the form p−n|σξ|coincides (up to sign) with the integral of this form over the image h(S):

∫

|ξ|=1

p−n(x, ξ)|σξ| = ±∫

h(S)

p−n(x, ξ)

since h(S) is homologous to S (the sign ”plus” corresponds to the case when hpreserves the orientation while ”minus” corresponds to the opposite case). Hence

∫

S

p−n(x, ξ)|σξ| = ±∫

S

h∗ (p−n(x, ξ)|σξ|) = | det h|∫

S

p−n(x, hξ)|σhξ|.


Setting y := ψ(x), ξ = tψ′(x)η, we obtain the following formula for the transformedresidue:

∫

|ξ|=1

p(x, ξ)|σξ||dnx| = | det ψ′(x)|∫

|η|=1

p−n(x, tψ′(x)η)|ση||dnx| =∫

|η|=1

p−n(x, tψ′(x)η)|ση||dny| =∫

|η|=1

p−n(y, η)|ση||dny|

where we have used the fact that the cycles |ξ| = 1 and |η| = 1 are homologous toeach other for a fixed x and the terms, consisting of derivatives, do not contribute tothe last integral. Then the latter formula implies that the density resxP is correctlydefined so the integral determined by this density does not depend on choice of localcoordinates.

Denote by P(M) the quotient of the algebra of classical pseudodifferential oper-ators on the manifold M with respect to the ideal of smoothing classical operators.The Wodzicki residue determines a trace on the algebra P(M) and (for n > 1) thistrace is uniquely defined (up to multiplication by a non-zero constant). This is as-serted by another Wodzicki theorem the proof of which can be found, for example,in [3], Theorem 7.6. It suggests the idea that Wodzicki residue should be relatedwith the Dixmier trace introduced above. We shall return to this question in thenext section.

Let us give an example of computation of Wodzicki residue for the Laplaceoperator on a compact Riemannian manifold.

Example 8. Let P = ∆ be the Laplace–Beltrami operator on a compact Rieman-nian n-dimensional manifold (M, g). Then

Res∆−n/2 = Ωn

where Ωn = 2πn/2

Γ(n/2)is the area of the sphere Sn−1.

Indeed, since ∆ is an operator of the second order the operator ∆−n/2 has theorder −n and the principal symbol of this operator has the form (gij(x)ξiξj)

−n/2

where (gij) is the metric tensor of the manifold M , (gij) is the inverse matrix of (gij).After the change of variables y = ψ(x), ξ = tψ′(x)η, for which ψ′(x) = (det g)1/2,the principal symbol will transform to |η|−n, and the residue density will be equalto

resxP = Ωn|dny| = Ωn det ψ′(x)|dnx| = Ωn|νg|where |νg| is the Riemannian density. It follows that Res∆−n/2 = Ωn.

2.1.5 Connes trace theorem

We have pointed out already that the Wodzicki residue and Dixmier trace of pseu-dodifferential operators should be related with each other. The concrete expressionfor this relation is given by the Connes trace theorem.

Theorem 15. Let P be an elliptic pseudodifferential operator of degree −n on acompact Riemannian manifold (M, g). Then the operator P belongs to the space


L1,∞ and is measurable. Moreover, its Dixmier trace is related to the Wodzickiresidue by the formula

Tr+ P =1

n(2π)nResP.

The proof of this theorem, which is omitted here, may be found in the originalbook [2] and [3], Theorem 7.18.

The Connes theorem implies the following

Corollary 6. For an arbitrary smooth function a ∈ C∞(M) the following equality∫

M

a(x)|νg| = n(2π)n

Ωn

Tr+(a∆−n/2g )

holds.

Proof. The operator a∆−n/2 is a pseudodifferential operator of degree −n with theprincipal symbol a−n(x, ξ) := a(x)(gijξiξj)

−n/2 so (cf. Example 8) the density ofWodzicki residue for this operator has the form

resx(a∆−n/2) = Ωna(x)|νg|.Hence, the left hand side of the required equality coincides with Ω−1

n Res(a∆−n/2).Now the assertion of the corollary follows from the trace theorem.

2.2 Noncommutative differential calculus

2.2.1 Universal differential algebra

Let E be a bimodule over a unital algebra A.

Definition 42. A derivation of the algebra A with values in the bimodule E is alinear map D : A → E satisfying the Leibniz rule

D(ab) = (Da)b + a(Db).

This definition immediately implies that D(1A) = 0 since D(1A) = 2D(1A).Denote by Der(A, E) the set of all derivations of the algebra A with values in E .

Any element s ∈ E determines a derivation from Der(A, E) by the formula

(ad s)a := sa− as.

Such derivation is called inner .The set Der(A) ≡ Der(A,A) of derivations of the algebra A is a Lie algebra since

the commutator of two derivations is again a derivation.Our next goal is to construct a bimodule Ω1A given together with a derivation

d : A → Ω1A which has the following universal property: for any derivation D of thealgebra A with values in the bimodule E there exists a unique bimodule morphismi : Ω1A → E making the following diagram commutative:

Ω1A

i²²ÂÂÂ

A

d==zzzzzzzz

D// E .

2.2. NONCOMMUTATIVE DIFFERENTIAL CALCULUS 69

In other words, the linear map

HomA(Ω1A, E) −→ Der(A, E)

given by the formula ϕ 7−→ ϕ d should be an isomorphism.

Construction of the bimodule Ω1A

Let A ⊗ A be the tensor product of the algebra A with itself, considered as anA-bimodule provided with the action of elements of A given on pure tensors by theformulas:

a(b⊗ c) ≡ (a⊗ 1A)(b⊗ c) = (ab)⊗ c,

(a⊗ b)c ≡ (a⊗ b)(1A ⊗ c) = a⊗ (bc).

Define a derivation d : A → A⊗A of the algebra A with values in A⊗A by theformula:

da := 1A ⊗ a− a⊗ 1A

(later on we shall omit the lower index A in the notation 1A when it does not leadto misunderstanding). Then

d(ab) = 1⊗ (ab)− (ab)⊗ 1 = a⊗ b− (ab)⊗ 1 + 1⊗ (ab)− a⊗ b = adb + (da)b

since

adb = a(1⊗ b)− a(b⊗ 1) = (a⊗ 1)(1⊗ b)− (a⊗ 1)(b⊗ 1) = a⊗ b− (ab)⊗ 1,

(da)b = (1⊗ a)b− (a⊗ 1)b = (1⊗ a)(1⊗ b)− (a⊗ 1)(1⊗ b) = 1⊗ (ab)− a⊗ b.

So d is indeed a derivation of the algebra A with values in A⊗ A.Denote by Ω1A the submodule in A⊗A generated by elements of the form adb.

It coincides with the kernel of the map

m : A⊗ A −→ A, a⊗ b 7−→ ab.

Indeed, if an element∑

k ak ⊗ bk ∈ A⊗ A belongs to Kerm, i.e.∑

k akbk = 0, then∑

k

ak ⊗ bk =∑

k

ak(1⊗ bk − bk ⊗ 1) =∑

k

akdbk

which implies the required assertion.Introduce the structure of an A-bimodule on Ω1A by setting

a(bdc) := (ab)dc, (adb)c := ad(bc)− (ab)dc.

The constructed bimodule is called the bimodule of universal 1-forms over the algebraA.

Suppose now that E be an arbitrary bimodule over the algebra A and D : A → Eis a derivation of A with values in E . Define the map i : Ω1A → E by setting itequal on pure tensors from A⊗ A to

i(a⊗ b) := a(Db)

and restricting then to Ω1A ⊂ A⊗A. This map is an A-bimodule morphism whichimplies that the bimodule Ω1A indeed has the universal property formulated in thebeginning of this section.

Differential graded algebra


Definition 43. A differential graded algebra (briefly: DG-algebra) (R•, δ) is anassociative algebra

R• =∞⊕

n=0

Rn

which is provided with a graded product , i.e. the product having the propertyRm ·Rn j Rm+n, and a differential δ, i.e. a linear map satisfying the conditions:

1. δ is a map of degree +1, i.e. it sends Rn → Rn+1,

2. δ2 = 0,

3. δ is an odd derivation, i.e. it satisfies the Leibniz rule of the form

δ(ωnη) = (δωn)η + (−1)nωnδη

where ωn ∈ Rn.

Our goal is to construct a DG-algebra

Ω•A =∞⊕

n=0

ΩnA

with differential d for which the two first summands have the form: Ω0A = A,Ω1A defined above, and the differential d extends the constructed derivation fromthe algebra A to Ω1A. Moreover, we would like to have the DG-algebra havinfthe following universal property: if (R•, δ) is another DG-algebra then any algebrahomomorphism ψ : A → R0 should extend to an algebra homomorphism ψ : Ω•A →R• of degree zero intertwining the differentials d and δ.

Denote by A the quotient algebra A := A/C and by a the image of an elementa ∈ A under the projection to A. The introduced bimodule Ω1A may be identifiedwith

Ω1A ∼= A⊗ A

by the map: a ⊗ b 7→ adb. This identification is correctly defined since d(1A) = 0.If we introduce in A⊗ A the left and right multiplication by elements c ∈ A by theformulas

c(a0 ⊗ a1) = (ca0)⊗ a1,

(a0 ⊗ a1)c = a0 ⊗ a1c− (a0a1)⊗ c

then the map A⊗ A → Ω1A becomes an isomorphism since

c(a0 ⊗ a1) = (ca0)⊗ a1 7−→ (ca0)⊗ da1,

(a0 ⊗ a1)c = a0 ⊗ a1c− (a0a1)⊗ c 7−→ a0 ⊗ da1c− (a0a1)⊗ dc = (a0da1)c.

We set now by definition

Ω2A := Ω1A⊗A Ω1A = (A⊗ A)⊗A (A⊗ A) = A⊗ A⊗ A.

More generally, define

ΩnA := Ω1A⊗A . . .⊗A Ω1A (n times)


so thatΩnA = A⊗ A⊗n.

The differential d : A⊗ A⊗n → A⊗ A⊗(n+1) is given by the shift

d(a0 ⊗ a1 ⊗ . . .⊗ an) := 1A ⊗ a0 ⊗ a1 ⊗ . . .⊗ an.

Then d2 = 0 since 1A = 0 in the algebra A.Identifying, as before, A⊗ A⊗n with (Ω1A)⊗n we shall have

a0 ⊗ a1 ⊗ . . .⊗ an = a0da1 . . . dan.

Introduce on Ω•A the structure of an A-bimodule. The left multiplication isgiven by the evident formula:

c(a0da1 . . . dan) = (ca0)da1 . . . dan.

In order to define the right multiplication we use the Leibniz rule: da·b = d(ab)−adb.Then

(a0da1 . . . dan)c = a0da1 . . . dan−1d(anc)− a0da1 . . . dan−1andc = . . .

= (−1)n(a0a1)da2 . . . danc +n−1∑j=1

(−1)n−ja0a1 . . . d(aiaj+1) . . . dandc+

+ a0da1 . . . dan−1d(anc).

At last, we define the product in Ω•A by setting:

(a0da1 . . . dan)(b0db1 . . . dbm) := (a0da1 . . . dan · b0) db1 . . . dbm.

Thus, Ω•A becomes a DG-algebra called the universal DG-algebra over the al-gebra A.

We have the following useful formulas:

d(a0da1 . . . dan) = 1Ada0da1 . . . dan = da0da1 . . . dan

anda0[d, a1] . . . [d, an] · 1A = a0da1 . . . dan.

The first of them rephrases the definition of differential using the identification ofA⊗ A⊗n with (Ω1A)⊗n, and for the proof of the one we note that

[d, an] · 1A = dan − and1A = dan,

[d, an−1]dan = d(an−1dan) = dan−1dan

and so on by induction.We check now the universal property for the constructed DG-algebra Ω•A. Sup-

pose that we are given another DG-algebra (R•, δ) and an algebra homomorphismψ : A → R0. Then its extension to a morphism ψ : Ω•A → R• is given by theformula

ψ(a0da1 . . . dan) := ψ(a0)δ (ψ(a1)) . . . δ (ψ(an)) .


2.2.2 Cycles and Fredholm modules

Cycles

Definition 44. A cycle of dimension n is the DG-algebra

Ω• =n⊕

k=0

Ωk

given together with the integral∫

, i.e. a linear map∫

: Ω• → C such that:

1.∫

ωk = 0 for k < n;

2.∫

dωn−1 = 0;

3.∫

ωkωl = (−1)kl∫

ωlωk.

A cycle over an algebra A is a cycle (Ω•, d,∫

) together with a homomorphismA → Ω0.

The standard examples of cycles of dimension n are given by the de Rhamcomplex over an n-dimensional smooth compact manifold and the algebra of smoothmatrix functions on Rn with appropriate growth condition at infinity where theintegral is given by

∫ωn :=

∫tr ωn. Less trivial examples are related to Fredholm

modules which we are going to define.

Fredholm modules and cycles defined by such modules

Definition 45. An odd Fredholm module over a C∗-algebra A is an involutive rep-resentation σ of this algebra A in a Hilbert space H provided with a symmetryoperator , i.e. a linear operator S such that S = S∗ and S2 = I, satisfying thecondition

[S, σ(a)] ∈ K(H) for all a ∈ A.

An even Fredholm module is given by a representation σ = σ0 ⊕ σ1 of the algebraA in a Z2-graded Hilbert space H = H0 ⊕ H1 with an odd symmetry operator Ssatisfying the same conditions as in the odd case.

A Fredholm module (A,H, S) generates a cycle over the algebra A with Ω0 = Awhile the symmetry operator S determines a Z2-grading on the algebra of boundedlinear operators L(H). Indeed, we can write down an arbitrary linear operator T inthe form

T = T+ + T− where T± =T ± STS

2.

Then

(TR)+ = T+R+ + T−R− and (TR)− = T+R− + T−R+.

Moreover,

T = T+ ⇐⇒ T = STS and T = T− ⇐⇒ T + STS = 0.


In order to define the integral we have to impose on the considered Fredholmmodules an additional condition of summability of order n:

[S, σ(a)] ∈ Ln+1(H),

where the number n is assumed to be odd for odd Fredholm modules and even foreven Fredholm modules.

Differentials

Assuming that the summability condition is satisfied we define the differentialby setting:

da = i[S, σ(a)] = 2iSσ(a)−

for a ∈ A. We shall omit further on the symbol ”σ” so that the last formula will bewritten in the form

da = i[S, a] = 2iSa−.

In other words, the differential d chooses the S-odd part of the element a and thesummability condition may be rewritten in the form da ∈ Ln+1.

The multiplier i is introduced in order to ensure that the differential d commuteswith the involution:

d(a∗) = (da)†

where in the right hand side we use the Hermitian conjugation.Having defined the differential d, we can also introduce the differentials of higher

orders . For that consider the space of 1-forms on the algebra of bounded linearoperators L(H). By these we mean the operators of the form a0da1, where a0, a1 ∈ A,and their linear combinations. The differential of the second order on 1-forms isdefined by the formula

d(a0da1) = i[S, a0da1] = i[S, a0]i[S, a1] = da0da1

(check that the second equality indeed holds!). Note that in this formula, as alsoin the next ones, we mean by the commutator the supercommutator so that thecommutator [S, a0da1] is in fact the anti-commutator since a0da1 is a 1-form.

From this definition it follows that

d(a0da1) = 2iS(a0da1)+,

i.e. the differential of the 2nd order, opposite to the differential of the 1st order,chooses the S-even part of the form a0da1, belonging to Ln+1 ·Ln+1 ⊂ L(n+1)/2, whilethe S-odd part of the form a0da1, equal to (a0)+(da1)− = (a0)+da1 belongs to Ln+1.(Further on we omit the prefix S speaking on the oddness or evenness of the forms.)

Consider next the space of 2-forms generated by the elements a0da1da2 wherea0, a1, a2 ∈ A. The differential of the 3rd order will choose the odd part of a0da1da2,belonging to L(n+1)/3, while the even part of a0da1da2 will belong to L(n+1)/2.

In the general case we consider the space Ωk of k-forms generated by the operators

a = a0da1 . . . dak with a0, a1, . . . , ak ∈ A.

If k = 2r, i.e. a ∈ Ω2r, then a+ ∈ L(n+1)/2r, a− ∈ L(n+1)/(2r+1). In the case whenk = 2r − 1, i.e. a ∈ Ω2r−1, we shall have: a+ ∈ L(n+1)/2r, a− ∈ L(n+1)/(2r−1).


The product of forms coincides with their composition, and the differential isgiven by the formula

d(a0da1 . . . dak) = i[S, a0da1 . . . dak] = i[S, a0[iS, a1] . . . i[S, ak]] = da0da1 . . . dak.

This implies the general formula

dω = i[S, ω] for ω ∈ Ω•.

We use now the polarization of the Hilbert space H generated by the symmetryoperator S:

H = H+ ⊕H−

whereH± is the (±1)-eigenspace of operator S. We shall write down linear operatorsacting in H in the block form so that

S =

(1 00 −1

)and a =

(a++ a+−a−+ a−−

).

Introduce also the following notations:

a+ =

(a++ 00 a−−

)and a− =

(0 a+−

a−+ 0

).

The formula for the differential of the 1st order may be rewritten in the form

da = i[S, a] = 2i

(0 a+−

−a−+ 0

).

Integral

We turn now to the definition of the integral. Introduce first of all the conditionaltrace of operator T ∈ L(H) by setting

Tr′T := Tr T+.

Note that Tr′T = Tr T if T ∈ L1 due to the cyclicity of the usual trace.Assume first that n is odd. Then (ωn)+ ∈ L1+1/n ⊂ L1 so the integral

∫ωn := Tr′ωn = − i

2Tr(Sdωn)

is well defined. The second equality in this formula is implied by the following chainof equalities

Sdωn = iS[S, ωn] = iS(Sωn + ωnS) = i(ωn + SωnS) = 2i(ωn)+.

For the forms ωk with k < n we set∫

ωk = 0 by definition.Let us show that the constructed integral has the properties listed in Definition

44. First of all ∫dωn−1 = − i

2Tr(Sd2ωn−1) = 0.


Secondly, consider the forms ωk, ωl with k + l = n. Assume for definiteness thatk is odd and l is even. Then

∫ωkωl = − i

2Tr(Sd(ωkωl)) =

= − i

2Tr(Sdωkωl − Sωkdωl) = − i

2Tr(−dωlSωk − ωlSdωk) =

= − i

2Tr(Sdωlωk + Sωldωk) = − i

2Tr(Sd(ωlωk)) =

∫ωlωk.

In the third equality we have used the cyclicity property: Tr(TR) = Tr(RT ) foroperators T ∈ Lp, R ∈ Lq with 1/p + 1/q = 1. In our case p = k/(n + 1),q = (l + 1)/(n + 1).

Assume now that n is even and again k + l = n. Then (ωn)− ∈ L1. Denote by χthe grading operator on H having the (±1)-eigenspaces coinciding respectively withH0 and H1. In this case we define the integral as

∫ωn := Tr′(χωn) = − i

2Tr(χSdωn)

and set:∫

ωk = 0 for forms ωk with k < n. The property of closedness is againevident: ∫

dωn−1 = − i

2Tr(χSd2ωn−1) = 0,

and the permutation property

∫ωkωl = (−1)kl

∫ωlωk

is checked as above taking into account the equality: χωk = (−1)kωkχ.

Example 9 (Hilbert transform). The Hilbert transform of a function h ∈ L2(R),defined on the real line, is the integral of the form

Sh(x) :=i

πlim

ε→0+

∫

|t|>ε

h(x− t)

tdt =:

i

πP.V.

∫

R

h(x− t)

tdt.

The Fourier transform of this function coincides with

F(Sh)(ξ) = (sgn ξ)Fh(ξ).

The Hilbert transform is the only linear bounded operator in L2(R) which commuteswith translations and dilatations.

Example 10 (Riesz operators). The Riesz operators are the multi-dimensional ana-logues of the Hilbert transform. The Riesz operators Rj, 1 ≤ j ≤ n, act in L2(Rn)by the formula

Rjh(x) :=2i

Ωn+1

P.V.

∫

Rn

tjh(x− t)

|t|n+1dt


where Ωn+1 is the volume of the unit sphere Sn equal to

Ωn+1 =2π(n+1)/2

Γ(n+12

).

The Fourier transform of this function is equal to

F(Rjh)(ξ) =ξj

|ξ|Fh(ξ).

The Riesz operators also commute with translations and dilatations, moreover

n∑j=1

R2j = 1.

The Fredholm module, associated with Riesz operators, is constructed from thecollection of (N ×N)-matrices γ1, . . . , γn such that

γiγj + γjγi = 2δij.

These matrices γj coincide with the Dirac matrices generating the spin representa-tion of the Clifford algebra ClC(Rn) in the space CN where N = 2[n/2]. Having suchcollection of matrices γj, we can define the symmetry operator by the formula

S :=n∑

j=1

γjRj.

The matrices γj can be constructed in the following way. For n = 1 we set

γ(1)1 = 1 and for all odd n > 1 define the collections of functions γ

(n)j by induction

setting:

γ(n)j :=

(0 γ

(n−2)j

γ(n−2)j 0

)for j = 1, . . . , n− 2,

and

γ(n)n−1 :=

(0 −ii 0

), γ(n)

n :=

(1 00 −1

).

In particular, for n = 3 we get the Pauli matrices. For even n we set: γ(n)j := γ

(n+1)j ,

j = 1, . . . , n.


2.2.3 Connections

Definition and existence of connections

Definition 46. Let E be a right A-module over an algebra A. Consider the rightA-module E ⊗A Ω1A. A connection on E is a linear map

∇ : E −→ E ⊗A Ω1A

satisfying the Leibniz rule:

∇(sa) = (∇s)a + s⊗ da

where s ∈ E , a ∈ A.

The operator, determined by the connection ∇, uniquely extends to an operatorof degree +1 on the whole graded algebra E ⊗A Ω•A by the formula:

∇(s⊗ ω) = ∇s⊗ ω + s⊗ dω

where s ∈ E , ω ∈ Ω•A and we identify (E ⊗A Ω1A)⊗A ΩnA with E ⊗A Ωn+1A.Considering E ⊗A Ω•A as a right (Ω•A)-module we obtain the Leibniz rule of the

form∇(σω) = (∇σ)ω + (−1)kσdω

where σ ∈ E ⊗A ΩkA, ω ∈ Ω•A.

Proposition 16. A right A-module admits a connection if and only if it is projective.

Proof. Consider the exact sequence of right A-modules

0 −→ E ⊗A Ω1Aj−→ E ⊗C A

m−→ E −→ 0

where j(s⊗da) = s⊗a−sa⊗1A, m(s⊗a) = sa and E⊗CA is considered as a free A-module with the basis given by a basis of E . With any linear map ∇ : E → E⊗AΩ1Awe can associate the linear map f which is the right inverse of the map m definedby the formula

f(s) := a⊗ 1A + j(∇s).

We look for the condition under which this map is a homomorphism of A-modules.We have the following formula

f(sa)− f(s)a = j (∇(sa)−∇s · a− s⊗ da) .

Indeed,f(sa) = sa⊗ 1A + j (∇(sa)) ,

f(s)a = (s⊗ 1A)a + j(∇s)a = sa⊗ 1A + j(∇s)a.

But (∇s)a = ∇s · a + s⊗ da which implies that

f(sa)− f(s)a = j (∇(sa)−∇s · a− s⊗ da) .

So the map f is a homomorphism of A-modules if ∇ satisfies the Leibniz rule. If thisrule is fulfilled than the map f splits the above exact sequence which implies that themodule E is a right summand in the free A-module E ⊗CA and so is projective.


Examples of connections

Note, first of all, that on the tensor product of two A-modules E and F over acommutative algebra A, provided with connections ∇E and ∇F respectively, we candefine the connection which is the tensor product of connections ∇E and ∇F . Thisconnection is defined by the formula

∇E⊗AF := ∇E ⊗ 1F + 1E ⊗∇F .

Example 11 (connection on An). Recall that we denote by An the free A-moduleconsisting of columns with entries from A. Then An⊗AΩ1A is identified with (Ω1A)n

andd t(a1 . . . an) := t(da1 . . . dan).

If ∇ is a connection on An then ∇− d is a A-linear map from An to (Ω1A)n, hence∇ may be written in the form

∇ = d + α

where α is an (n×n)-matrix with entries from Ω1A. If ujnj=1 is the standard basis

in An then duj = 0 so ∇uj =∑n

j=1 ujαij. The change of basis ujnj=1 7→ ujn

j=1,determined by the matrix b = (bij) according to the formula uj =

∑nj=1 uibij, leads

to the replacement of the matrix α by the matrix α equal to

α = b−1αb + b−1db.

Example 12 (Levi-Civita connection). Let E = eAn be a finitely generated projec-tive A-module and connection ∇ is given by the composition

E i−→ An d−→ An ⊗A Ω1Ae−→ E ⊗A Ω1A

where i : E → An. Identifying E with a submodule in An, we write down theintroduced connection in the form

∇s = e ds.

The constructed connection is called the Levi-Civita connection.

Example 13 (Hermitian connections). Working with C∗-modules it is natural touse the connections ∇ compatible with the inner product (· , ·), i.e. satisfying thecondition

(∇s, t) + (s,∇t) = d(s, t)

for all s, t ∈ E . Here we suppose that the inner product (· , ·) is extended to E⊗AΩ1Aas a sesquilinear pairing with values in Ω1A by the formula:

(s, t⊗ adb) := (s, t)adb.

In the case of Levi-Civita connection this means that the corresponding idempotente should be selfadjoint, i.e. a projector.

The difference ∇ = ∇1 −∇2 of two Hermitian connections on E belongs to thespace of homomorphisms HomA(E , E ⊗A Ω1A) and is a skew-Hermitian map, i.e.

(∇s, t) + (s,∇t) = 0.


If E ∼= pAm, where p is a projector in Matm(A), then

pAm ⊗A Ω1A⊗AmAp = pMatm(Ω1A)p.

The involution in Ω1A is given by the formula (adb)∗ := d(b∗)a∗ = d(b∗a∗)− b∗da∗.So the skew-Hermitian operator α ∈ HomA(E , E ⊗A Ω1A) may be identified with amatrix α ∈ Matm(Ω1A) consisting of 1-forms so that

α = pα = αp = pαp

where α∗ = −α. Then the Hermitian connection ∇ will be written in the form∇ = pd + α where α satisfies the above conditions.

Curvature of a connection

Consider the linear map

∇2 : E ⊗A Ω•A −→ E ⊗A Ω•+2A.

It satisfies the relation

∇2(sω) = ∇(∇s ω + sdω) = (∇2s)ω −∇s dω +∇s dω + sd2ω = (∇2s)ω

which means that ∇2 is a homomorphism of (Ω•A)-modules which is completelydetermined by its restriction to E . This homomorphism is called the curvature ofthe connection ∇ and is denoted by K∇.

Let us compute the curvature of the connection d+α in the free module An. Wehave

K∇s = ∇(ds + αs) = d2s + d(αs) + αds + α2s =

= dα s− αds + αds + α2s = (dα + α2)s.

Lemma 14. Let E0 and E1 be projective modules over a commutative algebra Aprovided with connections ∇0 and ∇1 with curvatures K0 and K1 respectively. Thenthere is an associated connection ∇ in the A-module HomA(E0, E1) given by theformula

(∇T )s := ∇1(Ts)− T (∇0s)

with curvature∇2T = K1T − TK0.

Proof. Rewrite the formula for ∇ in the form

∇1(Ts) = (∇T )s + T (∇0s).

Using this formula, it is easy to show that ∇ satisfies the Leibniz rule. Moreover,the same formula implies that

∇1 ((∇T )s) = (∇2T )s−∇T (∇0s).

Hence

(∇2T )s = ∇1 ((∇T )s) +∇T (∇0s) =

= ∇1 (∇1(Ts)− T (∇0s)) +∇1 (T (∇0s))− T (∇0s) =

= ∇21Ts− T∇2

0s = (K1T − TK0)s.


In particular, any connection ∇ in E in the case of a commutative algebra Agenerates a connection in EndAE given by the formula

∇T := ∇ T − T ∇.

2.2.4 Chern character

Let M be a smooth compact manifold and E → M is a vector bundle over M .Denote by Γ∞(M, E) ≡ Γ∞(E) the module of smooth sections of this bundle. Sincethis bundle may be embedded as the right summand into the trivial bundle of rankN over M this module may be represented in the form p [C∞(M)]N where p is aprojector in the free module [C∞(M)]N .

We provide M with a Riemannian metric g and denote by R the curvature ofthis metric. If s ∈ Γ∞(E) is a smooth section of the bundle E → M , i.e. ps = s,then

Rs = (∇g)2s = (pd)(pd)s = pdpds.

Differentiating the relation p2 = p, we obtain: pdp + dp p = dp, whence

pdp = dp(1− p) and dp p = (1− p)dp

so that pdp p = 0. If s = ps then ds = dp s+pds which implies that dp s = (1−p)ds.From these relations we get

Rs = pdpds = dp(1− p)ds = dpdp s = dpdp ps.

Hence,R = dpdp p = dp(1− p)dp = pdpdp.

Definition 47. The Chern character of a projector p ∈ Matm(A) over a commuta-tive algebra A is the following quantity

ch p := tr(exp R) =∞∑

k=0

ch2k(p) :=∞∑

k=0

1

k!tr p(dp)2k.

In the case when A coincides with C∞(M) we can consider dp as a matrix consistingof 1-forms, i.e. dp ∈ MatN(Ω1(M)), so that every term ch2k(p) ∈ Ω2k(M), inparticular, the sum in the definition of the character is finite.

Proposition 17. The Chern character in the case of algebra A = C∞(M) representsa de Rham cohomology class.

Proof. For the proof of proposition we should check that all forms ch2k(p) are closed.Indeed,

d(tr p(dp)2k

)= tr (dp)2k+1 = tr

(p(dp)2k+1

)+ tr

((1− p)(dp)2k+1

).

Both terms in the right hand side are equal to zero. For instance,

tr(p(dp)2k+1

)= tr

(p2(dp)2k+1

)= tr

(p(dp)2k+1p

)= tr

(p(1− p)(dp)2k+1

)= 0.


Proposition 18. The Chern class [ch2k(p)] depends only on the class [p] in thegroup K0(A).

Proof. It is sufficient to show that the map p 7→ ch2k(p) is homotopy invariant.Indeed, let pt, 0 ≤ t ≤ 1, is a smooth family of projectors. Denote: pt := dpt

dt. We

have to show that the form

d

dttr

(pt(dpt)

2k)

= tr(pt(dpt)

2k)

+ tr

(pt

d

dt(dpt)

2k

)

is exact.Let us show that the first summand in the right hand side vanishes. Indeed, as in

the formulas in the beginning of this section, we have: ptpt = pt(1− pt) ptptpt = 0.Hence the expressions in the right hand side of the following formula

tr(pt(dpt)

2k)

= tr(ptpt(dpt)

2k)

+ tr((1− pt)pt(dpt)

2k)

vanish. For instance,

tr(ptpt(dpt)

2k)

= tr(ptpt(dpt)

2kpt

)= tr

((1− pt)ptpt(dpt)

2k)

= 0.

On the other side,

tr

(pt

d

dt(dpt)

2k

)=

2k−1∑j=0

tr

(pt(dpt)

j d

dt(dpt)(dpt)

2k−j−1

)=

=∑.j

tr

((dpt)

2k−1ptd

dt(dpt)

)+

∑.j

tr

((dpt)

2k−1(1− pt)d

dt(dpt)

)=

2ktr

((dpt)

2k−1 d

dt(dpt)

)=

d

dttr(dpt)

2k,

i.e. in the right hand side we have an exact form which implies the required assertion.

Proposition 19. The Chern character is the ring homomorphism from K0(C∞(M))

to HevdR(M).

Proof. We have to check that for arbitrary vector bundles E, F over M the followingproperties

ch(E ⊕ F ) = ch E + ch F, ch(E ⊗ F ) = (ch E))(ch F )

hold. The first equality follows from

ch2k(p⊕ q) =1

k!tr

(p(dp)2k ⊕ q(dq)2k

)= ch2k(p) + ch2k(q).

To prove the second equality we use the evident relation

Γ∞(E ⊗ F ) ∼= (p⊗ q)Amn

where A = C∞(M). Note that the curvature of the tensor product ∇E⊗F = ∇E ⊗1 + 1⊗∇F of connections ∇E on E and ∇F on F is equal to K∇E ⊗ 1 + 1⊗K∇F .Hence

ch(p⊗ q) = tr (exp(pdpdp⊗ 1 + 1⊗ qdqdq)) = (ch p)(ch q).


2.2.5 Hochschild homology and cohomology

Chain complexes

Let (C•, d) be a chain complex of Abelian groups.

Definition 48. A chain map f : (C•, d) → (C ′•, d

′) from a chain complex (C•, d)to another chain complex (C ′

•, d′) is a family of maps fn : Cn → C ′

n making thefollowing diagram commutative:

Cndn−−−→ Cn−1

fn

yyfn−1

C ′n −−−→

d′n−1

C ′n−1 .

The maps fn send cycles to cycles and boundaries to boundaries, hence theyinduce homomorphisms Hnf : Hn(C) → Hn(C ′).

Definition 49. A chain homotopy between two chain maps f, g : (C•, d) → (C ′•, d

′)is a sequence of maps hn : Cn → C ′

n+1 satisfying the relations

hn−1dn + d′n+1hn = fn − gn

which may be written in the concise form: hd + d′h = f − g.

If the maps f, g are chain homotopic then Hnf = Hng since the closedness ofthe chain c: dc = 0, implies the equality: f(c)− g(c) = d′h(c).

Definition 50. A chain complex (C•, d) is called acyclic if

Hn(C) = 0 for n > 0.

Let now (C•(A), b) be a chain complex of algebras Cn(A) = A⊗(n+1), where A isa unital algebra, with the boundary map b given on Cn(A) by the formula

bn(a0 ⊗ . . .⊗ an) =n−1∑j=0

(−1)ja0 ⊗ . . .⊗ ajaj+1 ⊗ . . .⊗ an+

(−1)nana0 ⊗ a1 ⊗ . . .⊗ an−1 (2.6)

and b0 = 0 on C0(A) = A. For instance, b1(a0 ⊗ a1) = a0a1 − a1a0 and

b2(a0 ⊗ a1 ⊗ a2) = a0a1 ⊗ a2 − a0 ⊗ a1a2 + a2a0 ⊗ a1.

It is easy to show that bnbn−1 = 0 for any n.

Hochschild homology

Definition 51. The Hochschild homology of an algebra A is the homology of thecomplex (C•(A), b) denoted by H•(A,A) or HH•(A).

Lemma 15. HH0(C) = C, HHn(C) = 0 for n > 0.


Proof. For the algebra A = C we have: Cn(C) = C⊗(n+1) ∼= C and a0⊗ a1⊗ . . .⊗ an

coincides with the usual product a0a1 . . . an. The boundary map is determined bythe formula:

bn(1) =n∑

j=0

(−1)j =

1 for even n

0 for odd n.

Thus, in this case the Hochschild complex reduces to the exact sequence

. . .1−→ C 0−→ C 1−→ C 0−→ C

and so has trivial homology in all dimensions except for n = 0.

In analogous way one can define the Hochschild homology of an algebra A withvalues in an arbitrary A-bimodule E . For that it is sufficient to put Cn(A, E) :=E ⊗A An. The homology of the obtained complex are denoted by H•(A, E).

Any homomorphism of algebras f : A → A′ generates the chain map and homo-morphism HH•f : HH•(A) → HH•(A′) of zero degree. In other words, HHn is afunctor from the category of unital algebras into the category of vector spaces.

The chains of the form a = a0 ⊗ . . . ⊗ 1A ⊗ . . . ⊗ an with ak = 1A for some kgenerate a subcomplex D•A since a ∈ DnA implies that bna ∈ Dn−1A (why?). Weintroduce the boundary maps b′n given by the formula

b′n(a0 ⊗ . . .⊗ an) =n−1∑j=0

(−1)ja0 ⊗ . . .⊗ ajaj+1 ⊗ . . .⊗ an

which is obtained from the Formula (2.6) for the boundary map bn by cancellingthe last term. Taking the composition of b′n+1 with the map s0 : a0 ⊗ . . . ⊗ an 7→1A ⊗ a0 ⊗ . . .⊗ an, we get the formula

b′n+1s0(a0⊗ . . .⊗an) = a0⊗ . . .⊗an +n−1∑j=0

(−1)j+11A⊗a0⊗ . . .⊗ajaj+1⊗ . . .⊗an =

= (1− s0b′n)(a0 ⊗ . . .⊗ an).

In particular, it implies that (bn+1s0 + s0bn)a = a for any a ∈ DnA with an = 1.Applying in analogous way the map sj, inserting the unit 1A at the jth place ina0 ⊗ . . . ⊗ an, we obtain that (bn+1sj + sjbn)a = a for any a ∈ DnA with aj = 1.Thus, the family of maps sj determines a chain homotopy between the zero andidentity maps on Dn(A). Hence, the complex D•(A) is acyclic, i.e. Hn(D•(A), b) = 0for n > 0.

Using that we can define the quotient complex C•(A)/D•(A) coinciding in factwith Ω•(A). The boundary operator b on Ω•(A) takes on the following form

bn(a0da1 . . . dan) = a0a1da2 . . . dan +n−1∑j=0

(−1)ja0da1 . . . d(ajaj+1) . . . dan+

(−1)nana0da1 . . . an−1.

Otherwise, it can be written as

bk(ωkda) = (−1)k[ωk, a] (2.7)

for ωk ∈ ΩkA, a ∈ A.


Example 14 (homology H0). H0(A, E) = E/[E , A] since the boundary map b : E ⊗A → E is given by the formula: b(a⊗a) = sa−as. In particular, HH0(A) = A/[A,A]which coincides with A if the algebra A is commutative. If E is a symmetric bimoduleover a commutative algebra A then H0(A, E) = E .

Hochschild cohomology

Definition 52. A Hochschild n-cochain on an algebra A is an (n + 1)-linear func-tional on the algebra A or n-linear form on A with values in the dual space A′. Notethat A′ is an A-bimodule with respect to the operation

ϕ ∈ A′ 7−→ (bϕc)(a) := ϕ(cab).

The coboundary operator b is dual to the homology boundary operator:

bnϕ(a0, . . . , an+1) =n∑

j=0

(−1)jϕ(a0, . . . , ajaj+1, . . . , an+1)+

+ (−1)n+1ϕ(an+1a0, . . . , an).

The cohomology of the obtained cochain complex are called the Hochschild coho-mology of the algebra A and denoted by HH•(A) or H•(A,A′).

In particular, a 0-cocycle τ on the algebra A coincides with the trace sinceτ ∈ A′ = Hom(A,C) and

τ(a0a1)− τ(a1a0) =: b1τ(a0, a1) = 0.

In a more general way, one can define the Hochschild cohomology of the algebra Awith values in an arbitrary A-bimodule E . For that we denote by Cn(A, E) the vectorspace of n-linear maps ϕ : An → E considered as an A-bimodule with respect tothe operation: (bϕc)(a1, . . . , an) := bϕ(a1, . . . , an)c where b, c ∈ A. The coboundarymap in this case is given by the formula

bnϕ(a1, . . . , an+1) = a1ϕ(a2, . . . , an+1) +n∑

j=0

(−1)jϕ(a1, . . . , ajaj+1, . . . , an+1)+

+ (−1)n+1ϕ(a1, . . . , an)an+1.

Definition 53. An n-cochain ϕ on an algebra A is called cyclic if λϕ = ϕ where

λϕ(a0, . . . , an) := (−1)nϕ(an, a0, . . . , an−1).

For example, a cyclic 1-cocycle ϕ satisfies the relations: ϕ(a0, a1) = −ϕ(a1, a0)

ϕ(a0a1, a2)− ϕ(a0, a1a2) + ϕ(a2a0, a1) = 0,

while a cyclic 1-coboundary ϕ = bψ is determined by the equality

ϕ(a0, a1) = bψ(a0, a1) = ψ([a0, a1]),

i.e. it is a linear function of commutators.

Chern character


Definition 54. Suppose that it is given an n-dimensional cycle (Ω•, d,∫

) over analgebra A. Its Chern character is an (n + 1)-linear functional on A given by theformula

τ(a0, . . . , an) :=

∫a0da1 . . . dan.

Note first of all that τ is a cocycle, i.e. bτ = 0. Indeed,

∫ n∑j=0

(−1)ja0da1 . . . d(ajaj+1) . . . dan+1) + (−1)n+1

∫an+1a0da1 . . . dan =

= (−1)n

∫(a0da1 . . . dan)an+1 + (−1)n+1

∫(an+1a0da1 . . . dan) = 0

since∫

aωn =∫

ωna for any a ∈ A, ωn ∈ Ωn.We note next that the cocycle τ is cyclic since

τ(a0, a1, . . . , an) = (−1)n−1

∫dana0da1 . . . dan−1 =

= (−1)n

∫anda0da1 . . . dan−1 = (−1)nτ(an, a0, . . . , an−1).

Moreover, τ(1, a1, . . . , an) =∫

da1 . . . dan = 0.

Proposition 20. An (n + 1)-linear functional τ : An+1 → C, vanishing on C⊕An,is a cyclic n-cocycle if and only if it coincides with the Chern character of somen-dimensional cycle over A.

Proof. We have shown already that the Chern character of n-dimensional cycle overA has these properties. Conversely, if an (n + 1)-linear functional τ : An+1 → C is acyclic cocycle, vanishing on C ⊕ An, then we can construct an n-dimensional cycleover A, for which this cocycle will coincide with its Chern character, in the followingway.

Let Ω• =⊕n

k=0 ΩkA be the universal DG-algebra with the universal differentiald on ΩkA for k < n. We extend this definition to k = n by setting: d|ΩnA = 0. Wedefine next the integral

∫: ΩnA → C by setting

∫a0da1 . . . dan := τ(a0, a1, . . . , an).

We shall prove that this integral indeed determines the Chern character by showingthat (Ω•, d,

∫) is an n-dimensional cycle over A.

We have: ΩnA = A ⊗ A⊗n which implies that the form a0da1 . . . dan does notchange if some of its elements aj with 1 ≤ j ≤ n is replaced by aj + λj1A withλj ∈ C. In order to show that the introduced integral is correctly defined we have tocheck that τ(a0, a1, . . . , an) = 0 if one of the elements aj = 1. But this follows fromthe relation τ(1, a1, . . . , an) = 0 which is satisfied by the hypothesis and cyclicity ofτ . The same relation shows that

∫da1 . . . dan = 0 (closedness of

∫).

It remains to check the permutation property:∫

ωkωn−k = (−1)k(n−k)

∫ωn−kωk.


Consider first the case when k = n assuming that ω0 =: a ∈ A. By Formula (2.7)we have:

ωna− aωn = [ωn, a] = (−1)nb(ωnda).

Since bτ = 0 it implies that∫

ωna =∫

aωn.If ωn−1 ∈ Ωn−1A and da ∈ Ω1A then

ωn−1da− (−1)n−1daωn−1 = [ωn−1, da] =

= (−1)n−1(d[ωn−1, a]− [dωn−1, a]

)= (−1)n−1d[ωn−1, a] + b(dωn−1da).

Since bτ = 0 and the integral∫

is closed we have

∫ωn−1da = (−1)n−1

∫daωn−1.

Using successively these two cases we show also in the general case that

∫ωn−ka0da1 . . . dak = (−1)n−1

∫dakω

n−kda1 . . . dak−1 = . . .

(−1)k(n−1)

∫a0da1 . . . dakω

n−k = (−1)k(n−k)

∫a0da1 . . . dakω

n−k.

Chapter 3

SPINOR GEOMETRY

3.1 Spinor algebra

3.1.1 Clifford algebras

Definition

Let V be an n-dimensional Euclidean space provided with an orthonormal basisein

i=1.

Definition 55. A Clifford algebra is an associative algebra Cl(V ) over the field Rwith generators 1, e1, . . . , en satisfying the relations

e2i = −1, eiej + ejei = 0 for i 6= j.

We shall also denote it by Cl(n).

It follows from the given definition that V ⊂ Cl(V ) and

uv + vu = −2(u, v) for u, v ∈ V.

As a real vector space, Cl(V ) has dimension 2n and can be provided with the basisgiven by 1 and elements of the form

eI := ei1 · ei2 · . . . · eik

where I = i1, i2, . . . , ik is a strictly increasing subset of indices with |I| := kelements taken from the set 1, 2, , . . . , n, i.e. 1 ≤ i1 < . . . < ik ≤ n. In particular,any element x ∈ Cl(V ) can be written in the form

x =∑

I

xIeI

where we add to the collection I of sets of indices the subset I = 0 and put e0 := 1.Using this representation, we can introduce a natural inner product on Cl(V )

defined by the formula

(x, y) :=∑

I

xIyI

87

88 CHAPTER 3. SPINOR GEOMETRY

(which does not depend on the choice of the orthonormal basis eini=1).

Denote by Clk(V ) the subset Cl(V ) consisting of elements of degree k which arelinear combinations of basis elements eI with |I| = k (assuming that I = 0 fork = 0). We introduce also the following subsets of Cl(V ):

Clev(V ) :=⊕

k even

Clk(V ), Clod(V ) :=⊕

k odd

Clk(V ).

Then Clev(V ) will be a unital subalgebra in Cl(V ) and

Cl(V ) = Clev(V )⊕ Clod(V )

which provides Cl(V ) with the structure of a superalgebra.

Universal property

The notion of the Clifford algebra Cl(V ) does not depend in fact on the choice ofthe orthonormal basis ein

i=1 due to the following universal property of this algebrawhich may be taken for its definition.

Proposition 21. The Clifford algebra Cl(V ) is a unique associative R-algebra withunit which contains the Euclidean space V and has the following property: for anyassociative R-algebra A with unit 1A and any linear map f : V → A, satisfying thecondition

f(v) · f(v) = −|v|2 1A,

there exists a unique extension of f to an algebra homomorphism f : Cl(V ) → Asuch that the following diagram

V

i²²

f // A

Cl(V )f

<<yy

yy

is commutative.

Proof. To prove the formulated universal property of the Clifford algebra we use thefollowing equivalent definition of this algebra. Denote by

T (V ) =∞⊕

k=0

V ⊗k

the tensor algebra of the space V and consider the ideal J (V ) of this algebra gen-erated by the elements of the form

v ⊗ v + |v|2 · 1.

The Clifford algebra Cl(V ) coincides with the quotient of the tensor algebra T (V )by this ideal:

Cl(V ) ∼= T (V )/J (V )

3.1. SPINOR ALGEBRA 89

(check this assertion!).Return to the proof of the universal property of the algebra Cl(V ). Any linear

map f : V → A extends uniquely to an algebra homomorphism

f : T (V ) −→ A.

By assumption, this homomorphism vanishes on the ideal J (V ) so f can be pusheddown to an algebra homomorphism f : Cl(V ) → A.

Examples of Clifford algebras

1. Cl(R) = C with e1 = i.

2. Cl(R2) = H with e1 = i, e2 = j, e1e2 = k.

3. Cl(R4) = Mat2(H) is the space of quaternion 2× 2-matrices.

Multiplicative group

Denote by Cl×(V ) the group of invertible elements of the Clifford algebra Cl(V ). Itis a Lie group which contains V \ 0 since for any element v ∈ V \ 0 the inverseelement v−1 can be given by the formula

v−1 = − v

|v|2 .

The Lie algebra of the group Cl×(V ) is the algebra cl(V ) which coincides as aset with Cl(V ) and is provided with the Lie bracket of the form

[x, y] := xy − yx.

The group Cl×(V ) acts on the algebra Cl(V ) by the adjoint representation

w 7−→ Adwx := wxw−1, w ∈ Cl×(V ).

The differential of this action is a Lie algebra homomorphism

ad : cl(V ) −→ Der Cl(V )

from the algebra cl(V ) to the algebra Der Cl(V ) of derivations of Cl(V ) given by theformula

adyx := [y, x], y ∈ cl(V ), x ∈ Cl(V ).

For any u ∈ V \ 0 the map

−Aduv = v − 2(u, v)

|u|2 u , v ∈ V,

is the reflection of V with respect to the hyperplane u⊥ orthogonal to u. In orderto get rid of the minus sign in the left hand side of the last equality it is convenientto use, instead of the adjoint representation Ad, the action of the group Cl×(V ) onthe algebra Cl(V ) given by the twisted adjoint representation of the form

w 7−→ πw(x) := χ(w)xw−1, x ∈ Cl(V ), w ∈ Cl×(V )


where χ is the grading map defined on the homogeneous elements of degree k fromthe group Cl×(V ) by the formula

χ(w) := (−1)deg ww = (−1)kw.

Then the map πu : V → V , determined by an element u ∈ V \ 0, will coincidewith the reflection with respect to the hyperplane u⊥.

Taking into account these remarks, we can consider the subgroup of multiplica-tive group Cl×(V ) consisting of the elements x ∈ Cl×(V ) which have the property:πx(V ) = V . As it was pointed out before, this property has any element v ∈ V \0so it is natural to introduce the following group.

Definition 56. The Clifford group Γ(V ) ≡ Γ(n) is the subgroup of multiplicativegroup Cl×(V ) generated by the elements v ∈ V \ 0.

Every element of the group Γ(V ) generates a non-degenerate linear transform ofthe space V so we have a homomorphism

π : Γ(V ) −→ GL(V ).

This homomorphism takes values in the orthogonal group O(V ). Indeed, since anyelement x ∈ Γ(V ) may be represented as the product x = v1 · . . . vk, where vi ∈ V \0, the corresponding transform πx is the composition of reflections associated withelements vi, i.e. belongs to O(V ). Moreover, the homomorphism π : Γ(V ) → O(V )is an epimorphism since any orthogonal transform is the composition of reflections.

The homomorphism π : Γ(V ) → O(V ) may be included into the exact sequenceof group homomorphisms of the form

1 −→ R× −→ Γ(V )π−→ O(V ) −→ 1.

We set alsoSΓ(V ) := Γ(V ) ∩ Clev(V ).

Generalization of the Clifford construction to other fields and quadraticforms

The given definition of Clifford algebras immediately extends to the case of vectorspaces V over an arbitrary field K (we consider here only the fields K = R,C)provided with a non-degenerate symmetric bilinear form B.

In this case the Clifford algebra Cl(V, B) is defined again as

Cl(V, B) = T (V )/J (V, B)

where the ideal J (V, B) is generated by the elements of the form

u⊗ v + v ⊗ u + 2B(u, v) · 1 where u, v ∈ V.

Otherwise, we can define Cl(V, B) as an associative algebra with unit generated bythe elements u ∈ V satisfying the relations

uv + vu + 2B(u, v) · 1 = 0.


This algebra has again the universal property: if A is another associative algebrawith unit 1A and f : V → A is an arbitrary linear map with the property:

f(u)f(v) + f(v)f(u) = −2B(u, v) · 1A

for all u, v ∈ V , then this map extends uniquely to an algebra homomorphismf : Cl(V, B) → A.

Complexified Clifford algebra

The construction, presented in the previous section, is interesting for us, first ofall, in the case when K = C and the complex vector space is provided with anon-degenerate symmetric bilinear form (defined uniquely up to multiplication by anonzero complex number).

We introduce also the complexified Clifford algebra Cl(V ) of an n-dimensionalreal vector space V by setting

Cl(V ) := Cl(V )⊗R C.

This algebra is isomorphic to the Clifford algebra Cl(V C) of the complexified vectorspace V C := V ⊗R C provided with the complexified quadratic form.

3.1.2 Spinor groups

The group Pin

Definition 57. The group Pin(V ) is defined as the subgroup of the Clifford groupΓ(V ) generated by the unit vectors from V , i.e. by vectors v ∈ V with |v| = 1.

As in the case of the Clifford group, we have a homomorphism

π : Pin(V ) −→ O(V )

which is included into the exact sequence of group homomorphisms

1 −→ Z2 −→ Pin(V )π−→ O(V ) −→ 1.

The group Spin

Definition 58. The group Spin(V ) is the identity connected component of the groupPin(V ). It can be also defined as

Spin(V ) = Pin(V ) ∩ Clev(V ).

As in the case of the group Pin(V ), there is an exact sequence of group homo-morphisms

1 −→ Z2 −→ Spin(V )π−→ SO(V ) −→ 1.

For n > 2 the group Spin(n) is a simply connected covering group of the groupSO(V ).


Examples of Spin-groups

1. Spin(2) = U(1) = SO(2).

2. Spin(3) = SU(2).

3. Spin(4) = SU(2)× SU(2).

3.1.3 Relation to the exterior algebra

Definition

The exterior algebra

Λ(V ) =n⊕

k=0

Λk(V )

of the n-dimensional real vector space V may be also defined as the quotient of thetensor algebra T (V ) by the ideal I(V ) generated by the elements of the form

u⊗ v + v ⊗ u.

Otherwise, Λ(V ) is an associative unital algebra, containing V , with the product ∧satisfying the relation

u ∧ v + v ∧ u = 0 for all u, v ∈ V.

For a fixed orthonormal basis eini=1 of V we can define the subspace Λk(V ) of

elements of order k generated by 1 and elements of the form

εI := ei1 ∧ . . . ∧ eik

where I = i1, . . . , ik is a strictly increasing subset of indices from the set 1, . . . , n:1 ≤ i1 < . . . < ik ≤ n. We also extend this definition by setting I = 0 and e0 = 1 fork = 0. The dimension of Λk(V ) is equal to

(nk

)which implies that the dimension of

Λ(V ) is equal to 2n, i.e. coincides with the dimension of the Clifford algebra Cl(n).So we can expect that for the n-dimensional Euclidean space V it should exist aclose relation between Λ(V ) and Cl(V ) be established in the next section.

The exterior algebra can be also defined as a graded superalgebra having thefollowing universal property. If A is another graded superalgebra and f : V → A1

is an arbitrary linear map with the property:

f(u)f(v) + f(v)f(u) = 0 for all u, v ∈ V

then f extends uniquely to a homomorphism of graded superalgebras f : Λ(V ) → A.

Derivations and transposition

We consider now the derivations of Λ(V ) paying special attention to the inner andexterior ones.


Define first the inner product of a form and an element ξ ∈ V ′ of the dual spaceV ′ by setting ι(ξ)1 = 0 and

ι(ξ)(v1 ∧ . . . ∧ vk) =k∑

j=1

ξ(vj)v1 ∧ . . . ∧ vj ∧ . . . ∧ vk

where v1, . . . , vk ∈ V , and the “hat” over the symbol vj means that this symbolshould be omitted. Then for any ξ, η ∈ V ′ we shall have

ι(ξ)ι(η) + ι(η)ι(ξ) = 0.

Using now the universal property of the algebra Λ(V ′), we can extend the con-structed map ι : V ′ → EndΛ(V ) to the whole exterior algebra Λ(V ′) obtaining amap ι : Λ(V ′) → End Λ(V ).

On the other hand, we have on Λ(V ) the operation of exterior product generatinga homomorphism ε : Λ(V ) → End Λ(V ).

These operations are related by the commutation relations:

[ε(u), ε(v)] = ε(u)ε(v) + ε(v)ε(u) = 0,

[ι(ξ), ι(η)] = ι(ξ)ι(η) + ι(η)ι(ξ) = 0,

[ι(ξ), ε(v)] = ι(ξ)ε(v) + ε(v)ι(ξ) = ξ(v)

fulfilled for all u, v ∈ V , ξ, η ∈ V ′.The tensor algebra T (V ) has a special anti-automorphism generating the identity

transform on V . This automorphism is called the transposition and is uniquelydetermined by the formula

(v1 ⊗ . . .⊗ vk)T := vk ⊗ . . .⊗ v1.

Since the introduced operation preserves the ideal I(V ) it can be pushed down tothe exterior algebra Λ(V ). Moreover,

ηT = (−1)k(k−1)/2η

on the forms η ∈ Λk(V ).In the complex case, if we denote by V C = V ⊗R C the complexification of the

space V then we shall have on V C the operation of complex conjugation: v 7→ vwhich extends in a natural way to the whole exterior algebra Λ(V C). Define theinvolution in Λ(V C) by setting

η 7−→ η∗ := (η)T .

This map generates a conjugate-linear anti-automorphism of the algebra Λ(V C).

Relation between the exterior algebra and Clifford algebra

Consider the map s : V → EndΛ(V ) given by the formula

s(v) = ε(v) + ι(v′), v ∈ V,


where v′ ∈ V ′ is the functional dual to v which is defined as

v′ : u ∈ V 7−→ (v, u).

The map s extends to a homomorphism of the whole tensor algebra s : T (V ) →EndΛ(V ). The commutation relations for the inner and exterior derivations implythat

s(u)s(v) + s(v)s(u) = −2(u, v)1.

Hence s vanishes on the ideal J (V ) and so can be pushed down to a homomorphism

s : Cl(V ) −→ EndΛ(V ).

Consider next the map σ : Cl(V ) → Λ(V ) called the symbol and given by theformula

σ(x) := s(x)1

where 1 is considered as an element of Λ0(V ). In low degrees this map is easy tocompute explicitly:

σ(1) = 1,

σ(v) = v,

σ(v1v2) = v1 ∧ v2 + (v1, v2),

σ(v1v2v3) = v1 ∧ v2 ∧ v3 + (v2, v3)v1 − (v1, v3)v2 + (v1, v2)v3.

The inverse map of σ, sending

Alt : Λ(V ) −→ Cl(V ),

is called the alternation and is given by the formula

Alt(v1 ∧ . . . ∧ vk) =1

k!

∑p∈Sk

(−1)sgn pvp(1) · . . . · vp(k)

where the summation is taken over all permutations p ∈ Sk of the set 1, . . . , k,and sgn p denotes the parity of permutation p.

Involution and volume element

The transposition map on the tensor algebra T (V ), introduced in Sec.3.1.3, preservesthe ideal J (V ) and so can be pushed down to an anti-automorphism of the Cliffordalgebra Cl(V ) having the property:

(v1 · . . . · vk)T = vk · . . . · v1.

Note that the alternation map Alt intertwines the transpositions in Λ(V ) and Cl(V ).In the complex case we can define the involution on the Clifford algebra Cl(V )

by the formula:x 7−→ x∗ := (x)T

where x 7→ x is the complex conjugation on Cl(V ). Again the alternation map Altintertwines the involutions in Λ(V ) and Cl(V ).


Using the transposition, we can introduce one more useful map

N : x 7−→ xT x

defined on elements x of the Clifford group Γ(V ). Since any element x ∈ Γ(V ) maybe represented in the form x = v1 · . . . · vk it follows that N takes values in R×. Thismap generates a group homomorphism

N : Γ(V ) −→ R×, x 7−→ xT x

called the norm, having the property: N(λx) = λ2N(x) for λ ∈ R×.The alternation of the volume element ω ∈ Λn(V ) of the n-dimensional vector

space V yields an element ω ∈ Cl(V ) (denoted by the same letter ω) which is calledthe volume element of the Clifford algebra or its chiral element .

The square of ω is a scalar which is easy to show by choosing an orthonormalbasis ein

i=1 of the Euclidean space V and setting ω = e1 · . . . · en. Then

ω2 = (−1)n(n+1)/2.

Moreover, for odd n the element ω is central while for even n we have: xω = ωχ(x)for any x ∈ Cl(n).

3.1.4 The group Spinc

Definition

Consider the complexified Clifford algebra Cl(V ) = Cl(V C), where V C is the com-plexification of the space V , and provide it with the Hermitian inner product bysetting it equal to

〈z, w〉 := (z, w)

on elements z, w ∈ V C and extending then to the whole algebra Cl(V C). Recall thatthis algebra is provided with the involution defined by the equality: z∗ = (z)T onelements z ∈ Cl(V ).

Let Γ(V C) be the Clifford group of the space V C consisting of elements z ∈Cl×(V C) satisfying the condition: πz(V

C) = V C so that πz ∈ O(V C).

Definition 59. Introduce the following groups:

Γc(V ) = z ∈ Γ(V C) : z∗z ∈ R+,Pinc(V ) = z ∈ Γ(V C) : z∗z = 1,

Spinc(V ) = Pinc(V ) ∩ SΓ(V C).

It can be shown that the element z ∈ Γ(V C) belongs to the subgroup Γc(V )if and only if the transform πz ∈ O(V C) preserves the real subspace V . In otherwords, Γc(V ) coincides with the inverse image (with respect to π) of the subgroupO(V ) ⊂ O(V C).

The exact sequence for the group Γ(V C) transforms into the exact sequence ofgroup homomorphisms

1 −→ C× −→ Γc(V ) −→ O(V ) −→ 1


inducing the exact sequences

1 −→ U(1) −→ Pinc(V ) −→ O(V ) −→ 1,

1 −→ U(1) −→ Spinc(V ) −→ SO(V ) −→ 1,

since C× ∩ Pinc(V ) = C× ∩ Spinc(V ) = U(1). It follows from these sequences thatthe groups Pinc(V ) and Spinc(V ) are central extensions of the groups O(V ) andSO(V ) respectively by U(1).

Exact sequences for the group Spinc(V )

One can define the group Spinc(V ) in a different way as the subgroup of Γc(V ) ofthe form

Spinc(V ) = z = xeiθ : x ∈ Spin(V ), θ ∈ R.In other words, the group Spinc(V ) is the quotient of the group Spin(V )× U(1) bythe equivalence relation: (x, eiθ) ∼ (−x,−eiθ). The latter assertion may be writtenin the form of the exact sequence

1 −→ Spin(V ) −→ Spinc(V )δ−→ U(1) −→ 1

where δ : xeiθ 7→ e2iθ. Otherwise,

Spinc(V ) = Spin(V )×Z2 U(1).

The norm homomorphism

N : Γc(V ) −→ C×, z 7−→ zT z,

defined on the Clifford group Γ(V C), takes values in U(1) when applied to theelements from Pinc(V ). The combination of this homomorphism with the aboveexact sequences yields:

1 −→ Z2 −→ Γc(V )(π,N)−→ O(V )× C× −→ 1,

1 −→ Z2 −→ Pinc(V )(π,N)−→ O(V )× U(1) −→ 1,

1 −→ Z2 −→ Spinc(V )(π,N)−→ SO(V )× U(1) −→ 1

where the homomorphism π is given by the map z 7→ πz(v) = χ(z)vz−1.The Lie algebra of the group Spinc(V ) coincides with

spinc(V ) = cl2(V )⊕ iR

where cl2(V ) is the quadratic component (coinciding as a set with Cl2(V )) of theClifford algebra Cl(V ) provided with the commutator as the Lie bracket.

Examples of the Spinc-groups

1. Cl(R) = C⊕ C, the group Spinc(R) coincides with the group U(1) embeddedinto C⊕ C with the help of diagonal map.

2. Cl(R2) = Mat2(C), the group Spinc(R2) coincides with the group U(1)×U(1)consisting of the unitary diagonal matrices in Mat2(C).


3.1.5 Spin representation

Clifford modules

Definition 60. A Clifford representation is a homomorphism

ρ : Cl(V ) −→ EndS

from the Clifford algebra Cl(V ) into the algebra of linear operators acting in acomplex vector space S called the Clifford module over Cl(V ) or the spinor spacefor the algebra Cl(V ). We shall assume that S is provided with an Hermitian innerproduct.

Otherwise, the Clifford representation may be defined as a linear map ρ : V →EndS having the property:

ρ(u)ρ(v) + ρ(v)ρ(u) + 2(u, v)1 = 0

for all u, v ∈ V . Then by universal property such map extends to a representationρ : Cl(V ) → EndS.

The standard definitions and properties from the representation theory of asso-ciative algebras apply also to Clifford representations.

The action of the representation ρ on the space S is often denoted by

ρ(x)s := x · s

for x ∈ Cl(V ), s ∈ S, and called the Clifford multiplication.

Semispinor spaces

The Clifford algebra Cl(n) of n-dimensional Euclidean vector space V , providedwith an orthonormal basis ein

i=1, has the volume element ω = e1 · . . . · en. In thecomplex case we can introduce, together with ω, also a complex volume element ωc

defined by the formulaωc := i[(n+1)/2]ω

where [ · ] denotes the integral part of a number.For odd n the elements ω and ωc belong to the center of the Clifford algebra.

Moreover,ω2 =1 for n ≡ 3, 4 mod 4,

(ωc)2 = 1 for all n.

Consider first the real volume element and suppose that n ≡ 3, 4 mod 4 so thatω2 = 1. The the elements π± := 1±ω

2are mutually orthogonal idempotents and

π+ + π− = 1. Hence the Clifford algebra Cl(V ) admits the decomposition of theform

Cl(V ) = Cl+(V )⊕ Cl−(V )

where Cl±(V ) := π±Cl(V ).In the same way any Clifford module S over Cl(V ) can be represented in the

formS = S+ ⊕ S−


where S± are the eigenspaces of the operator ρ(ω) with eigenvalues ±1 so thatS± = π±S. The subspaces S± are called the semispinor spaces .

In the complex case for any odd n we shall have an analogous decomposition ofthe complexified Clifford algebra

Cl(V ) = Cl+(V )⊕ Cl−(V )

where Cl±(V ) := (1± ωc)Cl(V ).

Description of irreducible Clifford modules

Proposition 22. Let ρ : Cl(n) → EndS be an irreducible representation of theClifford algebra and n = 4m + 3. Then

ρ(ω) = ±Id .

Moreover, both possibilities are realized and the corresponding Clifford representa-tions are not equivalent. An analogous assertion holds for the complexified Cliffordalgebra Cl(n) for any odd n.

Proof. We represent the spinor space S as the direct sum

S = S+ ⊕ S−

of (±1)-eigenspaces of operator ρ(ω). These subspaces are invariant under Cliffordmultiplication because the element ρ(ω) is central. Since the representation S isirreducible we have either S = S+, or S = S− which proves the first assertion of theproposition.

These irreducible representations are not equivalent to each other because ifρ(ω) = ±Id, i.e. it is a multiple of Id, then this property is valid also for anyequivalent representation ρ1, moreover ρ1(ω) is proportional to Id with the sameproportionality coefficient as for ρ(ω).

In order to see that both possibilities are realized it is sufficient to consider theaction of the Clifford algebra Cl(V ) on Cl±(V ) by left multiplication.

The proof of proposition in the complex case we leave as an exercise.

Before passing to the study of irreducible representations of the Clifford algebrain the case n = 4m, we prove the following lemma relating the Clifford algebraCl(n− 1) with the even part of the Clifford algebra Cl(n).

Lemma 16. For any n > 1 we have an algebra isomorphism

Cl(n− 1) −→ Clev(n).

Proof. Choose an orthonormal basis e1, . . . , en−1, en in the space Rn and denote byRn−1 the subspace spanned by the first n− 1 vectors of this basis.

Consider the map f : Rn−1 → Clev(n) defined on the basis elements by theformula:

f(ei) = en · ei, i = 1, . . . , n− 1.


Compute the value f(x)2 on an arbitrary element x =∑n−1

i=1 xiei from the spaceRn−1. We have

f(x)2 =

(∑i

xienei

)·(∑

j

xjenej

)=

∑i,j

xixjeneienej =

∑i,j

xixjeiej (since enei = −eien and e2n = −1)

= x · x = −‖x‖2 · 1.So by the universal property the constructed map extends to an algebra homomor-phism

f : Cl(n− 1) −→ Clev(n).

Considering it on the basis elements it is easy to see that this map is an isomorphism.

Proposition 23. Let ρ : Cl(n) → EndS be an irreducible representation of theClifford algebra and n = 4m. Consider the decomposition of the space S into thedirect sum of semispinor spaces

S = S+ ⊕ S−.

Then each of the subspaces S± is invariant under the even part Clev(n) of the Cliffordalgebra Cl(n). These subspaces correspond to two different representations of thealgebra Cl(n− 1) ∼= Clev(n).

An analogous assertion holds for the complexified Clifford algebra Cl(n) for anyeven n.

Proof. The subspaces S± are invariant under Clev(n) since for even n the volumeelement ω commutes with any element from Clev(n). Under the isomorphismCl(n−1) ∼= Clev(n) from the proved Lemma 16 the volume element ω′ of the algebraCl(n−1) is mapped to the volume element ω ∈ Clev(n) of the algebra Cl(n). Hence,ρ(ω′) = Id on S+ and ρ(ω′) = −Id on S−. By the previous proposition theserepresentations are not equivalent to each other.

The proof of proposition in the complex case we leave as an exercise.

Spin representation

Definition 61. The real spin representation of the group Spin(n) is a group homo-morphism

∆n : Spin(n) −→ GL(S)

obtained by the restriction of an irreducible representation ρ : Cl(n) → EndS of theClifford algebra Cl(n) to the group Spin(n) ⊂ Clev(n) ⊂ Cl(n).

Proposition 24. For n = 4m + 3 the representation ∆n is irreducible and bothirreducible representations of the algebra Cl(n) give the same representation whenrestricted to Spin(n).

For n = 4m the representation ∆4m admits the decomposition

∆4m = ∆+4m ⊕∆−

4m


into the direct sum of two non-equivalent irreducible representations of the groupSpin(n).

Proof. If n = 4m + 3 then the grading automorphism χ permutes the summandsCl+(n) and Cl−(n) in the decomposition

Cl(n) = Cl+(n)⊕ Cl−(n)

since χ(ω) = −ω. Hence, the elements of the subalgebra Clev(n), which are bydefinition invariant under χ, should have the form (x, χ(x)) ∈ Cl+(n) ⊕ Cl−(n),i.e. the subalgebra Clev(n) is embedded into Cl+(n)⊕ Cl−(n) diagonally. Since thetwo irreducible representations of the algebra Cl(n) differ from each other by theautomorphism χ they become equivalent after restriction to Clev(n). This provesthe first assertion of proposition.

If n = 4m then the restriction of the representation of the algebra Cl(n) to theeven part Clev(n) decomposes into the direct sum of two non-equivalent representa-tions. Each of these representations in its turn restricts to the group Spin(n) as anirreducible one because Spin(n) contains the basis of the algebra Clev(n) consideredas a vector space.

Definition 62. The complex spin representation of the group Spin(n) is a grouphomomorphism

∆cn : Spin(n) −→ GL(S,C)

obtained by restricting of an irreducible representation ρ : Cl(n) → EndS of thecomplexified Clifford algebra Cl(n) to the group Spin(n) ⊂ Clev(n) ⊂ Cl(n).

Proposition 25. For odd n the representation ∆cn is irreducible and both irreducible

representations of the algebra Cl(n) give the same representation when restricted toSpin(n).

For even n = 2m the representation ∆c2m admits the decomposition

∆c2m = (∆c

2m)+ ⊕ (∆c2m)−

into the direct sum of non-equivalent irreducible representations of the group Spin(n).

The proof is analogous to the proof of the previous proposition.

Examples of spin representations

1. The representation ∆3 has dimension 2 yielding a homomorphism (not de-pending on the choice of an orthonormal basis)

Spin(3) −→ SU(2)

which is an isomorphism by the dimension counting, i.e. Spin(3) ∼= SU(2).

2. Both representations ∆±4 are 2-dimensional. As in previous case, the represen-

tation ∆+4 ⊕∆−

4 generates an isomorphism

Spin(4) ∼= SU(2)× SU(2).

3.2. SPINOR GEOMETRY 101

Spin representation in the complex case

Let V be the (n = 2m)-dimensional Euclidean vector space identified with them-dimensional complex vector space. We shall assume that this space is Kahler,i.e. it is provided also with an Hermitian inner product 〈·, ·〉 compatible with theEuclidean one. Denote by V ∗ the space dual to V with respect to this inner product.Its complexification V ∗

C = V ∗⊗RC may be represented in the form: V ∗C = V 1,0⊕V 0,1.

So for every vector v ∈ V the dual covector v∗ is decomposed into the sum

v∗ = v1,0 + v0,1.

We shall construct a canonical representation of the Clifford algebra Cl(n) in thespace

Scan := Λ0,∗V ∗ =m⊕

q=0

Λ0,q(V ∗).

The representation ρcan : V → EndScan in this space is determined by the formula

ρcan(v)η = v0,1 ∧ η − v0,1y η

where v ∈ V, η ∈ Λ0,q(V ∗). It can be checked that

ρcan(v) ρcan(v)η = −‖v‖2η

so by universal property the map ρcan : V → EndScan extends to a representationof the Clifford algebra

ρcan : Cl(n) −→ EndScan.

The semispinor spaces coincide in this case with

S+can = Λ0,ev(V ∗), S−can = Λ0,od(V ∗).

3.2 Spinor geometry

3.2.1 Spin structures

Spin structures on vector bundles

Let p : E → M be a real vector bundle of rank n over a smooth compact orientedRiemannian manifold M . We shall assume that this bundle is Riemannian, i.e. ineach fibre Ex, x ∈ M , it is given a positive definite inner product (·, ·) smoothlydepending on the point x ∈ M .

Moreover, we suppose that E is orientable , i.e. in each fibre Ex it is givenan orientation which depends smoothly on the point x ∈ M . In contrast with theRiemannian structure, existing on any smooth vector bundle, for the orientabilityof E it is necessary to impose the following topological condition.

Proposition 26. A bundle p : E → M is orientable if and only if its 1st Stiefel–Whitney class w1(E) = 0.


For the proof cf. [7], II.1, Theorem 1.2. Basic properties of characteristic classesmay be found in the book [8].

So we assume that p : E → M is an orientable Riemannian vector bundle ofrank n and PSO(E) is the bundle of oriented orthonormal bases (briefly: frames) infibres of E. Suppose that n ≥ 2 so that the homomorphism π : Spin(n) → SO(n) isa double covering.

Definition 63. The spin structure on the vector bundle p : E → M is a principalSpin(n)-bundle PSpin(E) together with a double covering bundle morphism

Π : PSpin(E) −→ PSO(E)

which is Spin-equivariant in the sense that

Π(sg) = Π(s)π(g)

for any s ∈ PSpin(E), g ∈ Spin(n). The action of the group Spin(n) on PSO(E) isgiven by the homomorphism π : Spin(n) → SO(n).

The given definition may be rewritten in the form of the following commutativediagram

PSpin(E) Π //

$$IIIIIIIIIPSO(E)

vvvv

vvvv

v

M

in which the restriction of Π to the fibers coincides with the homomorphism π.

Proposition 27. The spin structure on a vector bundle p : E → M exists if andonly if the 2nd Stiefel–Whitney class w2(E) = 0. If this condition is satisfied thendifferent spin structures on E are numerated by the elements of the cohomology groupH1(M,Z2).

The proof of the proposition cf. in [7], II.1, Theorem 1.7.

Definition 64. An oriented Riemannian manifold M is called the spin manifold ifits tangent bundle TM admits a spin structure.

Examples of spin manifolds

1. A complex manifold M is spin if and only if its 1st Chern class is even, i.e.c1(M) ≡ 0 mod 2. This fact follows from the relation w2(M) ≡ c1(M) mod 2.

2. A complex projective space CPn is spin if and only if n is odd.

3. Let Σg be a compact Riemann surface of genus g. Then on Σg there exist 22g

different spin structures.


Spinor and Clifford bundles

Recall, first of all, a general construction of the adjoint bundle. Let π : P → M be aprincipal G-bundle over a manifold M and F is another smooth manifold providedwith a smooth action of the group G from the right, i.e. with a homomorphismρ : G → Diff(F ). Then we can construct a new bundle P ×ρ F with fibre F andstructure group G.

For that consider the left action of the group G on the product P × F given bythe formula:

g(s, f) := (sg−1, ρ(g)f)

where s ∈ P, f ∈ F, g ∈ G. The quotient P × F by this action is denoted byP ×ρ F and consists of the orbits [s, f ] of elements (s, f) ∈ P × F . The projectionπρ : P ×ρ F → M is defined by the formula: πρ([s, f ]) := π(s).

If ρ : G → GL(V ) is a linear representation of the group G in a vector space Vthen the associated bundle P ×ρ F is a vector bundle over M .

Example 15. Let M be an oriented Riemannian manifold of dimension n andP = PSO(M) is the principal SO(n)-bundle of frames on M . Denote by ρ thestandard action of the group SO(n) on the space Rn. Then

TM = PSO(M)×ρ Rn.

In a more general way, if E → M is an oriented Riemannian vector bundle over Mthen

E = PSO(E)×ρ Rn.

Return to the Clifford algebras. Every orthogonal transform of the space V = Rn

generates an automorphism of the Clifford algebra Cl(n). Indeed, such transformmaps the tensor algebra T (V ) into itself and preserves the idealJ (V ), determiningthe Clifford algebra, so it can be pushed down to the Clifford algebra Cl(n). Thus,we have a representation

cl : SO(n) −→ AutCl(n).

Definition 65. Let E → M be an oriented Riemannian vector bundle. Its Cliffordbundle is a bundle of the form

Cl(E) := PSO(E)×cl Cl(n).

In other words, Cl(E) is the bundle of Clifford algebras associated with the bun-dle E of Euclidean vector spaces. Hence, all constructions which we have inventedfor Clifford algebras, may be extended to the Clifford bundles. In particular, we canintroduce on Cl(V ) a natural inner product.

Definition 66. Let E be an oriented Riemannian vector bundle provided with aspin structure Π : PSpin(E) → PSO(E), and ∆n : Spin(n) → GL(S) is the real spinrepresentation. A real spinor bundle over E is the bundle of the form

S(E) := PSpin(E)×∆n S.

If ∆cn : Spin(n) → GL(Sc,C) is the complex spin representation then the bundle

Sc(E) := PSpin(E)×∆cn

Sc

is called the complex spinor bundle.


Further on we shall assume that S and Sc are provided with Hermitian structures.Having the definition of spinor bundles, we can introduce the semispinor bundles.

For that consider the section ωc of the bundle Cl(E) of rank n = 2m given at thepoint x ∈ M by the formula

ωc = ime1 · . . . · e2m

for an oriented orthonormal basis ej of the space Ex. Then (ωc)2 = 1 and we candefine the semispinor bundles Sc

±(E) as the (±1)-eigenbundles of the operator ofClifford multiplication by ωc. Thus,

Sc±(E) = PSpin(E)×(∆c

2m)± Sc.

For n = 4m an analogous construction goes through in the real case. Namely, let ωbe the section of the bundle Cl(E) given by the formula

ω = e1 · . . . · e4m

in terms of an orthonormal basis ej of the space Ex. Then ω2 = 1 and we havethe decomposition

S(E) = S+(E)⊕ S−(E)

into the direct sum of (±1)-eigenbundles of the operator of Clifford multiplicationby ω.

3.2.2 Spinor connections

In this section we give a short exposition of the theory of connections. A moredetailed presentation and the proofs of formulated results may be found in the book[7].

Connections in principal bundles

Let π : P → M be a principal G-bundle over a smooth manifold M of dimensionn. Denote by V the distribution in the tangent bundle TP formed by the tangentspaces Vp of the fibers of the bundle P at points p ∈ P : Vp = v ∈ Pp : π∗(v) = 0.The subspace Vp is called the vertical subspace at p.

Note that for any principal bundle π : P → M there exists a homomorphism ofthe Lie algebra g of the Lie group G into the Lie algebra Vect(P ) of smooth vectorfields on P generated by the right action of the group G on P . Namely, we associatewith an element ξ of the Lie algebra g the vector field X given at point p ∈ P bythe formula

Xp = p∗(ξ) :=d

dt(p · exp(tξ)) |t=0.

The vector fields X, constructed in this way, are vertical, moreover, for any v ∈ Vp

there exists a vector field X of the described type such that Xp = v. In other words,the assignment g 3 ξ 7→ Xp ∈ Vp allows to identify the vertical subspace Vp withthe Lie algebra g.


Definition 67. A connection in a principal bundle π : P → M is a smooth dis-tribution H : P 3 p → Hp of subspaces Hp ⊂ TpP , called horizontal , having thefollowing properties:

1. the tangent map π∗ : Hp → Tπ(p)M is an isomorphism of vector spaces for anyp ∈ P ;

2. the distribution H is G-invariant, i.e. g∗Hp = Hpg for any p ∈ P , g ∈ G wherewe denote by g∗ the map tangent to the right action of g on P .

In other words, the tangent space TpP at any point p ∈ P admits a decompositionTpP = Vp ⊕ Hp into the direct sum of vertical and horizontal subspaces, i.e. aconnection is a G-invariant method of choosing the distribution H supplement tothe vertical distribution V .

At any point p ∈ P the horizontal subspace Hp determines the projection TpP →Vp parallel to Hp. Using the isomorphism Vp

∼= g, defined above, we can constructa linear map ωp : TpP → g. It determines a 1-form ω on P with values in the Liealgebra g.

This 1-form ω is called the connection form on P and has the following properties:

1. ω is vertical, i.e. it vanishes on horizontal vectors;

2. for any element ξ ∈ g the following equality

ω(Xp) = ξ

holds for any p ∈ P where Xp = p∗(ξ);

3. the form ω is equivariant in the sense that g∗(ω) = Adg−1ω.

From any smooth 1-form ω with these properties we can reconstruct the connec-tion H by setting:

Hp := Kerωp .

Definition 68. The curvature of a connection H in a principal bundle π : P → Mis a smooth 2-form Ω on P with values in the Lie algebra g defined by the equality:

Ω = dω + [ω, ω].

This form is horizontal, i.e. it vanishes on any pair of vectors if at least one ofthem is vertical, and equivariant, i.e. it transforms under the action of elementsg ∈ G by the formula:

g∗Ω = Adg−1Ω.

Riemannian connections

Let PSO(E) be the principal SO(n)-bundle of frames (oriented orthonormal bases)of an oriented vector bundle E of rank n over an oriented Riemannian manifoldM . The Lie algebra so(n) of the Lie group SO(n) coincides with the space of realskew-symmetric (n × n)-matrices so the connection form ω on this bundle may beconsidered as an (n × n)-matrix ω = (ωij), composed of 1-forms ωij satisfying therelation: ωij + ωji = 0.


The curvature Ω in this case will be given by a matrix Ω = (Ωij) composed of2-forms

Ωij = dωij +n∑

k=1

ωik ∧ ωkj,

and the map Ad acts by the formula: Adgω = gωg−1.

Definition 69. The covariant derivative in E is a linear map

∇ : Γ(E) −→ Γ(T ∗M ⊗ E)

satisfying the Leibniz rule:

∇(fs) = df ⊗ s + f∇s

for any f ∈ C∞(M), s ∈ Γ(E).If X is a smooth tangent vector field on M then the pairing with X generates a

linear map∇X : Γ(E) −→ Γ(E)

called the covariant derivative along X.

Let ω be a connection form on PSO(E) and e = e1, . . . , en is a local orthonormalbasis of sections of the bundle E in a neighborhood U of a point x0 ∈ M . Then edetermines a local section

e : U −→ PSO(E)

of the bundle PSO(E) over U . The dual tangent bundle

e∗ : T ∗(PSO(E)) −→ T ∗U

allows to transport the connection form ω = (ωij) to U by setting:

ω := e∗ω.

With this remark we can formulate the following

Proposition 28. Let ω be a connection form on the bundle PSO(E). Then it deter-mines uniquely a covariant derivative ∇ on the bundle E given in terms of a localorthonormal basis of sections of the bundle E by the formula

∇ei =n∑

j=1

ωij ⊗ ej (3.1)

in which ω = e∗ω.This covariant derivative is compatible with Riemannian structure on E in the

sense thatX〈s, s′〉 = 〈∇Xs, s′〉+ 〈s,∇Xs′〉

for any smooth tangent vector fields X and any smooth sections s, s′ ∈ Γ(E) where〈· , ·〉 denotes the inner product on E.

This covariant derivative is called Riemannian.Conversely, any covariant derivative ∇ on E, compatible with Riemannian struc-

ture, determines a unique connection with connection form ω on PSO(E) determinedby Formula (3.1).


The proof cf. in [7], II.4, Proposition 4.4.Let ∇ be a covariant derivative on the bundle E. Consider the composition

Γ(E)∇−→ Γ(T ∗M ⊗ E)

∇−→ Γ(Λ2(T ∗M)⊗ E)

where ∇ is a natural extension of the covariant derivative ∇ to the sections of thebundle T ∗M ⊗ E of the form η ⊗ s where η is a 1-form on M and s is a section ofE. This extension is defined by the formula

∇(η ⊗ s) := dη ⊗ s− η ∧∇s.

The composition R := ∇ ∇ is called the Riemannian curvature.

Proposition 29. In the notation of Proposition 28 denote by Ω = (Ωij) the curva-ture of the connection form ω. Then in terms of a local orthonormal basis of sectionsof the bundle E the Riemannian curvature will be given by the formula

Rei =n∑

j=1

Ωij ⊗ ej

where Ω = (Ωij) and Ω = e∗Ω. For any smooth tangent vector fields X, Y on M wehave

RX,Y s = (∇X∇Y −∇Y∇X −∇[X,Y ])s.

The map RX,Y : Γ(E) → Γ(E), called the curvature transform, has the followingsymmetry property:

〈RX,Y s, s′〉+ 〈s, RX,Y s′〉 = 0.

The proof cf. in [7], II.4, Propositions 4.5, 4.6.

Connections in Clifford and spinor bundles

The construction of connections in Clifford bundles is based on the following idea.Let π : P = PG → M be a principal G-bundle over a manifold M . Suppose that

it is given a faithful representation ρ : G → SO(n) of the group G in the space Rn.Denote by E = Eρ the Riemannian vector bundle over M associated with P . Inother words:

E = Eρ = P ×ρ Rn.

Then for any given connection H in the bundle P we can construct an inducedcanonical connection Hρ in the principal bundle

P (E) = P (Eρ) = P ×ρ SO(n)

where the group G acts from the left on SO(n) by the homomorphism ρ.In order to construct the desired connection Hρ in P (E) we extend the given

connection H in the bundle P trivially to the direct product P×SO(n) and then pushdown the obtained connection to P ×ρ SO(n) using the invariance of the connectionH. This defines the connection Hρ in the bundle P (Eρ).


Note that we have a canonical G-equivariant map i : P → P (E) given by theformula

P 3 p 7−→ [p, Id]

where [p, h] denotes the class of the pair (p, h) ∈ P × SO(n) in the quotient P ×ρ

SO(n). This map is an embedding due to the faithfulness of the representation ρ.

Proposition 30. Let ω be the connection form of a connection H in the bundle PG

with curvature Ω. Denote by ωρ the form of the induced connection Hρ in the bundleP (Eρ) with curvature Ωρ. Then

ωρ|P = ρ∗ω, Ωρ|P = ρ∗Ω

where ρ∗ : g → so(n) is the Lie algebra homomorphism tangent to the representationρ : G → SO(n).

The proof cf. in [7], II.4, Proposition 4.7.We apply this construction to Clifford bundles. Let E → M be an oriented

Riemannian bundle of rank n provided with a Riemannian connection. Recall thatthe Clifford bundle Cl(E) is the bundle associated with the principal SO(n)-bundlePSO(E) by the homomorphism cl : SO(n) → Aut(Cl(n)), i.e.

Cl(E) = PSO(E)×cl Cl(n).

Hence, in accordance with the above construction, the given Riemannian connectionin E generates in a canonical way a connection in the Clifford bundle Cl(E). The co-variant derivative, corresponding to this connection, has the following characteristicproperty.

Proposition 31. The covariant derivative ∇ of the constructed connection in theClifford bundle Cl(E) acts by the derivation of sections of the Clifford bundle, i.e.

∇(σ · τ) = (∇σ) · τ + σ · (∇τ)

for any sections σ, τ ∈ Γ(Cl(E)).

Note that under the canonical identification of Cl(E) with the bundle Λ∗(E)the covariant derivative ∇ will correspond to the derivation of Λ∗(E) preserving thesubbundles Λk(E). It implies, in particular, that ∇ preserves also the subbundlesClev(E) and Clod(E), and the volume element ω = e1 · . . . ·en is parallel with respectto ∇, i.e. ∇ω = 0. It follows that for n ≡ 3, 4 mod 4 the subbundles Cl±(E) are alsopreserved by the derivative ∇.

The proof of these assertions and the following proposition may be found in [7],II.4, Propositions 4.8, 4.10.

Proposition 32. For any pair of tangent vector fields X, Y in a neighborhood of apoint x ∈ M the curvature transform

RX,Y : Cl(Ex) −→ Cl(Ex)

is a derivation, i.e.

RX,Y (σ · τ) = RX,Y (σ) · τ + σ ·RX,Y (τ)

for any sections σ, τ ∈ Cl(Ex).


This transform preserves also the subspaces Clev(Ex), Clod(Ex) and Cl±(Ex).Suppose now that the bundle E is provided with a spin structure, i.e. it is given

a bundle epimorphismΠ : PSpin(E) −→ PSO(E).

Consider the associated spinor bundle

S(E) = PSpin(E)×∆n S

constructed with the help of the spin representation ∆n : Spin(n) → S. Then theconnection in the bundle PSO(E) may be pulled up with the help of the epimorphismΠ to a connection in the bundle PSpin(E). The obtained connection in its turngenerates in a canonical way a connection in the spinor bundle S(E). For thisconnection the analogues of Propositions 31 and 32 hold.

The covariant derivative of the constructed connection and its curvature may becomputed explicitly in terms of a local orthonormal basis of sections of the bundlePSO(E). We shall give here only the formulation of this result referring for its proofto [7], II.4, Theorems 4.14, 4.15.

Proposition 33. Let ω be a connection form on the bundle PSO(E) and S(E) isthe spinor bundle associated with E. Then the covariant derivative of the connec-tion in S(E), constructed from the connection form ω, is given in terms of a localorthonormal basis e = ei of sections of the bundle E by the formula

∇sk =1

2

∑i<j

ωij ⊗ eiej · sk

where s = sk is a local section of the bundle PSO(S(E)), determined by the sectione, and ω = e∗ω.

If Ω is the curvature of the connection ω then the curvature R of the spinorconnection ∇ in S(E) is given by the formula

Rs =1

2

∑i<j

Ωij ⊗ eiej · s

where s is a section of the bundle S(E), and Ω = e∗Ω. In particular, for any pairof tangent vector fields X,Y in a neighborhood of a point x ∈ M the curvaturetransform

RX,Y : S(Ex) −→ S(Ex)

is given by the formula

RX,Y s =1

2

∑i<j

〈RX,Y (ei), ej〉eiej · s.

In the case when E = TM we denote PSO(M) := PSO(TM) and Cl(M) :=Cl(TM). If ∇ is a connection in the bundle PSO(M) then we can construct withits help a tensor field T associating with a pair of tangent vector fields X,Y in aneighborhood of a point x ∈ M the quantity

TX,Y := ∇XY −∇Y X − [X,Y ]

called the torsion tensor of the connection ∇. The tensor T determines a 2-form onthe manifold M with values in the tangent bundle TM .


Theorem 16. Let M be a Riemannian manifold and PSO(M) be the bundle offrames on TM . Then there exists a unique connection in PSO(M) with vanishingtorsion tensor.

This connection is called the canonical Riemannian connection or the Levi-Civitaconnection on M . It induces a canonical connection in the Clifford bundle Cl(M).In the case when M admits a spin structure this connection induces also a canon-ical Riemannian connection in the bundle PSpin(M) hence, in any spinor bundleassociated with PSpin(M).

The curvature tensor R of the canonical Riemannian connection on M has thefollowing symmetry properties:

RX,Y Z + RY,ZX + RZ,XY = 0,

〈RX,Y Z, U〉 = 〈RZ,UX, Y 〉

for any tangent vector fields X,Y, Z, U in a neighborhood of a point x ∈ M .

3.2.3 Dirac operator

Basic definitions

Let M be an oriented Riemannian manifold. Using the Riemannian metric, wecan identify the tangent space TxM with the cotangent space T ∗

xM at any pointx ∈ M . Denote by Cl(M) the Clifford bundle over M . Suppose that S is a bundleof Clifford modules over M so that every fibre Sx is a module over the Cliffordalgebra Cl(TxM). We shall assume also that S is Riemannian and is providedwith a Riemannian connection ∇. Denote by ρ : Γ(Cl(M)) ⊗ S → S the Cliffordmultiplication map on S.

Definition 70. The Dirac operator in S, generated by the connection ∇, is a lineardifferential operator D : Γ(S) → Γ(S) of the form

Γ(S)∇−→ Γ(T ∗M ⊗ S)

j⊗1−→ Γ(Cl(M)⊗ S)ρ−→ Γ(S)

where the embedding j is generated by the identification of Λ(T ∗M) = Λ(TM) withCl(TM).

In terms of a local orthonormal basis e1, . . . , en of the bundle TM the Diracoperator D is given by the formula

Ds =n∑

j=1

ej · ∇ejs

where s ∈ Γ(S). The operator D2 is called the Dirac Laplacian.

Lemma 17. Let f ∈ C∞(M) be a smooth function on M considered as the multi-plication operator, acting on sections from Γ(S). Then

[D, f ] = ρ(df).


Proof. Since the multiplication by f commutes with the Clifford multiplication, wehave for any s ∈ Γ(S):

[D, f ]s = ρ∇(fs)− fρ(∇s) = ρ(df ⊗ s) = ρ(df)s.

Recall that the principal symbol of a differential operator D : Γ(E) → Γ(E),given in local coordinates x in a neighborhood of a point x0 ∈ M by the formula

D =∑

|α|≤m

aα(x)∂|α|

∂xα,

is the map associating with every x ∈ M and every covector ξ =∑

ξjdxj ∈ T ∗xM \

0 the linear map σξ(D) : Ex → Ex of the form

σξ(D) = im∑

|α|=m

aα(x)ξα.

The operator D is called elliptic if the map σξ(D) : Ex → Ex is non-degenerate forany x ∈ M and ξ ∈ T ∗

xM \ 0.

Lemma 18. For any ξ ∈ T ∗xM \0 the principal symbols of the Dirac operator and

Dirac Laplacian are equal to

σξ(D) = iξ and σξ(D2) = ‖ξ‖2

where both operators act on S by the Clifford multiplication. In particular, bothoperators D and D2 are elliptic.

Proof. Fix a point x0 ∈ M and a local basis e1, . . . , en of the bundle TM in aneighborhood of x0. Choose the local coordinates in a neighborhood of x0 so thatx0 = 0 and ej = ∂/∂xj at x0. Then for any local trivialization of the bundle S ina neighborhood of x0 the covariant derivative ∇ej

in a neighborhood of x0 will bewritten in the form

∇ej= ∂/∂xj + terms of 0th order,

and the Dirac operator will have the form

D =∑

j

ej∂/∂xj + terms of 0th order.

So for any covector ξ =∑

ξjdxj ∈ T ∗xM \ 0 in a neighborhood of x0 the following

relations hold:σξ(D) = i

∑j

ejξj = iξ,

σξ(D2) = σξ(D)2 = −ξ · ξ = ‖ξ‖2.


General properties

We impose now some natural extra conditions on the bundle S of Clifford modules.First of all, we shall assume that the bundle S is provided with a Riemannian

structure compatible with the Clifford multiplication in the sense that the Cliffordmultiplication by unit vectors from TM is an orthogonal transform of S so that forany x ∈ M the following relation

〈es1, es2〉 = 〈s1, s2〉 (3.2)

holds for any unit vector e ∈ TxM and any sections s1, s2 ∈ Γ(S) at x.Moreover, we shall suppose that the bundle S is provided with a Clifford con-

nection ∇ satisfying the following Leibniz rule

∇(σ · s) = (∇σ) · s + σ · (∇s)

where σ ∈ Γ(Cl(M)), s ∈ Γ(S), and ∇σ is determined by the action of the canonicalRiemannian connection on the Clifford bundle.

The bundles S of Clifford modules, having these properties, we call briefly theDirac bundles .

Introduce on S an inner product given by the formula

(s1, s2) :=

∫

M

〈s1, s2〉 vol

where s1, s2 ∈ Γ(S), vol is the volume form on M .

Proposition 34. The Dirac operator D on a Dirac bundle S is formally selfadjoint,i.e.

(Ds1, s2) = (s1, Ds2)

for any smooth sections s1, s2 ∈ C∞(S) with compact supports on M .

Proof. Fix a point x0 ∈ M and a local basis e1, . . . , en of the bundle TM ina neighborhood of the point x0 such that ∇ei

ej = 0 at x0. (Such basis may beconstructed by choosing it first at x0 and then extending to a neighborhood of x0

by the parallel transport along the geodesics with initial point at x0.) Then thecomputation at x0 will give:

〈Ds1, s2〉 =∑

j

〈ej∇ejs1, s2〉 = (Formula (3.2))

−∑

j

〈∇ejs1, ej · s2〉 = (compatibility with Riemannian structure)

−∑

j

ej〈s1, ej · s2〉 − 〈s1, (∇ej

ej)s2〉 − 〈s1, ej∇ejs2〉

=

−∑

j

ej〈s1, ej · s2〉+ 〈s1, Ds2〉.

Introduce a vector field X defined by the equality:

〈X, Y 〉 = −〈s1, Y s2〉


for an arbitrary tangent vector field Y . In terms of this vector we can rewrite thefirst term in the last formula in the above chain of equalities in the form

−∑

j

ej〈s1, ej · s2〉 = −∑

j

ej〈X, ej〉 = (adding the zero term −∑

j

〈X,∇ejej〉)

∑j

ej〈X, ej〉 − 〈X,∇ej

ej〉

= (compatibility with Riemannian structure)

∑j

〈∇ejX, ej〉 =: divX.

Thus, we have established that

〈Ds1, s2〉 = divX + 〈s1, Ds2〉.

Due to the compactness of supports of s1, s2, the first term on the right hand sidewill vanish after the integration over M and we obtain the required assertion.

Remark 15. In the case of a manifold M with boundary ∂M the above argumentyields the following Stokes formula

(Ds1, s2)− (s1, Ds2) =

∫

∂M

〈ν · s1, s2〉 vol

where ν is the exterior normal to ∂M .

Remark 16. It follows from the general theory of elliptic operators D that any weaksolution of the equation Ds = 0 is in fact C∞-smooth. If the manifold M is compactthen this theory implies also that the space of solutions of the equation Ds = 0 isfinite-dimensional.

Denote by L2(S) the space of L2-sections of the bundle S obtained by the com-pletion of the space Γ∞0 (S) of smooth sections of S with compact supports withrespect to the L2-norm, introduced above. The Dirac operator D is a symmetric op-erator on Γ∞0 (S) so it admits the completion in L2(S)-norm. The obtained operatoris an unbounded selfadjoint operator in L2(S).

Recall that the Dirac operator D has the principal symbol iξ, and its square D2

has the principal symbol ‖ξ‖2 coinciding with the principal symbol of the Laplace–Beltrami operator on M . If we introduce the spinor Laplacian ∆S given by theformula

∆S := −Trg(∇S ∇S)

where ∇S is a spinor connection on S then this operator will be related to the DiracLaplacian by the following formula

D2 = ∆S +1

4scalg

where scalg is the scalar curvature of (M, g). This formula, called the Lichnerowiczformula (cf. [7], II.8, Theorem 8.8), is a particular case of a general Weitzenbockformula.


Classical Dirac operator

Let M = Rn and S = Rn×S0 where S0 is a Clifford module over Cl(n). In this casethe Dirac operator D is a differential operator with constant coefficients of the form

D =n∑

j=1

γj∂

∂xj

,

acting on S0-valued functions defined on Rn. Here, γj are the Dirac matrices , i.e.linear maps γj : S0 → S0 satisfying the relations

γjγk + γkγj = −2δjk

for all j, k = 1, . . . , n. These relations imply that D2 = ∆ ·Id where ∆ = −∑nj=1

∂2

∂x2j

is the Laplacian on Rn.


Particular cases

1. For n = 1 the Clifford algebra Cl(1) = C, and the Dirac operator D = i ∂∂x1

.

2. For n = 2 the Clifford algebra Cl(2) = H = C⊕ C = Clev(2)⊕ Clod(2) and Dpermutes Clev(2) and Clod(2). Introduce on H the real coordinates by writingquaternions q ∈ H in the form q = x01 + x1e1 + x2e2 + x3e3. If we identifyClev(2) and Clod(2) with C using the maps u+e2e1 ↔ u+ iv ↔ ue1 +ve2 thenthe operator D = e1∂/∂x1 + e2∂/∂x2 will be given by the matrix

D =

(0 −∂/∂z

∂/∂z 0

)

where ∂/∂z = ∂/∂x1 + i∂/∂x2. In other words, the restriction of this operatorto the space Clev(2) coincides with Cauchy–Riemann operator.

3. For n = 3 the Clifford algebra Cl(3) = H⊕H, S0 = H. This algebra has tworepresentations in H acting in the following way. Identify the space R3 withthe space ImH of imaginary quaternions by introducing in the space ImH thestandard basis formed by the imaginary units i, j, k. Then the action of theserepresentations in H will be given by the multiplication by basic quaternionsfrom the left or from the right. Choosing the left action we shall see that theDirac operator coincides with the operator of the form

D = i∂

∂x1

+ j∂

∂x2

+ k∂

∂x3

acting on H-valued functions on the space R3 = ImH.

4. For n = 4 the Clifford algebra Cl(4) = Mat2(H), and S0 = H⊕H = Clev(4)⊕Clod(4). As in the case n = 2, the operator D permutes Clev(4) and Clod(4).We identify the space R4 with H by choosing the standard basis in H formed by1, i, j, k. Introduce the quaternion analogue of the Cauchy–Riemann operator,acting on functions H→ H, by the formula:

∂

∂q=

∂

∂x0

+ i∂

∂x1

+ j∂

∂x2

+ k∂

∂x3

or in terms of Pauli matrices

σ0 =

(1 00 1

), σ1 =

(i 00 −i

), σ2 =

(0 −11 0

), σ3 =

(0 ii 0

)

by the formula

∂

∂q= σ0

∂

∂x0

+ σ1∂

∂x1

+ σ2∂

∂x2

+ σ3∂

∂x3

.

Then the Dirac operator D, acting on functions with values in S0 = H ⊕ H,will be given by the matrix of the form

D =

(0 −∂/∂q

∂/∂q 0

).


Other examples of Dirac operators

1. Let S = Cl(M) be the Clifford bundle over TM considered as a bundle ofClifford modules over Cl(M) provided with the Clifford multiplication fromthe left. Then the corresponding Dirac operator coincides with the square rootof the Hodge Laplacian and is called the Dirac–Hodge operator .

2. Let M be a spin manifold and S is a spinor bundle over M provided with aRiemannian connection. The arising Dirac operator is called the Atiyah–Singeroperator and plays a key role in the Atiyah–Singer index theorem.

3. Using the isomorphism Cl(M) ∼= Λ∗(M) ≡ Λ∗(T ∗M), we can consider onΛ∗(M), along with the exterior derivation operator d : Λ∗(M) → Λ∗(M), theformally adjoint operator d∗ : Λ∗(M) → Λ∗(M) given on sections of Λp(M) bythe formula

d∗ = (−1)np+n+1 ∗ d∗where ∗ is the Hodge operator defined by the equality: µ ∧ ∗ν = 〈µ, ν〉 vol.Then the Dirac operator on Cl(M) ∼= Λ∗(M) will coincide with the operatorD = d + d∗, and the operator D2 with the Hodge Laplacian ∆ = dd∗ + d∗d.

3.2.4 Spinc-structures

Spinc-structures on principal bundles

Let M be a compact oriented Riemannian manifold of dimension n and PSO → M isthe principal SO(n)-bundles of frames (orthonormal bases) on M . Then the Spinc-structure on PSO → M is the pull-back of this bundle to a principal Spinc(n)-bundleover M .

More formally,

Definition 71. The Spinc-structure on the principal bundle PSO → M is a principalSpinc(n)-bundle PSpinc → M together with a Spinc(n)-equivariant bundle epimor-phism

PSpinc //

""FFFFFFFFPSO

M

where Spinc(n) acts on the bundle PSO → M by the homomorphism π : Spinc(n) →SO(n).

Associate with the bundle PSpinc → M a principal U(1)-bundle PU(1) → Mtogether with a Spinc(n)-equivariant bundle epimorphism so that the following dia-gram

PSpinc //

""FFFFFFFFPU(1)

||yyyy

yyyy

M

is commutative where Spinc(n) acts on the bundle PU(1) → M by the homomorphismδ : Spinc(n) → U(1). The complex line bundle L → M , associated with the principal


bundle PU(1) → M , is called the characteristic bundle, and its Chern class c1(L)is called the characteristic class of the considered Spinc-structure. In terms of theintroduced bundle the Spinc-structure may be also defined in the following equivalentway.

Definition 72. Let PSO → M be the principal SO(n)-bundle of frames on M .Then a Spinc-structure on PSO → M is determined by a principal Spinc(n)-bundlePSpinc → M and principal U(1)-bundle PU(1) → M together with a Spinc(n)-equivariant bundle epimorphism

PSpinc //

""FFFFFFFFPSO × PU(1)

yyrrrrrrrrrr

M

where Spinc(n) acts on the bundle PSO × PU(1) by the homomorphism (π, δ).

The given definition admits an extension to arbitrary oriented Riemannian vec-tor bundles E → M of rank n over a compact oriented Riemannian manifold Massociated with the principal bundle PSO → M , i.e.

E = PSO ×SO(n) Rn.

Definition 73. The Spinc-structure on a bundle E → M is an extension of itsstructure group from SO(n) to Spinc(n). In other words, the bundle E → Madmits a Spinc-structure if it is a bundle associated with a principal Spinc(n)-bundlePSpinc → M , i.e. there exists a Spinc(n)-equivariant bundle epimorphism makingthe following diagram

PSpinc ×Spinc(n) Rn //

''OOOOOOOOOOOOO E

ÄÄ~~~~

~~~~

M

commutative where the group Spinc(n) acts on Rn by the homomorphism π :Spinc(n) → SO(n).

If, in particular, we take for E the tangent bundle TM of an n-dimensionalRiemannian manifold M then a Spinc-structure on TM is called the Spinc-structureon the manifold M .

Examples of Spinc-structures

Proposition 35. A principal SO(n)-bundle PSO → M admits a Spinc-structure ifand only if its 2nd Stiefel–Whitney class w2(PSO) is the mod 2-reduction of someintegral class c ∈ H2(M,Z), i.e.

w2(PSO) ≡ cmod 2.

An analogous assertion holds for oriented Riemannian vector bundles E → Mover M .

Recall that a bundle PSO → M admits a Spin-structure if and only if its 2ndStiefel–Whitney class is vanishes: w2(PSO) = 0. This implies


Example 16. Any principal bundle PSO → M , provided with a spin structure, hasa canonical Spinc-structure. In this case the principal Spinc-bundle PSpinc → M isdefined as

PSpinc = PSpin ×δ U(1)

where U(1) denotes the trivial U(1)-bundle over M , and the group Spinc(n) acts onthe bundle in the right hand side as Spinc(n) = Spin(n)×Z2 U(1).

Example 17. Any complex vector bundle bundle E → M has a canonical Spin-structure. Indeed, in this case w2(E) ≡ c1(E) mod 2 so the existence of this Spin-structure follows from Proposition 35 (more precisely, its analogue for vector bun-dles).

Complex spinor bundles

Let M be a Spinc-manifold of dimension n. The complex spinor bundle over M is acomplex vector bundle S of the form

S = PSpinc ×∆cn

S0

where S0 is a Clifford module, and ∆cn : Spinc(n) → GL(S0,C) is the spin represen-

tation. The bundle S is called fundamental if the representation ∆cn is irreducible.

If n is even then there are two irreducible representations of the complexifiedClifford algebra which became equivalent after restriction to Spinc(n). So on anySpinc-manifold there exists only one fundamental spinor bundle.

Spinc-structures on complex manifolds

If a Spinc-manifold M is complex than it has a canonical Spinc-structure

Scan = Λ∗C(TM).

The Clifford multiplication ρ : Cl(M) → End Scan acts in the following way. Asso-ciate with an arbitrary tangent vector v the linear map ρ(v) : Λ∗C(TM) → Λ∗C(TM)given by the formula:

ρ(v)ξ = v ∧ ξ − ξy v∗

where v∗ is the covector Hermitian dual to v. Then ρ(v)(ρ(v)ξ) = −‖v‖2ξ so themap ρ extends by universal property to a representation of the whole Clifford algebraCl(M).

Other Spinc-structures on the manifold M provided with the canonical Spinc-structure, may be constructed by multiplying tensorially the canonical spinor bundleScan by some complex line bundle L → M , i.e. by setting

S(L) := Scan ⊗ L

andPSpinc(L) := PSpin ×δ PU(1)(L)

where PU(1)(L) is the principal U(1)-bundle associated with L, and the action of thegroup Spinc(n) on the bundle in the right hand side is given by the homomorphism(π, δ) : Spinc(n) → SO(n)× U(1).


Hence, there is an action of the group H2(M,Z), parameterizing the equivalenceclasses of complex line bundles over M , on the space of Spinc-structures. Thequotient by this action, i.e. the space of different Spinc-structures on M , is identifiedwith the group H1(M,Z2).

Relation to spin structure

The definition of Spinc-connections in the bundles with Spinc-structures is com-pletely analogous to the definition of Spin-connections in spinor bundles. Suchconnections admit the following description.

Proposition 36. Let M be a Spinc-manifold which is also spin. Then from anySpinc-structure on M , corresponding to a complex line bundle L → M , and arbitraryU(1)-connection in the associated principal bundle PU(1) → M , we can constructa canonical connection in PSpinc → M which is the pull-up of the connection inPSO × PU(1) given by the tensor product of the canonical Riemannian connection inPSO and given U(1)-connection in PU(1).

The proof cf. in [7], Proposition D.11.In fact, the spinority assumption, imposed on M , is superfluous. The given

construction extends to the case of general Spinc-manifolds M if one replaces theU(1)-connection on L by the so called virtual connection on the virtual bundle L1/2

(cf. [7], pp.396-398).In conclusion we consider the following question: how to describe the spin struc-

ture in terms of Spinc-structure. Let M be a Spinc-manifold and S is the corre-sponding complex spinor bundle provided with Hermitian metric.

Proposition 37. The manifold M is spin if and only if there exists an anti-linearisometry C : S → S having the following properties:

1. C(sf) = (Cs)f for s ∈ Γ(S), f ∈ C∞(M);

2. C(σs) = χ(σ)(Cs) for s ∈ Γ(S), σ ∈ Γ(Cl(M));

3. 〈Cs, Cs′〉 = 〈s′, s〉 for s, s′ ∈ Γ(S).

The proof cf. in [3], Theorem 9.6.


Chapter 4

NONCOMMUTATIVE SPINORGEOMETRY

4.1 Spectral triples

Definition 74. The spectral triple for an algebra A is a triple (A,H, D) consisting ofa Hilbert space H, a representation π of the algebra A acting in A by bounded linearoperators, and a selfadjoint operator D in H with compact resolvent, satisfying thefollowing property: the commutator [D, a] of the operator D with any element a ∈ A(more precisely, with the representation operator π(a) defined by this element) is abounded linear operator in H.

Definition 75. The real spectral triple for an algebra A with involution is thespectral triple with an anti-unitary operator C in the space H for which the map

b 7−→ Cb∗C−1

determines an action of the opposite algebra Ao on H commuting with the actionof the algebra A, i.e.

[a, Cb∗C−1] = 0 (4.1)

for all a, b ∈ A.

Recall that an operator C : H → H is called anti-unitary if it defines an anti-linear bijection H → H having the following anti-isometric property :

(Cξ, Cη) = (η, ξ)

for all ξ, η ∈ H and C2 = ±1.

If π is an action of the algebra A on H then the action πo of the opposite algebraAo on H is given by the formula

πo(b) := Cπ(b∗)C−1.

So Condition (4.1) means that representations π and πo commute with each other.

121

122 CHAPTER 4. NONCOMMUTATIVE SPINOR GEOMETRY

4.2 Definition of the noncommutative spinor ge-

ometry

In this section we formulate conditions determining the noncommutative spinorgeometry.

Dimension

There is a nonnegative integer n, called the dimension of geometry , forwhich D−1 ∈ Ln+(H) but D−1 /∈ Ln+

0 (H). It implies, in particular, thatthe operator |D|−n has finite Dixmier trace not equal to zero.

In the case when the algebra A and the Hilbert space H are finite-dimensionalthe dimension of geometry is set to zero.

Note that the dimension of geometry is uniquely defined by the above condition.Indeed, if an operator T ∈ Lp(H) with p ≤ n then |T |n is of trace class, hence itsDixmier trace is equal to zero. In particular, if D−1 ∈ Lr+(H) with r < n thenD−1 ∈ Lp(H) with p = n+r

2so D−1 ∈ Ln+

0 (H) violating our assumption.

4.2.1 Regularity

The given spectral triple (A,H, D) should be regular . It means that thealgebra AD := A ∪ [D,A], generated by the algebra A and all operatorsof the form [D, a] with a ∈ A, should satisfy the following condition

AD = A ∪ [D,A] ⊂ Dom∞δ

where δ(T ) := [|D|, T ].

The formulated condition means, in other words, that the algebra AD belongs tothe smooth definition domain Dom∞δ of the derivation operator δ. In particular, theoperators of the form [D, a] belong to the definition domains Dom δk of all naturalpowers δk of operator δ.

Let us consider this condition in more detail. For a regular spectral triple(A,H, D) we can introduce the Sobolev scale of Hilbert spaces

Hs := Dom |D|s, s ∈ R,

with the norm:‖ξ‖2

s := ‖ξ‖2 + ‖ |D|sξ‖2.

For s > t there is a continuous embedding Hs ⊂ Ht and the intersection

H∞ :=⋂

s∈RHs =

∞⋂

k=0

Hk = Dom∞|D|

is a Frechet space with the metric given by the family of seminorms ‖ · ‖k, k ∈ N.Denote by Opr

D the space of operators of rth order , i.e. linear operators T :H∞ → H∞ such that for any s ∈ R there exists a positive constant Cs for which

‖Tξ‖s−r ≤ Cs‖ξ‖s.

4.2. DEFINITION OF THE NONCOMMUTATIVE SPINOR GEOMETRY 123

In other words, the operator T extends to a bounded operator Hs → Hs−r.The regularity condition implies that AD ⊂ Op0

D, and the operators b−|D|b|D|−1 ∈Op−1

D for any b ∈ AD.

4.2.2 Finiteness

The algebra A is a pre-C∗-algebra and the space of smooth vectors

H∞ =∞⋂

k=0

Dom Dk

is a finitely generated projective A-module.

Starting from the first condition, recall that the pre-C∗-algebra is a subalgebraA in a C∗-algebra B which is complete with respect to some locally convex topol-ogy, being finer than the topology of B, and close with respect to the holomorphicfunctional analysis.

In more detail, since the algebra A ⊂ Dom∞δ we can introduce on it a Frechettopology generated by the seminorms a 7→ ‖δk(a)‖, k ∈ N. If the algebra A iscomplete in this topology then

A =⋂n

An

where An is the Banach algebra obtained by the completion of the algebra A withrespect to the norm a 7→ ∑n

k=0 ‖δk(a)‖. In this case the algebra A evidently satisfiesthe first condition. Motivated by this observation, we introduce this requirementinto the finiteness condition also in the general case.

Consider now the second requirement in the formulation of the finiteness condi-tion. According to it, we can find a number m ∈ N and an idempotent e ∈ Matm(A)for which there is an isomorphism H∞ → mAe of left A-modules. Replacing theidempotent e by the projector p from the algebra Matm(A) by Kaplansky formulawe shall obtain that H∞ = mAp. The algebra of endomorphisms EndAH∞ will thenidentify with the algebra

pMatm(A)p = pAm ⊗AmAp.

4.2.3 Reality

A key role in the formulation of this condition plays the notion of real spectral datafor an algebra A with involution τ which depends crucially on the parity of thenumber j ≡ n mod 8.

Definition 76. The real spectral data of index j ∈ Z8 for an algebra A with invo-lution τ consist for even j of (A,H, D, C, χ), and for odd j of (A,H, D, C) where:

1. (A,H, D) is a spectral triple for the algebra A.

2. C is an anti-linear isometry of the space H compatible with involution τ ; thelatter means that CaC−1 = τ(a) for all a ∈ A.


3. for even j: χ is the grading operator on H which anti-commutes with D.

4. the operators D, C, χ satisfy the commutation relations of the form

C2 = ±1, CD = ±DC, Cχ = ±χC

where the signs depend on the number j and are listed in two following tables:

j mod8 0 2 4 6C2 = ±1 + – – +

CD = ±DC + + + +Cχ = ±χC + – + –

and

j mod 8 1 3 5 7C2 = ±1 + – – +

CD = ±DC – + – +

The choice of the signs depends crucially on the representation theory of realClifford algebras Clp,q associated with non-degenerate quadratic forms of signature(p, q). This theory is out of the scope of our lectures, we refer the interested readerto the detailed discussion of Clifford algebras Clp,q and real spectral triples to [3],Secs.5.1,9.5.

Recall that in the Hilbert space H we have representations π of the algebra Aand πo of the opposite algebra Ao. Consider the representation π⊗πo of the algebraA⊗ Ao, being the tensor product of the algebras A and Ao, given by the formula

a⊗ bo 7−→ aCb∗C−1

where ∗ is the involution in the algebra A. We can introduce an involution in A⊗Ao

by the formulaτ(a⊗ bo) := b∗ ⊗ (a∗)o.

The element of the algebra A⊗Ao in the right hand side corresponds to the operatorin H acting by the formula:

b∗CaC−1 = CaC−1b∗ = C(aC−1b∗C)C−1.

Since the conjugation operator C satisfies the condition C2 = ±1 we have C−1b∗C =Cb∗C−1 and the last off-line formula may be rewritten as the relation

b∗CaC−1 = C(aCb∗C−1)C−1

which means that the conjugation operator C is compatible with involution τ . Sowe are now in the situation to which applies Definition 76 of real spectral data forthe algebra A⊗ Ao.

The conjugation operator C satisfies the commutation relations

C2 = ±1, CD = ±DC, Cχ = ±χC

so that (A,H, D, C, χ) constitute real spectral data of index j ≡ n mod8for the algebra A⊗ Ao with involution τ .

We shall also call the conjugation operator C the real structure for the spectraltriple (A,H, D).

4.2. DEFINITION OF THE NONCOMMUTATIVE SPINOR GEOMETRY 125

4.2.4 First order

The representation πo of the algebra Ao commutes not only with rep-resentation π of the algebra A but also with all operators of the form[D, a] with a ∈ A, i.e.

[[D, a], Cb∗C−1

]= 0

for all a, b ∈ A.

This definition is symmetric with respect to A and Ao since the Jacobi identityimplies that

[[D, a], Cb∗C−1

]+

[a, [D,Cb∗C−1]

]=

[D, [a, Cb∗C−1]

]= 0

whence [a, [D,Cb∗C−1]

]= 0.

4.2.5 Orientation

Using the first order condition, we can construct a representation of Hochschildcochains on A with values in the algebra A⊗Ao. Note, first of all, that this algebrais an A-bimodule with a natural bimodule structure given by the relation

a′(a⊗ bo)a′′ := a′aa′′ ⊗ bo.

The mentioned representation is given on the homogeneous Hochschild k-cochainsfrom Ck(A,A⊗ Ao) by the formula

πD ((a⊗ bo)⊗ a1 ⊗ . . .⊗ ak) := aCb∗C−1[D, a1] . . . [D, ak].

Now we can formulate a condition determining the volume form.

There exists a Hochschild cycle c ∈ Zn(A,A⊗ Ao) such that

πD(c) = χ

in the case of even dimension n of our geometry. In the odd case thiscondition reduces to the relation πD(c) = 1.

4.2.6 Poincare duality

This condition is a reformulation of the classical Poincare duality in terms of K-theory. Namely, using the index map for the operator D, it is possible to construct(cf. [3], p.485) additive pairings on the K-groups:

Ki(A)×Ki(A) −→ Z

where i = 0, 1. Then

the Poincare duality means that the constructed additive pairings on thegroups K0(A) and K1(A) are non-degenerate.


4.2.7 Definition of noncommutative spinor geometry

Definition 77. The noncommutative spinor geometry is a spectral triple (A,H, D)satisfying the seven conditions, formulated above.

4.3 Dirac geometry as a noncommutative spinor

geometry

Let M be a compact oriented Riemannian manifold provided with a Spinc-structure.In other words, it is given a Riemannian spinor bundle S together with an anti-linearisometry C : S → S.

Definition 78. The Dirac geometry on M is a five-tuple G = (A,H, D, C, χ) where(A,H, D) is a spectral triple with

1. A = C∞(M).

2. H = L2(M, S) is the Hilbert space of spinors obtained by the completion ofthe space of smooth sections Γ∞(M, S) with respect to the norm determinedby the inner product

(s, t) :=

∫

M

〈s, t〉 vol

where 〈· , ·〉 is the Euclidean inner product on S, vol is the volume form on M .

3. D is the operator onH obtained by the closure of the Dirac operator D = ρ∇on Γ∞(M,S) given by the composition of the Clifford multiplication ρ andspinor connection ∇ ≡ ∇S.

4. C is a conjugation operator determining the spin structure on M .

5. χ = ρ(ω) is the grading operator if the dimension of M is even, and χ = 1 ifthe dimension of M is odd.

We are going to show that the Dirac geometry on M is a noncommutative spinorgeometry in the sense of Definition 74, i.e. it satisfies the seven conditions listedin Sec.4.2. Since the algebra A = C∞(M) is commutative some of these conditionssimplify. For instance, the opposite algebra Ao in this case coincides with the originalalgebra A, and the representation πo of this algebra coincides with the representationπ of the algebra A. The relation [a, Cb∗C−1] = 0 transforms into the commutativitycondition [a, b] = 0 for the algebra A.

The Hochschild cycle c ∈ Zn(A,A ⊗ Ao), determining the orientation, may beconsidered in this case as an element of Zn(A). Indeed, the representation πD fromSec.4.2.5 by bounded operators, acting in H, reduces in this case to the representa-tion of the group of Hochschild chains Ck(A) in L(H), given by the formula

πD(a0 ⊗ . . .⊗ ak) = a0[D, a1] . . . [D, ak].

4.3. DIRAC GEOMETRY AS A NONCOMMUTATIVE SPINOR GEOMETRY127

The kernel of this map contains the subcomplex Dk(A), generated by the chainsof the form a0 ⊗ . . .⊗ 1⊗ . . .⊗ ak, for which some of the elements ai is equal to 1.Pushing down to the quotient

Ωk(A) = Ck(A)/Dk(A),

we can consider πD as an A-module homomorphism πD : Ωk(A) → L(H).

The main result of this section is the following

Theorem 17. The Dirac geometry is a noncommutative spinor geometry.

In other words, it satisfies the conditions, listed in Sec.4.2. We shall give herean idea of the proof referring for the detailed proof to [3], Theorem 11.1.

4.3.1 Dimension

The dimension of the geometry G coincides with the dimension of the manifold M .Indeed, the square of the Dirac operator D2 has the principal symbol

σ2(D2)(x, ξ) = ‖ξ‖2

and so coincides with the principal symbol of the Laplace–Beltrami operator ∆ onM . Hence

σ−n(|D|−n) =(‖ξ‖2

)−n/2 · Id = σ−n(∆−n/2) · Idwhere Id is the identity operator on S.

It implies that the operator |D|−n is a measurable operator of Dixmier class.Indeed, the Wodzicki residue in this case is equal to

Res(f |D|−n) = rankS · Res(f∆−n/2)

for f ∈ C∞(M). The noncommutative integral is written in the form∫

f |D|−n = cnTr+(f |D|−n)

where

cn =n(2π)n

2[n/2]Ωn

.

In particular, ∫|D|−n = cnTr+(|D|−n) = 1,

i.e. the operator |D|−n ∈ L1+(H) but does not belong to the space L1+0 (H).

4.3.2 Regularity

Since [D, f ] = ρ(df) by Lemma 17 we have

‖[D, f ]‖ = ‖ρ(df)‖ = ‖df‖,i.e. the operator [D, f ] is bounded in H for any f ∈ C∞(M).

For the proof of regularity of the spectral triple (A,H, D) we have to check alsothat the algebra AD, generated by A and [D,A], lies in the smooth definition domainDom∞δ where δ(T ) := [|D|, T ]. We refer for the proof of this fact to [3], p. 489.


4.3.3 Finiteness

The algebra A = C∞(M) is closed with respect to the holomorphic functional cal-culus since a function f ∈ C∞(M) is invertible in this algebra if and only if it hasno zeros but in this case the inverse function 1/f also belongs to C∞(M).

The smooth definition domain of the operator D coincides with H∞ = C∞(M).The latter space is a finitely generated projective module over C∞(M) by the Serre–Swan theorem (more precisely, by its smooth version).

4.3.4 Reality

The check of the fact that the five-tuple (A,H, D, C, χ) constitute real spectral datafor the algebra A = C∞(M) may be found in [3], pp. 406-407.

4.3.5 First order

This condition in the considered case takes the form

[[D, f ], g] = [df, g] = 0

for f, g ∈ C∞(M) and is evidently satisfied in the case of the commutative algebraA = C∞(M).

4.3.6 Orientation

The required Hochschild n-cycle c ∈ Zn(A) coincides in this case with the volumeform vol of the oriented Riemannian manifold M (the proof of this fact is given in[3], pp. 489-490).

4.3.7 Poincare duality

This condition is satisfied since in the case of the algebra A = C∞(M) it is reducedto the usual Poincare duality between the de Rham homology and cohomology ofthe manifold M (cf. [3], pp. 490-491).

4.4 Noncommutative spinor geometry over the al-

gebra A = C∞(M)

We call the noncommutative spinor geometry G = (A,H, D,C, χ) irreducible ifit cannot be represented as a nontrivial direct sum of two other noncommutativespinor geometries G1 = (A,H1, D1, C1, χ1) and G2 = (A,H2, D2, C2, χ2), i.e. it doesnot admit the decomposition of the form

G = (A,H1 ⊕H2, D1 ⊕ S2, C1 ⊕ C2, χ1 ⊕ χ2).

The main result of this section is the following

4.4. NONCOMMUTATIVE SPINOR GEOMETRY OVER THE ALGEBRA A = C∞(M)129

Theorem 18. Let G = (A,H, D,C, χ) be an irreducible noncommutative spinorgeometry of dimension n over the algebra A = C∞(M) where M is a compactoriented Riemannian manifold. Then

1. there exists a unique Riemannian metric g = g(D) on the manifold M withthe distance function

dg(p, q) = sup|f(p)− f(q)| : f ∈ C∞(M), ‖[D, f ]‖ ≤ 1.

2. M is a spin manifold and the Dirac operator D, corresponding to its spinstructure, differs from the the original operator D only by terms of 0th order.

In other words, the noncommutative spinor geometry over the algebra A =C∞(M) is a Dirac geometry.

As in the previous theorem we give here only an idea of the proof of Theorem18 referring for details to the book [3], Theorem 11.2.

4.4.1 Construction of the volume form

We define first the noncommutative integral using the following assertion.

Proposition 38. If G is a noncommutative spinor geometry of dimension n overthe algebra A = C∞(M) then the operator f |D|−n is measurable for any functionf ∈ C∞(M).

The proof is given in [3], Proposition 11.3.This proposition means, in other words, that the definition of the noncommuta-

tive integral ∫f |D|−n = Trω(f |D|−n)

does not depend on the choice of the form ω in the definition of Dixmier trace.It can be shown that (cf. [3], pp. 494-500) the introduced integral is positivelydefined in the sense that

∫f |D|−n > 0 for any positive element f of the algebra A.

In particular, this integral is non-degenerate which allows to introduce its densitydetermining the volume form on M .

4.4.2 Construction of the spin structure and metric

A good candidate for the role of spinor module is the space H∞ of smooth vectorswith respect to the action of operator D on H. This space by the finiteness propertyis a finitely generated projective A-module on which the algebra A = C∞(M) actsby the multiplication operators. By the Serre–Swan theorem H∞ coincides with themodule Γ∞(M, S) of smooth sections of some vector bundle S → M .

Proposition 39. On the space H∞ there exists a unique A-valued pairing ·, ·such that

(ϕ, ψ) =

∫ψ, ϕ|D|−n

for all ϕ, ψ ∈ H∞.


Proof. The space H∞ may be identified with the module mAp for some projectorp ∈ Matm(A). On mAp there is a standard Hermitian pairing

ap, bp′ ≡ ap b∗ =∑

j,k

ajpjkb∗k

so we can introduce on H∞ a new inner product by setting

(ϕ, ψ)′ :=∫ψ, ϕ′|D|−n.

This inner product is equivalent but, generally speaking, does not coincide with theoriginal inner product (ϕ, ψ). However,

(ϕ, ψ)′ = (ϕ, Tψ), ϕ, ψ ∈ H∞,

for some positive invertible operator T ∈ L(H). Then for any f ∈ C∞(M) we shallhave

(ϕ, Tfψ) = (ϕ, fψ)′ =∫fψ, ϕ′|D|−n =∫ψ, f ∗ϕ′|D|−n = (f ∗ϕ, ψ)′ = (f ∗ϕ, Tψ) = (ϕ, fTψ).

In other words, the operator T commutes with the action of the algebra A so wecan introduce a new inner product on the A-module H∞ by setting

ψ, ϕ := T−1ψ, ϕ′.

This A-valued pairing already satisfies the hypothesis of the proposition.To prove the uniqueness of the introduced pairing note that the difference of two

such pairings yields an A-valued bilinear map equal to zero on all functionals of theform g 7→ ∫

fg|D|n c f ∈ C∞(M). In particular,∫

ff ∗|D|n = 0 which is possibleonly if ff ∗ = 0 in A, i.e. f = 0.

We have identified H∞ with the module Γ∞(M, S) of smooth sections of thebundle S → M . The first order condition says that

[[D, f ], g] = 0

for all f, g ∈ C∞(M). Using the regularity property we can show that the operator[D, f ] preserves the space H∞. Hence the above equality implies that [D, f ] isthe 0th order operator on the space of sections Γ∞(M, S), i.e. it belongs to thespace Γ∞(M, EndS). In other words, this operator is a matrix-valued multiplicationoperator on H∞.

Then for arbitrary f, g ∈ C∞(M) and ψ ∈ H∞ we have

[D, fg]ψ = f [D, g]ψ + [D, f ]gψ = f [D, g]ψ + g[D, f ]ψ,

i.e.[D, fg] = f [D, g] + g[D, f ].


It means that D is a matrix-valued differential operator of 1st order acting on smoothsections of the bundle S → M .

The principal symbol of this operator is an operator-valued function σ1(D) de-fined on the cotangent bundle T ∗M . We compute it using the following formulafrom the theory of differential operators:

σ1(D)(x, ξ) = limt→∞

1

te−itf(x)Deitf(x)

fulfilled for any function f ∈ C∞(M) such that df(x) = ξ. Using the L’Hopital rulein the limit in the right hand side we can rewrite the latter formula as

σ1(D)(x, ξ) = limt→∞

d

dte−itf(x)Deitf(x).

Note that for any 1-form η ∈ Ω1(M) the map

x 7−→ σ1(D)(x, ηx)

determines a smooth section from the space Γ∞(M, EndS). Denote by ρ(η) thesection of the form

ρ(η)(x) := −iσ1(D)(x, ηx). (4.2)

Then for η = df , ξ = df(x) we shall have

ρ(df)(x) := −iσ1(D)(x, ξ) = −i limt→∞

d

dte−itf(x)Deitf(x) =

limt→∞

e−itf(x)[D, f ]eitf(x) = [D, f ](x)

where we have used the fact that [D, f ] is a multiplication operator in the lastequality. Hence,

[D, f ] = ρ(df).

The Formula (4.2) determines the Clifford action of 1-forms from Ω1(M) on thespace H∞. Moreover, −ρ(η)2 generates a non-degenerate metric associated with thequadratic form on T ∗M :

gx(ηx, ηx) = g−1(η, η)(x) := −ρ(ηx)2. (4.3)

This metric is Riemannian since

−ρ(η)2(x) = σ2(D2)(x, ηx)

coincides with the principal symbol of a positive definite operator.By Formula (4.3) the Clifford action ρ extends to the whole Clifford bundle

Cl(M). Thus, we have constructed the spinor bundle S → M with the Cliffordaction ρ : Cl(M) → EndS, i.e. a Spinc-structure on M .

We turn now to the distance function determined by the introduced metric g. Letf ∈ C∞(M) and dg is the distance function on M associated with the constructed


metric. If γ : [0, 1] → M is a piecewise smooth curve, connecting two points p andq, then

f(q)− f(p) = f(γ(1))− f(γ(0)) =

∫ 1

0

d

dtf(γ(t))dt =

∫ 1

0

dfγ(t)(γ(t))dt =

∫ 1

0

gγ(t)(gradγ(t)f, γ(t))dt.

Using the Cauchy inequality in the integrand we get

|f(q)− f(p)| ≤∫ 1

0

|gγ(t)(gradγ(t)f, γ(t))|dt ≤∫ 1

0

|gradγ(t)f | · |γ(t)|dt ≤ ‖gradf‖∞∫ 1

0

|γ(t)|dt = ‖gradf‖∞`(γ)

where `(γ) is the length of the curve γ. So for ‖gradf‖∞ ≤ 1

|f(q)− f(p)| ≤ `(γ)

for any piecewise smooth curve γ connecting the points p and q. Hence,

|f(q)− f(p)| ≤ dg(p, q) (4.4)

and

sup|f(q)− f(p)| : f ∈ C∞(M) ‖gradf‖∞ ≤ 1 ≤ dg(p, q).

In fact, Estimate (4.4) is satisfied for any absolutely continuous functions f for whichtheir gradient is defined almost everywhere as an essentially bounded vector field.

In order to show that the supremum in Formula (4.4) coincides with dg(p, q) takefor f the function

f(x) ≡ fp(x) := dg(p, x).

It is a Lipschitz function with Lipschitz norm equal to 1 (by the triangle inequality),and the supremum in Formula (4.4), equal to dg(p, q), is attained for f .

Taking this into account, we obtain

Proposition 40. The distance between the points p and q on the manifold M maybe computed by the formula

dg(p, q) = sup|f(q)− f(p)| : f ∈ C∞(M) with ‖[D, f ]‖ ≤ 1.

Proof. Since [D, f ] = ρ(df) we have

‖ρ(df)‖2∞ = sup

x∈M‖ρ(df)(x)‖2 = sup

x∈Mg−1

x (df(x), df(x)) =

supx∈M

gx(gradxf , gradxf) = ‖gradf‖2∞

which implies the assertion of the proposition.


Thus, the distance on M is determined completely in terms of the operator D.We turn now to the spinor structure on M defined in terms of the Spinc-structure

by the conjugation operator C. For the commutative algebra A = C∞(M), due tothe coincidence of representations π and πo, we should have the equality

Cf ∗C−1 = f,

i.e. the operator C determines an anti-linear automorphism of the bundle C.If η ∈ Ω1(M) is a real 1-form then the operator C intertwines ρ(η) with −ρ(η).

Moreover, C is an anti-untary operator with respect to the pairing ·, · (cf. [3],p.505). Thus, this operator satisfies the properties listed in Proposition 37, and sodoes define a spinor structure on M .

4.4.3 Dirac operator

The Dirac operator D for the introduced spinor structure on M differs, generallyspeaking, from the original operator D but both has the same principal symbol equalto

σ1(D)(x, ηx) = −iρ(η)(x) = σ1(D)(x, ηx).

Hence, these operators differ by a term of 0th order given by a matrix-valued mul-tiplication operator acting in the space H∞:

D = D −m (4.5)

where m ∈ Γ∞(M, EndS). The matrix-valued function m has the same propertiesas the Dirac operator, namely:

m∗ = m, χm = (−1)nmχ, CmC−1 = ±m. (4.6)

Since the operators D and D are elliptic the same is true for their powers so wecan consider the noncommutative integrals of the form

∫f |D|−n defined in terms of

Wodzicki residue by the formula:∫

f |D|−n = cnRes(f |D|−n)

where cn = 12[n/2]Ωn

. The operator f |D|−n for f ∈ C∞(M) has degree −n and itsprincipal symbol is equal to

f(x)σ−n(f |D|−n) = f(x)σ−n(∆−n/2) · Id.

The Wodzicki density is given by the formula

resx(f |D|−n) = c′nf(x)√

det gx dnx

where c′n = 2[n/2]Ωn, and νg =√

det gx dnx is the density of the Riemannian metricg. So the integral ∫

f |D|−n =

∫fνg

does not depend on the term m of 0th order.


For operators given by Formula (4.5) with the term m satisfying Relations (4.6),we can introduce an action given by the noncommutative integral of the form

S(D) =

∫|D|−n+2.

The direct computation of this integral, carried out in [3], pp. 507-512, shows thatthis action functional (as a function of m) attains its absolute minimum at m = 0and this minimum is equal to

S(D) = −n− 2

24

∫

M

scalgνg,

i.e. coincides with the Hilbert–Einstein action.

Bibliography

[1] H.Bass, Algebraic K-theory, Benjamin, New York, 1968.

[2] A.Connes, Noncommutative Geometry, Academic Press, London–San Diego,1994.

[3] J.M.Gracia-Bondia, J.C.Varilly, H.Figueroa, Elements of Noncommutative Ge-ometry, Birkhauser, Boston–Basel–Berlin, 2001.

[4] L.Hormander, The Analysis of Linear Partial Differential Operators, Springer,Berlin–Heidelberg, 2003.

[5] M.Khalkhali, Basic Noncommutative Geometry, European Mathematical Soci-ety, Zurich, 2013.

[6] G.Landi, An Introduction to Noncommutative Spaces and their Geometries,Springer, Berlin, 1997.

[7] H.Lawson, M.-L.Michelsohn, Spin Geometry, Princeton University Press,Princeton, New Jersey, 1989.

[8] J.Milnor, J.D.Stacheff, Characteristic Classes, Princeton University Press,Princeton, 1974.

[9] W.Rudin, Functional Analysis, McGraw–Hill, New York, 1991.

[10] A.G.Sergeev, Lectures on Functional Analysis, Steklov Institute, Moscow, 2014(in Russian).

[11] M.E.Taylor, Pseudodifferential operators, Springer, New York, 1996.

135

Index

(AB)-correspondence, 53A-Fredholm operator, 43A-compact operator, 28C∗-algebra, 7C∗-module, 24C∗-premodule, 23Spinc-structure on a Riemannian

manifold, 119Spinc-structure on a principal bundle,

118Spinc-structure on an associated bun-

dle, 119*-algebra, 51*-homomorphism, 11

abelianization of a group, 40acyclic complex, 84adjoint bimodule, 54adjoint bundle, 104adjoint operator, 30adjoint representation, 91adjointable operator, 30algebraic K0-group, 36algebraic K1-group, 40algebraic tensor product, 24alternation, 96anti-unitary operator, 123associativity condition, 52, 53Atiyah–Janich theorem, 43Atiyah–Singer operator, 118

Bott periodicity theorem, 39

calss of symbols Sd(U), 64canonical Riemannian connection,

111Cauchy–Hadamard formula, 10chain homotopy, 84chain map, 84character, 8

characteristic bundle of a Spinc-structure, 119

characteristic class of a Spinc-structure,119

Chern character, 82Chern character of a cycle, 87chiral element, 97classical pseudodifferential operator,

65classical symbol, 64Clifford algebra, 89Clifford bundle, 105Clifford group, 92Clifford module, 98Clifford multiplication, 99Clifford representation, 98commutant of a group, 40complex spin representation, 102complex spinor bundle, 105, 120complex vector bundle, 17complex volume element, 99complexified Clifford algebra, 93conditional trace, 76connection, 79connection form, 107connection in a principal bundle, 106covariant derivative, 107covariant derivative along a vector

field, 108cross property, 26cross-norm, 26curvature, 81curvature of a connection in a princi-

pal bundle, 107curvature transform, 109cycle, 74cycle over an algebra, 74cyclic cochain, 86cyclic vector, 14

136

INDEX 137

degree of an element of Clifford alge-bra, 90

dense inner product, 51density, 67derivation of an algebra, 70DG-algebra, 72differential, 75differential graded algebra, 72differential in a DG-algebra, 72differentials of higher orders, 75dimension of geometry, 124Dirac bundle, 114Dirac geometry, 128Dirac Laplacian, 112Dirac matrices, 116Dirac operator, 112Dirac–Hodhe operator, 118Dixmier trace, 62

elliptic operator, 64, 113equivalence bimodule, 53equivalence bimodules, 50equivalent idempotents, 36even Fredholm module, 74exterior algebra, 94

faithful functor, 19finitely generated module, 19Fredholm operator, 42free module, 19full C∗-module, 53full functor, 19function σN(T ), 58function σλ(T ), 59functor Γ, 18fundamental spinor bundle, 120

Gelfand tranform, 11Gelfand–Naimark embedding theo-

rem, 14Gelfand–Naimark theorem, 12GNS-construction, 13graded product, 72grading map, 92Grothendieck group, 35group Pin(V ), 93group Spin(V ), 93group Kn(A), 39

half-exactness, 38Hilbert transform, 77Hochschild cochain, 86Hochschild cohomology, 86Hochschild homology, 84horizontal subspace, 106

ideal Lp, 58ideal Lp,q, 59idempotent, 16index of a Fredholm operator, 42index of a regular operator, 47index of an A-Fredholm operator, 49inner derivation, 70inner product, 94integral, 74involution, 7involution on the Clifford algebra, 96involutive algebra, 51irreducible spinor geometry, 130

Kalkin algebra, 41Kaplansky formula, 34Kasparov absorption theorem, 46kernel of a pseudodifferential opera-

tor, 64ketbra-operator, 27

Levi-Civita connection, 80, 111Lichnerowicz formula, 115

measurable operator, 63Morita-equivalent C∗-algebras, 53Morita-equivalent algebras, 50multiplicative group of the Cliffors al-

gebra, 91

noncommutative Atiyah–Janich theo-rem, 50

noncommutative spinor geometry, 128norm on the Clifford group, 97nuclear C∗-algebra, 26nuclear operator, 58

odd derivation, 72odd Fredholm module, 74operator of A-finite rank, 28operator of rth order, 124opposite algebra, 51

138 INDEX

orientable vector bundle, 103

polarization, 76positive element, 12positive functional, 13principal symbol, 64, 113projective module, 19projector, 25projector equivalence, 34pseudodifferential oiperator, 65pseudodifferential operator, 64pseudoinverse operator, 44pure state, 13

real spectral data, 125real spectral triple, 123real spin representation, 101real spinor bundle, 105real structure on a spectral triple, 126regular operator, 45regular spectral triple, 124Riemannian covariant derivative, 108Riemannian curvature, 108Riemannian density, 67Riemannian vector bundle, 103Riesz operators, 77

section, 17self-adjoint element, 9semispinor bundles, 105semispinor spaces, 99Serre–Swan theorem, 22smoothing operator, 64spectral radius, 10spectral triple, 123spectrum od an algebra, 8spectrum of an algebra element, 9spin manifold, 104spin structure on a vector bundle, 103spinor Laplacian, 115spinor space, 98splitting exact sequence, 18stability, 38stabilization, 28stable C∗-algebra, 28stable equivalence, 28stable quasi-isomorphic modules, 47state, 13

Stone–Weierstrass theorem, 12summability, 75suspension of a C∗-algebra, 39symbol map on the Clifford algebra,

96symmetry operator, 74

tensor product of C∗-algebras, 26tensor product of C∗-modules, 27tensor product of a C∗-module and a

C∗-algebra, 27tensor product of a Hilbert space and

a C∗-algebra, 25tensor product of Hilbert spaces, 26topological K0-group, 35torsion tensor, 111trace, 13transition functions, 17transposition, 95trivial bundle, 17trivializing covering, 17twisted adjoint representation, 92

unital algebra, 7unitalization, 8unitary equivalent C∗-modules, 30unitary operator, 30universal 1-form, 71universal DG-algebra, 73universal property of the Clifford al-

gebra, 90

vertical subspace, 106volume element of the Clifford alge-

bra, 97

Wodzicki residue, 68

, 20

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