Topics in Cyclic Theory (London Mathematical Society Student Texts) 1108479618, 9781108479615

Table of contents :
Contents
Introduction
1 Background Results
1.1 Graded Algebras
1.2 Derivations
1.3 Commutators and Traces
1.4 Tensor Algebras
1.5 Real Clifford Algebras
1.6 Lie Bracket
1.7 The Poisson Bracket
1.8 Extensions of Algebras via Modules
1.9 Deformations of the Standard Product
2 Cyclic Cocycles and Basic Operators
2.1 The Chain Complex
2.2 The λ and b Operators
2.3 Cyclic Cocycles on a Manifold
2.4 Double Complexes
2.5 The b′ and N Operators
2.6 Hochschild Cohomology
2.7 Vector Traces
2.8 Bianchi’s Identity
2.9 Projective Modules
2.10 Singular Homology
3 Algebras of Operators
3.1 The Gelfand Transform
3.2 Ideals of Compact Operators on Hilbert Space
3.3 Algebras of Operators on Hilbert Space
3.4 Fredholm Operators
3.5 Index Theory on the Circle via Toeplitz Operators
3.6 The Index Formula for Toeplitz Operators
3.7 Wallach’s Formula
3.8 Extensions of Commutative C*-Algebras
3.9 Idempotents and Generalized Toeplitz Operators
4 GNS Algebra
4.1 Idempotents and Dilations
4.2 GNS Theorem for States on a C*-Algebra
4.3 GNS Algebra
4.4 Stinespring’s Theorem
4.5 The Generalized Stinespring Theorem
4.6 Uniqueness of GNS(ρ)
4.7 Projective Hilbert Modules
4.8 Algebras Associated with the Continuous Functions on the Circle
4.9 Algebras Described by Universal Mapping Properties
4.10 The Universal GNS Algebra of the Tensor Algebra
4.11 The Cuntz Algebra
4.12 Fredholm Modules
5 Geometrical Examples
5.1 Fredholm Modules over the Circle
5.2 Heat Kernels on Riemannian Manifolds
5.3 Green’s Function
5.4 Maxwell’s Equation
5.5 Dirac Operators
5.6 Theta Summable Fredholm Modules
5.7 Duhamel’s Formula
5.8 Quantum Harmonic Oscillator
5.9 Chern Polynomials and Generating Functions
6 The Algebra of Noncommutative Differential Forms
6.1 Kahler Differentials on an Algebraic Curve
6.2 Homology of Kahler Differential Forms
6.3 Noncommutative Differential Forms ΩA
6.4 Ω1A as an A-bimodule
6.5 The Cuntz Algebra with Fedosov’s Product
6.6 Cyclic Cochains on the Cuntz Algebra
6.7 Tensor Algebra with the Fedosov Product
6.8 Completions
7 Hodge Decomposition and the Karoubi Operator
7.1 Hodge Decomposition on a Compact Riemann Surface
7.2 The b Operator and Hochschild Homology
7.3 The Karoubi Operator
7.4 Connes’s B-operator
7.5 The Hodge Decomposition
7.6 Harmonic Forms
7.7 Mixed Complexes in the Homology Setting
7.8 Homology of the Reduced Differential Forms
7.9 Cyclic Cohomology
7.10 Traces on RA and Cyclic Cocycles on A
8 Connections
8.1 Connections and Curvature on Manifolds
8.2 The Chern Character
8.3 Deforming Flat Connections
8.4 Universal Differentials
8.5 Connections on Modules over an Algebra
8.6 Derivations and Automorphisms
8.7 Lifting and Automorphisms of QA
9 Cocycles for a Commutative Algebra over a Manifold
9.1 Poisson Structures on a Manifold
9.2 Weyl Algebras
9.3 Representations of the Heisenberg Group
9.4 Quantum Trace Formula
9.5 The Poisson Bracket and Symbols
9.6 Cocycles Generated by Commutator Products
10 Cyclic Cochains
10.1 Traces Modulo Powers of an Ideal
10.2 Coalgebra
10.3 Quotienting by the Commutator Subspace
10.4 Bar Construction
10.5 Cochains with Values in an Algebra
10.6 Analogue of Ω1R for the Bar Construction
10.7 Traces Modulo an Ideal
10.8 Analogue of the Quotient by Commutators
10.9 Connes’s Chain and Cochain Bicomplexes
11 Cyclic Cohomology
11.1 Connes’s Double Cochain Complex
11.2 Connes’s S Operator
11.3 Connes’s Long Exact Sequence
11.4 A Homotopy Formula for Cocycles Associated with Traces
11.5 Universal Graded and Ungraded Cocycles
11.6 Deformations of Fredholm Modules
11.7 Homotopy Formulas
11.8 Cyclic Cocycles over the Circle
11.9 Connections over a Compact Manifold
11.10 The Trivial Bundle
11.11 Cocycles Arising from the Connection
11.12 Super Connections and Twisted Dirac Operators
12 Periodic Cyclic Homology
12.1 The X Complex and Periodic Cyclic Homology
12.2 X(A) for Commutative Differential Graded Algebras
12.3 The Canonical Filtration
12.4 The Hodge Approximation to Cyclic Theory
References
List of Symbols
Index of Subjects

Citation preview

L O N D O N M AT H E M AT I C A L S O C I E T Y S T U D E N T T E X T S Managing Editor: Ian J. Leary, Mathematical Sciences, University of Southampton, UK 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Introduction to Banach algebras, operators and harmonic analysis, GARTH DALES et al Computational algebraic geometry, HAL SCHENCK Frobenius algebras and 2-D topological quantum field theories, JOACHIM KOCK Linear operators and linear systems, JONATHAN R. PARTINGTON An introduction to noncommutative Noetherian rings (2nd Edition), K. R. GOODEARL & R. B. WARFIELD, JR Topics from one-dimensional dynamics, KAREN M. BRUCKS & HENK BRUIN Singular points of plane curves, C. T. C. WALL A short course on Banach space theory, N. L. CAROTHERS Elements of the representation theory of associative algebras I, IBRAHIM ASSEM, DANIEL SIMSON & ´ ANDRZEJ SKOWRONSKI An introduction to sieve methods and their applications, ALINA CARMEN COJOCARU & M. RAM MURTY Elliptic functions, J. V. ARMITAGE & W. F. EBERLEIN Hyperbolic geometry from a local viewpoint, LINDA KEEN & NIKOLA LAKIC Lectures on K¨ahler geometry, ANDREI MOROIANU ¨ AN ¨ ANEN ¨ Dependence logic, JOUKU VA Elements of the representation theory of associative algebras II, DANIEL SIMSON & ´ ANDRZEJ SKOWRONSKI Elements of the representation theory of associative algebras III, DANIEL SIMSON & ´ ANDRZEJ SKOWRONSKI Groups, graphs and trees, JOHN MEIER Representation theorems in Hardy spaces, JAVAD MASHREGHI ´ PETER ROWLINSON & An introduction to the theory of graph spectra, DRAGOSˇ CVETKOVIC, ´ SLOBODAN SIMIC Number theory in the spirit of Liouville, KENNETH S. WILLIAMS Lectures on profinite topics in group theory, BENJAMIN KLOPSCH, NIKOLAY NIKOLOV & CHRISTOPHER VOLL Clifford algebras: An introduction, D. J. H. GARLING Introduction to compact Riemann surfaces and dessins d’enfants, ERNESTO GIRONDO & ´ GABINO GONZALEZ-DIEZ The Riemann hypothesis for function fields, MACHIEL VAN FRANKENHUIJSEN Number theory, Fourier analysis and geometric discrepancy, GIANCARLO TRAVAGLINI Finite geometry and combinatorial applications, SIMEON BALL ¨ The geometry of celestial mechanics, HANSJORG GEIGES Random graphs, geometry and asymptotic structure, MICHAEL KRIVELEVICH et al Fourier analysis: Part I - Theory, ADRIAN CONSTANTIN ˘ Dispersive partial differential equations, M. BURAK ERDOGAN & NIKOLAOS TZIRAKIS Riemann surfaces and algebraic curves, R. CAVALIERI & E. MILES ¨ Groups, languages and automata, DEREK F. HOLT, SARAH REES & CLAAS E. ROVER Analysis on Polish spaces and an introduction to optimal transportation, D. J. H. GARLING The homotopy theory of (∞,1)-categories, JULIA E. BERGNER The block theory of finite group algebras I, M. LINCKELMANN The block theory of finite group algebras II, M. LINCKELMANN Semigroups of linear operators, D. APPLEBAUM Introduction to approximate groups, M. C. H. TOINTON Representations of finite groups of Lie type (2nd Edition), F. DIGNE & J. MICHEL Tensor products of C*-algebras and operator spaces, G. PISIER

London Mathematical Society Student Texts 97

Topics in Cyclic Theory DA N I E L G . Q U I L L E N University of Oxford and

G O R D O N B L OW E R Lancaster University

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108479615 DOI: 10.1017/9781108855846 © The Estate of Daniel G. Quillen and Gordon Blower 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. ISBN 978-1-108-47961-5 Hardback ISBN 978-1-108-79044-4 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Introduction

page 1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Background Results Graded Algebras Derivations Commutators and Traces Tensor Algebras Real Clifford Algebras Lie Bracket The Poisson Bracket Extensions of Algebras via Modules Deformations of the Standard Product

4 4 7 9 12 16 18 21 25 27

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

Cyclic Cocycles and Basic Operators The Chain Complex The λ and b Operators Cyclic Cocycles on a Manifold Double Complexes The b and N Operators Hochschild Cohomology Vector Traces Bianchi’s Identity Projective Modules Singular Homology

30 30 34 34 37 38 39 41 42 43 52

3 3.1 3.2 3.3

Algebras of Operators The Gelfand Transform Ideals of Compact Operators on Hilbert Space Algebras of Operators on Hilbert Space

55 55 59 64

vii

viii

Contents

3.4 3.5 3.6 3.7 3.8 3.9

Fredholm Operators Index Theory on the Circle via Toeplitz Operators The Index Formula for Toeplitz Operators Wallach’s Formula Extensions of Commutative C ∗ -Algebras Idempotents and Generalized Toeplitz Operators

69 71 76 78 82 86

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

87 87 88 90 91 94 96 96

4.9 4.10 4.11 4.12

GNS Algebra Idempotents and Dilations GNS Theorem for States on a C ∗ -Algebra GNS Algebra Stinespring’s Theorem The Generalized Stinespring Theorem Uniqueness of GNS(ρ) Projective Hilbert Modules Algebras Associated with the Continuous Functions on the Circle Algebras Described by Universal Mapping Properties The Universal GNS Algebra of the Tensor Algebra The Cuntz Algebra Fredholm Modules

102 102 103 104 105

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Geometrical Examples Fredholm Modules over the Circle Heat Kernels on Riemannian Manifolds Green’s Function Maxwell’s Equation Dirac Operators Theta Summable Fredholm Modules Duhamel’s Formula Quantum Harmonic Oscillator Chern Polynomials and Generating Functions

107 107 108 114 116 119 123 127 130 134

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

The Algebra of Noncommutative Differential Forms K¨ahler Differentials on an Algebraic Curve Homology of K¨ahler Differential Forms Noncommutative Differential Forms A 1 A as an A-bimodule The Cuntz Algebra with Fedosov’s Product Cyclic Cochains on the Cuntz Algebra Tensor Algebra with the Fedosov Product Completions

137 138 144 150 155 161 164 165 169

Contents

ix

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10

Hodge Decomposition and the Karoubi Operator Hodge Decomposition on a Compact Riemann Surface The b Operator and Hochschild Homology The Karoubi Operator Connes’s B-operator The Hodge Decomposition Harmonic Forms Mixed Complexes in the Homology Setting Homology of the Reduced Differential Forms Cyclic Cohomology Traces on RA and Cyclic Cocycles on A

171 172 175 178 181 183 188 189 194 195 196

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Connections Connections and Curvature on Manifolds The Chern Character Deforming Flat Connections Universal Differentials Connections on Modules over an Algebra Derivations and Automorphisms Lifting and Automorphisms of QA

201 201 206 213 215 217 221 223

9 9.1 9.2 9.3 9.4 9.5 9.6

Cocycles for a Commutative Algebra over a Manifold Poisson Structures on a Manifold Weyl Algebras Representations of the Heisenberg Group Quantum Trace Formula The Poisson Bracket and Symbols Cocycles Generated by Commutator Products

229 229 231 236 239 242 248

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Cyclic Cochains Traces Modulo Powers of an Ideal Coalgebra Quotienting by the Commutator Subspace Bar Construction Cochains with Values in an Algebra Analogue of 1 R for the Bar Construction Traces Modulo an Ideal Analogue of the Quotient by Commutators Connes’s Chain and Cochain Bicomplexes

252 252 254 256 258 261 264 266 266 268

x

Contents

11 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12

Cyclic Cohomology Connes’s Double Cochain Complex Connes’s S Operator Connes’s Long Exact Sequence A Homotopy Formula for Cocycles Associated with Traces Universal Graded and Ungraded Cocycles Deformations of Fredholm Modules Homotopy Formulas Cyclic Cocycles over the Circle Connections over a Compact Manifold The Trivial Bundle Cocycles Arising from the Connection Super Connections and Twisted Dirac Operators

269 269 270 272 273 275 278 281 283 284 285 286 289

12 12.1 12.2 12.3 12.4

Periodic Cyclic Homology The X Complex and Periodic Cyclic Homology X(A) for Commutative Differential Graded Algebras The Canonical Filtration The Hodge Approximation to Cyclic Theory

293 293 295 297 301

References

307

List of Symbols

312

Index of Subjects

317

To Daniel Quillen, in memoriam

Introduction

Daniel Gray Quillen gave graduate lectures at the Mathematical Institute in St Giles in Oxford as part of his duties as Waynflete Professor of Pure Mathematics. The audience consisted of postgraduate students, researchers and faculty members, mainly with interests in differential geometry, algebra and functional analysis. At the time, Yang–Mills theory and index theory for operators were pervasive topics in Oxford mathematics. The courses typically lasted for 16 lectures, and were intended as self-contained introductions to the research papers that Quillen was writing. Some of the material fed directly into the papers, whereas other ingredients included motivation and applications which do not feature in the published work. The latter is valuable to a newcomer to cyclic theory, and these notes emphasize the motivation and applications. Quillen kept a mathematical diary which records how he worked over topics to refine essential ideas and developed a profound understanding of diverse theories. The diaries show a remarkable breadth of interest, covering topics in several branches of mathematics. Quillen delivered lectures in traditional style, with calculations written in full detail, so the blackboard was soon filled with tensor products and commuting diagrams. As far as I am aware, he did not produce a collated set of lecture notes, and the record is primarily based upon my notes from the lectures. The material in Chapters 3 and 4 of this volume is taken from the course ‘Topics in K-theory and cyclic homology’ which Quillen gave in Hilary term 1989, followed by ‘Topics in K-theory and cyclic cohomology’ in Michaelmas 1989. The Hilary 1989 course also covered material which I defer until Chapters 8 and 11, since it required more by way of prerequisites. The title of the present book reflects the emphasis of the courses on cyclic theory and its connection with other branches of mathematics. A special issue of the Journal of K-Theory 11 (3) (2013) describes the mathematical legacy of Quillen’s

1

2

Introduction

work, particularly his contribution to K-theory. Segal [94] describes Quillen’s professional career. The original lectures were not always linearly ordered, as Quillen would revisit calculations and improve them, and digress into topics which were intended to provide motivation for forthcoming material. A significant simplification evidently emerged through his collaboration with Joachim Cuntz, and part of this work is produced here in Chapters 6 and 7, which is taken from the course ‘Cyclic cohomology and Karoubi operators’ given in Hilary term 1991, and updated in Trinity term 1992. The latter course also covered the material in Chapter 8, regarding connections, and Chapter 12 on Hodge decompositions. Cyclic theory is an aspect of noncommutative geometry. Classical geometry uses commutative algebra, particularly to describe geometrical objects with additional structure such as differentiability or smoothness. Noncommutative geometry involves noncommutative algebras and seeks to describe geometrical objects which are possibly rough, or topologically complicated. In order to understand the basic definitions of cyclic theory, it is helpful to review some of the definitions of differential geometry that point towards the noncommutative theory. These notes are intended as an introduction to some topics in cyclic theory and are assuredly not a systematic review. Apart from the historical interest in this material, Quillen’s lecture presentations seem to me more accessible than the papers and emphasize the motivation of, and output from, cyclic theory. Some readers might be surprised at the apparent lack of generality, particularly that all the calculations are carried out over fields of characteristic zero, with C as the default choice of field. Generally, the presentation emphasizes analysis and differential geometry, rather than topology and homological algebra. Quillen motivated some of his results on noncommutative differential forms by comparing the theory with the algebraic approach to K¨ahler differential forms on algebraic varieties, as we discuss in Chapter 6. This may reflect the interests of Quillen’s likely audience for the lectures, or his choice of contemporaneous reading material. His earliest researches were in the formal theory of partial differential equations, and he developed his interest in quantum mechanics and field theory. I have attempted to fill in some of the analytical details omitted from the lectures without making a meal of them, so the reader at least can appreciate what the analytical issues are and how they can be addressed. In so doing, I have used methods that can be readily understood from the viewpoint of a student; I have not introduced more difficult concepts such as metric tensor products, Lie group representations, diffusions on manifolds or Kasparov’s KK theory. There are some differences between Connes’s cyclic theory and Quillen’s which are worth highlighting. The Chern character is central to Connes’s

Introduction

3

approach, and his B operator has a natural interpretation as a boundary operator. In these notes, the Chern character is mainly discussed for commutative algebras that arise in geometry, and the definition in Chapter 8 is taken from differential geometry, using connections. Connections are fundamental to the development of cyclic theory in this book. In Chapter 7, B is introduced via the Karoubi operator κ, and the properties of B emerge from some simple algebraic computations which do not reveal the geometrical motivation or interpretation. Instead, B and κ are used to turn graded complexes of cochains into modules over principal ideal domains or local rings. Quillen’s approach to curvature and commutators fits neatly with the analytical theory of Helton and Howe, as mentioned in Chapters 3 and 9. Some of the results in Chapter 4 regarding algebraic approaches to dilations and extensions appear to be original contributions by Quillen that have the potential for further development. Chapters 6 and 7 emphasize both the similarities and differences between commutative differential forms and noncommutative differential forms. As motivation, we mention results about coordinate rings on algebraic varieties and Riemann surfaces in particular. Quillen was an assiduous craftsman of mathematics, who refined manuscripts through several stages before publication. The present notes have not undergone such a process and should be read as an outline of his ideas, rather than the finished product. My main purpose in writing these notes is to make the material available to another generation of mathematicians in the hope that they can realize the potential for further development. Some of the topics are of current research interest and there is a legacy of resistant problems which deserve further consideration. For instance, projective modules over Banach algebras are still mysterious. The references do not cover more recent results on cyclic theory or operator K-theory. The first nine chapters should be accessible to any reader with a basic graduate-level knowledge of commutative algebra, differential geometry and functional analysis. I am especially grateful to Jean Quillen for granting permission on behalf of the Quillen Estate to proceed with publication. Also, I am grateful to Glenys Luke and Roger Astley for encouragement with this project, to James Groves for inspecting the manuscript, and to Andrey Lazarev for helpful comments on an earlier draft. I record my belated but sincere thanks to Merton College Oxford, who supported the junior research fellowship that enabled me to attend Quillen’s lectures. More recently, the University of Macau supported the writing of these notes through a Senior Visiting Fellowship. Gordon Blower Lancaster University

1 Background Results

The contents of this chapter are classical, in that they refer to the algebraic structures that are used to describe classical mechanics and often involve modules over commutative algebras as in [8]. Some of the results are used in subsequent chapters, while other aspects will be generalized to quantum mechanics and to noncommutative algebras in subsequent chapters. Advanced readers can proceed to Chapter 2.

1.1 Graded Algebras A unital algebra over a field k can be regarded as triple (A,m,1) where m : A × A → A is the associative multiplication and 1 is the multiplicative unit. The associativity law (ab)c = a(bc) says that the following diagram commutes A⊗A⊗A ↓ A⊗A

−→ →

A⊗A ↓ A

a⊗b⊗c ↓ ab ⊗ c

→ a ⊗ bc ↓ ; → abc

(1.1.1)

the operation of scalar multiplication on the right gives a commuting diagram A⊗A ↓ A⊗k

−→ →

a ⊗ κ1 −→ ↓ a⊗κ →

A ↓ A

κa ↓; aκ

(1.1.2)

a similar diagram applies to multiplication from the left. Hence we identify a ⊗ κb with κa ⊗ b, and as a default we work with A ⊗k A, the tensor products over k. 4

1.1 Graded Algebras

5

An ideal I of A is a subalgebra such that am ∈ I and ma ∈ I for all a ∈ A and m ∈ I . The square of I is I 2 = spank {mp : m,p ∈ I }, which is also an ideal with I 2 ⊆ I ⊆ A, and we can proceed to form I 3,I 4, . . . likewise. Definition 1.1.1 (i) (Graded algebra) Let k be a field and A a unital algebra over k. Suppose that (An )∞ n=0 is a sequence of k-vector subspaces of A such A , in the sense that each a ∈ A has a unique expression as that A = ⊕∞ n n=0 ∞ a where aj ∈ Aj and the multiplication satisfies a finite sum a = j j =0 Am An ⊆ Am+n . We suppose that A0 = k1. (ii) (Principal ideal domain) A commutative ring R with 1 is an integral domain if xy = 0 implies x = 0 or y = 0. Suppose further that every ideal I in R has the form I = {rx : r ∈ R} = (x) for some x ∈ R. Then R is called a principal ideal domain (PID). (iii) An ideal I in a ring R is called nilpotent if I n = 0 for some n ∈ N. Example 1.1.2 (Polynomial algebra) (i) Let k be a field and t an indeter j minate. Then the polynomial algebra k[t] = { m j =0 aj t : aj ∈ k,m = 0, 1, . . .} with the usual addition and multiplication is a PID. Also with Am =  {am t m : am ∈ k} we obtain a graded algebra. Then m j =0 Aj is the space of polynomials of degree less than or equal to m, including the zero polynomial. (ii) Let R be a unital algebra which is an integral domain, and let t1, . . . ,tn be commuting indeterminates. Then the algebra of polynomials A = R[t1, . . . ,tn ] is a graded algebra where ⎧ ⎫ n ⎨ ⎬  am1,...,mn t1m1 . . . tnmn : am1,...,mn ∈ A, Am = mj = m (1.1.3) ⎩ ⎭ j =1

and where m is called the total degree, and the elements of Am are called homogeneous components of degree m. This R has derivatives ∂j such that ∂j : A → A is k-linear and mj −1

∂j am1,...,mn t1m1 . . . tnmn = mj am1,...,mn t1m1 . . . tj

. . . tnmn ,

(1.1.4)

so Leibniz’s rule holds ∂j (f g) = (∂j f )g + f (∂j g)

(f ,g ∈ A)

(1.1.5)

∂j : Am ⊆ Am−1 and ∂j ∂k = ∂k ∂j , and ∂j is an operator of degree (−1). This is the fundamental example of a commutative graded algebra, and may be compared to the following one. Example 1.1.3 (Formal power series) (i) For a given field k, let A = k[[h]] ¯ be the algebra of formal power series in h. ¯ Then there is a derivation δ : k[[h]] → k[[ h]] given by formal differentiation ¯ ¯

6

Background Results

δ:

∞ 

kn h¯ n →

n=0

∞ 

nkn h¯ n−1 .

(1.1.6)

n=1

Let I = h ¯ be the ideal generated by h. ¯ Then I n = h¯ n , and A/I n is n−1 } as a k-vector space. Then A/I n is an isomorphic to span{1, h¯ , . . . , h¯ Artinian algebra, which has a unique maximal ideal (h¯ + I n ), which is nilpotent, since (h¯ + I n )n = (0). Observe that I n I m ⊆ I m+n and A ⊃ I ⊃ I 2 ⊃ n · · · is an infinite strictly decreasing sequence of ideals with ∩∞ n=1 I = {0}. There is a derivative ∂ : A → A that satisfies ∂(f g) = (∂f )g + f ∂g, and ∂I n ⊂ I n−1 .  (ii) Now let A[h¯ −1 ] = { ∞ ¯ j ;n ∈ N;aj ∈ k} be the algebra of j =−n aj h formal Laurent series with coefficient in k that have only finitely many non zero terms. Then An = { ∞ ¯ j ;aj ∈ k} is the space of formal Laurent j =−n aj h series that have a pole of order at most n at h¯ = 0, and An is a module over A. Exercise 1.1.4 (Gradings by powers of an ideal) Let I be an ideal in a unital algebra A over a field k of characteristic zero. There is a natural filtration by powers of the ideal A ⊇ I ⊇ I2 ⊇ I3 ⊇ · · · n called the I -adic filtration. We also impose the condition ∩∞ n=1 I = {0}.

(i) Show that there is a natural multiplication map I n /I n+1 × I m /I m+1 → I n+m /I n+m+1 .

(1.1.7)

(ii) Deduce that with I 0 = A, there is a graded algebra gI (A) =

∞

I j /I j +1 .

(1.1.8)

j =0

(iii) Suppose that A is commutative and I is a maximal ideal. Show that k = A/I is a field, and gI (A) is a direct sum of k-vector spaces. (iv) Suppose that A is commutative and I is a finitely generated maximal ideal. Show that the summands in gI (A) are finitely generated. (See [8] Theorem 11.22 for more on the structure of gI (A).) (v) Show that I is an ideal of R with I 2 = 0, where  I=

0 0

 B : B ∈ Mn (C) 0

 R=

A 0

 B : A,B,D ∈ Mn (C) . D

1.2 Derivations

7

1.2 Derivations Let A and R be algebras over a field k. A homomorphism is a map ρ : A → R such that ρ(λa + μb) = λρ(a) + μρ(b), ρ(ab) = ρ(a)ρ(b)

(a,b ∈ A;λ,μ ∈ k).

(1.2.1) (1.2.2)

If A and R are unital, and ρ(1) = 1, then ρ is said to be unital. Many of the algebras we consider are not unital. The kernel (nullspace) {a : ρ(a) = 0} of any homomorphism ρ : A → R is an ideal, and conversely, any ideal arises from the kernel of some homomorphism. Given any ideal I of A, the quotient space R = A/I is an algebra with the usual multiplication and addition, and π : A → R : a → a + I is a homomorphism with kernel I ; we call this the canonical (quotient) homomorphism, and write 0 −→ I −→ A −→ R −→ 0.

(1.2.3)

Let M be a bimodule over A. This means that M is a k-vector space with operations A × M → M and M × A → M such that (i)

a(m + n) = am + an;

(m + n)a = ma + na;

(1.2.4)

(ii)

(a + b)m = am + bm;

m(a + b) = ma + bm;

(1.2.5)

(iii)

a(bm) = (ab)m;

(ma) = m(ab);

(1.2.6)

and (iv)

a(mb) = (am)b

(a,b ∈ A;m,n ∈ M).

(1.2.7)

If A has a unit 1 we generally suppose that 1m = m1 = m for all m ∈ M. In particular, A is a bimodule over A with the left and multiplication operations (a,b) → ab. Likewise, any ideal I of A is an A-bimodule. Given A-modules M and N , we write HomA (M,N ) for the space of left A-module maps ψ : M → N such that ψ(am + np) = aψ(m) + bψ(p) for all a,b ∈ A and m,p ∈ M. Definition 1.2.1 (Derivation)

A derivation is a map δ : A → M such that

δ(λa + μb) = λδ(a) + μδ(b), δ(ab) = aδ(b) + δ(a)b

(a,b ∈ A;λ,μ ∈ k).

(1.2.8) (1.2.9)

We write the space of such derivations as Derk (A,M) to emphasize that k is a field of constants. Given any bimodule M over A, we can choose m ∈ M and introduce the inner derivation δm : A → M by

8

Background Results δm (a) = [a,m] = am − ma.

(1.2.10)

It is of interest to determine the derivations that can be expressed in this form; see Section 2.6. Example 1.2.2 (Derivations on polynomials) Let k[t] be the polynomial algebra over k with the derivation ∂ : k[t] → k[t] given by the usual derivative   ∂( nj=0 aj t j ) = nj=1 j aj t j −1 . Next let dt be the formal infinitesimal with (dt)2 = 0, and introduce k[t,dt] = k[t] ⊕ k[t]dt with the multiplication    f0 (t)+f1 (t)dt g0 (t) +g1 (t)dt = f0 (t)g0 (t) +(f0 (t)g1 (t) +f1 (t)g0 (t))dt, so that R = k[t,dt] is a commutative algebra with ideal I = k[t]dt which has square zero in the sense that I 2 = 0, and we can identify A = k[t] with R/I . Also, let d(f0 (t) + f1 (t)dt) = ∂f0 (t)dt. Then R is an A-bimodule, and d : A → R gives a derivation. Example 1.2.3 (Pseudo-differential operators) (i) Let A be a unital commutative ring, and let ∂ : A → A be a derivation. Then the space of formal differential operators with coefficients in A is given by ⎧ ⎫ n ⎨ ⎬ DOA = aj ∂ j : n ∈ N,aj ∈ A ⎩ ⎭ j =0

which forms a ring under the composition rule ⎛ ⎞   n m    j⎠ k ⎝ aj ∂ bj ∂ = ◦ j =0

j,k, :0≤ ≤j ≤n,0≤k≤m

k=0

    j aj ∂ bk ∂ j +k−

k

so that A is a subring of DOA . Now we introduce the integration operator ∂ −1 , which is required to satisfy ∂ −1 ◦ a = a∂ −1 − (∂a)∂ −2 + (∂ 2 a)∂ −3 − · · · , and ∂ ◦ ∂ −1 = 1, so ∂ ◦ ∂ −1 ◦ a = a for all a ∈ A. As in [70], one can extend the composition rule to powers ∂ j with j ∈ Z via the generalized Leibniz rule. We introduce ⎧ ⎫ n ⎨  ⎬ aj ∂ j : aj ∈ A DOAn = (n ∈ Z). ⎩ ⎭ j =−∞

Then DOA = DOA ⊕ DOA−1 gives a graded algebra such that DOAn ⊆ j j +1 DOAn+1 with ∂ ◦ DOA ⊆ DOA and j

j +k

DOA ◦ DOAk ⊆ DOA

(j,k ∈ Z).

1.3 Commutators and Traces

9

For later use, we note that the commutator [X,Y ] = X ◦ Y − Y ◦ X has the special property j

j +k−1

[DOA,DOAk ] ⊂ DOA

(j,k ∈ Z),

so we lose a derivative when taking the commutator. There is an exact sequence of algebras 0 −→ DOA−1 −→ DOA0 −→ A −→ 0. Mulase [76] considered this as a graded algebra. (ii) (Weyl algebra) With A = C[z], the algebra of differential operators with polynomial coefficients DOA = C z,d/dz is known as the Weyl algebra; in Definition 9.2.1 we obtain this algebra via a more sophisticated construction. As in (i), we can extend DOA to DOA = C z,d/dz,(d/dz)−1 .

1.3 Commutators and Traces Definition 1.3.1 (i) (Commutators) Let A be an associative algebra over k, and M an A-bimodule. Then we write [a,m] = am − ma for the commutator of a ∈ A and m ∈ M, then let [A,M] = spank {[a,m];

a ∈ A,m ∈ M}

for the commutator subspace, which is a k-vector subspace of M. The commutator quotient space is M/[A,M]. (ii) (Traces) In particular, A is an A-bimodule for the standard multiplication, so [A,A] is the subspace of A spanned by the commutators. Then a trace τ : A → k is a k-linear map such that τ ([a,b]) = 0 for all a,b ∈ A equivalently, a trace is a linear function τ : A/[A,A] → k. Let I be an ideal in an associative algebra A, so that I is an A-bimodule. Then [I,I ] and [I,A] are commutator subspaces of I . A trace on I is a linear functional τ : I → k such that τ |[I,I ] = 0. Such a trace may have the stronger property that τ |[I,M] = 0. See Proposition 3.2.9(iv) for a significant example. Example 1.3.2 (Quaternions) (i) We consider an example of a noncommutative algebra, namely the quaternions. We introduce Pauli’s matrices







1 0 1 0 0 −i 0 1 σ0 = , σ1 = , σ2 = , σ3 = , 0 1 0 −1 i 0 1 0 (1.3.1)

10

Background Results

which satisfy σ02 = σ12 = σ22 = σ32 = I ;

σj σk = −iσ

(1.3.2)

for all cyclic permutations (j k ) of (123). Note that {σ0,σ1,σ2,σ3,iσ0,iσ1, iσ2,iσ3 } gives the quaternion group of eight elements. The space H = {a0 σ0 + ia1 σ1 + ia2 σ2 + ia3 σ3 : aj ∈ R} gives the algebra of quaternions, a fourdimensional noncommutative division ring over R. The elements with a0 = 0 are called pure quaternions. (ii) Let R3 have the standard unit vector basis {e1,e2,e3 } and let H → R × R3 be the linear map a0 e0 + a1 e1 + a2 e2 + a3 e3 → (a0,a) where a = (a1,a2,a3 ). We write (a0,a)∗ = (a0, − a), and introduce the multiplication (a0,a)(b0,b) = (a0 b0 − a · b,a0 b + b0 a + a × b)

(1.3.3)

where a · b is the usual scalar (dot) product of vectors and a × b is the usual vector (cross) product on R3 . This ∗ gives an anti-automorphism of the real skew ring H, such that (q1 + q2 )∗ = q1∗ + q2∗ ; Then we define a norm by (a0,a) = checks that



(q1 q2 )∗ = q2∗ q1∗ . a02 + a · a. By vector algebra, one

(a0,a)(b0,b) = (a0,a)(b0,b).

(1.3.4)

If (a0,a) ∈ H has (a0,a) = 1, then we say that (a0,a) is a unit quaternion. One checks that Sp(1) = {(a0,a) ∈ H : (a0,a) = 1} is a group. Now observe that

a0 + ia1 a2 + ia3 a0 σ0 + ia1 σ1 + ia2 σ2 + ia3 σ3 = −a2 + ia3 a0 − ia1 so that there is a bijective group homomorphism map between the group Sp(1) of unit quaternions and the group SU (2) of 2×2 unitary complex matrices that have determinant one; Sp(1) ∼ SU (2,C). (iii) (Rotations and spin) Let SO(3) be the group of all rotations of Euclidean space about the origin, namely {U ∈ M3 (R) : U t U = I, det U = 1} with matrix multiplication. There is an exact sequence of groups 0 → Z/(2) → Sp(1) → SO(3) → 0,

(1.3.5)

arising as follows. For all p ∈ Sp(1), there exists a real algebra automorphism of H given by conjugation αp (q) = pqp−1 . Note that αp ◦ αr = αpr , so p → αp is a group action; evidently the kernel is {±1}. Conversely, given any real automorphism α of H, α acts on the pure quaternions so that

1.3 Commutators and Traces

11

α(e1 ), α(e2 ) and α(e3 ) are unit pure quaternions, satisfying α(e1 )2 = α(e2 )2 = α(e3 )2 = −1 with α(e1 )α(e2 ) = α(e3 ), hence they may be identified with unit vectors α(e1 ),α(e2 ) and α(e3 ) = α(e1 ) × α(e2 ) ∈ R3 . We deduce that there exists A ∈ SO(3) that takes the frame {e1,e2,e3 } to {α(e1 ),α(e2 ),α(e3 )}. One can check that A arises from conjugation by some p ∈ Sp(1). Let Mn (H) = {[aj,k ]nj,k=1 : aj,k ∈ H}, and GLn (H) = {A ∈ Mn (H) : A ∗ ] invertible}. Then one can extend the ∗ operation to Mn (H) by [aj k ]∗ = [akj and consequently define the quaternionic symplectic group Sp(n) = {A ∈ GLn (H) : A∗ A = I }. Proposition 1.3.3 Let A be a unital algebra which is a finite-dimensional vector space over the field k and suppose that the only nilpotent ideal of A is zero. Let τ be a trace on A. Then there exist indices nj ∈ N for j = 1, . . . , and division rings Dj over k such that A=



(1.3.6)

Mnj (Dj )

j =1

and linear functionals φj : Dj → k such that τ=



φj ⊗ tracenj

(1.3.7)

j =1

where tracenj : Mnj (k) → k is the standard trace on nj × nj matrices. Proof The algebra A is semisimple, and the Artin–Wedderburn theory  applies to give the direct sum decomposition. We write τ = j =1 τj , where τj : Mnj (Dj ) → k is a trace. Let epq be the usual system of matrix units, so epq has entry 1 in place pq, and zeros elsewhere. Note that epq = ep1 e1q − e1q ep1 = [ep1,e1q ]

(p = q)

(1.3.8)

so τj (αpq epq ) = 0 for all αpq ∈ Dj ; also







0 1 1 0 0 1 0 0 = , 1 0 0 0 1 0 0 1 so one can deduce by considering suitable block matrices that τj (αpp epp ) = τj (αpp e11 ). We deduce that ⎛ ⎞ nj nj     τj ⎝ (1.3.9) αpq epq ⎠ = τj αpp e11 p,q=1

p=1

12

Background Results

so that τj = φj ⊗ tracenj for the k-linear map φj : Dj → k determined by φj (α) = τj (αe11 ) for α ∈ Dj . Remarks 1.3.4 Any finite-dimensional right A-module M is a direct sum of minimal A-modules, so M = ⊕ j =1 Mj , where Mj is isomorphic to a minimal right ideal of A, namely a row of Mnj (Dj ). Complement 1.3.5 Let k = R. Then each Dj in Proposition 1.3.3 is either the real numbers R, the complex numbers C or the quaternions H. See [57] for a discussion of division rings over R.

1.4 Tensor Algebras In this section, we introduce the tensor algebra, and then some related algebras with universal properties. Let k be a field of characteristic zero, which we later specialize to the complex numbers C or the real numbers R. Given a k-vector space V , the dual V ∗ is the space of all k-linear functionals φ : V → k. Let V ⊗0 = k, and then let V ⊗m = V ⊕ · · · ⊗ V be the m-fold tensor product of V with itself over k, so αv1 ⊗ v2 ⊕ · · · ⊗ vn = v1 ⊗ αv2 ⊕ · · · ⊗ vn = · · · = v1 ⊗ v2 ⊕ · · · ⊗ αvn (α ∈ k,vj ∈ V ). (1.4.1) Often for convenience, we abbreviate v1 ⊗ v2 ⊕ · · · ⊗ vn by (v1, . . . ,vn ), where such elementary tensors span V ⊗n . A multilinear functional on V ×m is equivalent to ϕ ∈ (V ⊗m )∗ . (i) Tensor algebra: As in Example 1.1.2 the tensor algebra over V generalizes the notion of the algebra of polynomials in several variables. Given a ⊗n where V ⊗0 = k. k-vector space V , the tensor algebra is T (V ) = ⊕∞ n=0 V The product is defined via concatenating elementary tensors V ⊗n ×V ⊗m → V ⊗(n+m) : (a1, . . . ,an ) · (b1, . . . ,bm ) = (a1, . . . ,an,b1, . . . ,bm ) (1.4.2) and taking linear combinations. The product is not generally commutative, but there is a unit 1T (V ) = (1,0,0, . . .).

(1.4.3)

The tensor algebra is characterized by the following universal property.

1.4 Tensor Algebras

13

Proposition 1.4.1 Let A be an algebra over k and φ : V → A a k-linear map. Then there exists a unique k-algebra homomorphism  : T (V ) → A such that |V = φ. Proof The basic idea is that in degree n, we can define (a1, . . . ,an ) = φ(a1 )φ(a2 ) . . . φ(an ), and extend this by linearity. See Jacobson [57] for more details. (ii) Symmetric algebra: The symmetric algebra S(V ) is T (V )/I where I = (v ⊗ w − w ⊗ v;v,w ∈ V ) is the ideal generated by all the antisymmetric tensors v ⊗ w − w ⊗ v with v,w ∈ V . Hence S(V ) is a commutative algebra. Let W = V ∗ be the dual space of linear maps V → k. Then S(V ) may be regarded as the algebra of polynomial functions on W . Lemma 1.4.2 C[t1, . . . ,tn ].

Let V = Cn . Then S(V ) is isomorphic to the algebra

Proof Let V have basis {e1, . . . ,en } and observe that S(V ) is spanned by 1 and products ej1 . . . ejr where 1 ≤ j1 ≤ · · · ≤ jr ≤ n are integers. We introduce a multi index α = (α1, . . . ,αn ) such that n

ej1 . . . ejr = e1α1 . . . enαn

(1.4.4)

and |α| = j =1 αj = r is the degree of α. Then the isomorphism S(V ) → C[t1, . . . ,tn ] is given by e1α1 . . . enαn → t1α1 . . . tnαn .

(1.4.5)

(iii) Clifford algebra: Let V be an n-dimensional vector space over k of characteristic zero. A quadratic form on V is a map q : V → k such that q(tv) = t 2 q(v) for all t ∈ k and v ∈ V , and   (1.4.6) β(v,w) = 2−1 q(v) + q(w) − q(v − w) is a bilinear form β : V × V → k. Evidently β(v,w) = β(w,v), so β is symmetric and determines q as q(v) = β(v,v). Then the Clifford algebra associated with q is Cq = T (V )/(v 2 − q(v);v ∈ V ),

(1.4.7)

which has the following universal property. Given any algebra A over k and k-linear map φ : V → A such that q(v)1A = φ(v)2 , there exists an algebra homomorphism  : T (V ) → A extending φ such that (v 2 ) = (v)2 = φ(v)2 = q(v)(1k ),

(1.4.8)

14

Background Results

so all elements of the form v 2 − q(v) are in the kernel; hence we can factor ˜ ◦ π , where π : T (V ) → Cq is the canonical homomorphism, and  =  ˜ ◦ π(v) for all v ∈ V . It ˜  : Cq → A is a homomorphism such that φ(v) =  follows easily from the universal property that Cq is unique.s We can obtain a more explicit description of Cq in specific cases. Proposition 1.4.3 (i) There exists a unique Clifford algebra Cq for any such q. (ii) There is an algebra automorphism ε : Cq → Cq given by ε(v) = −v for v ∈ V . (iii) For V of dimension n, Cq has dimension 2n . Proof Given a vector space V of dimension n over k, one can introduce a Clifford algebra as follows. Let {e1, . . . ,en } be any basis for V and q be any quadratic form on V , and introduce β as above. Then let A = spank {e1i1 e2i2 . . . enin : i1, . . . ,in ∈ {0,1}}

(1.4.9)

where we identify e10 e20 . . . en0 with 1 ∈ k. Then A has dimension 2n as a k-vector space. We multiply elements of A using associativity and the identities ej ej = 2−1 β(ej ,ej );

em ej = −ej em − 2β(ej ,em )

(j < m) (1.4.10)

to reduce the products to sums of monomials in A. By induction on n, one proves that the product A×A → A is well-defined and satisfies q(v) = v 2 . For each a ∈ A, let λa : A → A : b → ab for all b ∈ A. The map a → λa gives a representation of A as linear transformations on A, so λa may be expressed as a matrix in M2n (k). We have not yet assumed that q or β is non-degenerate, so the squares could all be zero; indeed, this is a significant case in geometry which we consider next. Lemma 1.4.4 The Clifford algebra with q = 0 is the Grassmann algebra with product (v,w) → v ∧ w so that v ∧ w = −w ∧ v. Proof In the setting of (iii) let q = 0, which leads to a case which is important in geometry. Then C0 has v 2 = 0 and vw = −wv, so C0 =

∗ 

V = k ⊕ V ⊕ ∧ 2 V ⊕ · · · ⊕ ∧n V ⊕ · · ·

(1.4.11)

is the Grassmann algebra over V . In this context, one often writes the product as (v,w) → v ∧ w, and one calls ∧k V the kth exterior power of V . Suppose

1.4 Tensor Algebras

15

that V is finite dimensional, and let {e1, . . . ,en } be a basis for V , and σ a permutation of {1, . . . ,n}. Then one checks that ej2 = 0,

ej ek = −ek ej

(1.4.12)

hence eσ (1) . . . eσ (n) = sign(σ )e1 . . . en , so the summand ∧n V has dimension one. Generally, the subspace ∧m V is spanned by ei1 . . . eim where  n  i1 < i 2 , and C0 < · · · < im with all ij ∈ {1, . . . ,n}, so ∧m V has dimension m n has dimension 2 . The highest exterior power of V is (V ) = ∧n V , all higher powers being zero. For a given permutation (σ1,σ2, . . . σn ), let N be the number of transpositions of adjacent terms needed to restore the natural order (1,2, . . . ,n); then let sgn(σ ) = (−1)N . We have eσ (1) eσ (2) . . . eσ (n) = sgn (σ )e1 e2 . . . en

(1.4.13)

for all permutations σ . Let T : V → V be a linear transformation, and consider the matrix of T with respect to the basis (ej ). Using the definition of the determinant of this matrix, one checks that T (e1 ) . . . T (en ) = det(T )e1 . . . en .

(1.4.14)

Exercise 1.4.5 (Determinants) Let V and W be k-vector spaces of the same finite dimension, and A : V → W a linear transformation. (i) Show that there is a k-algebra homomorphism (A) : T (V ) → T (W ) such that (A)|V = A. (ii) Show that there is a k-algebra homomorphism (A) : ∧ (V ) → ∧(W ) such that (A)|V = A. (iii) Let (V ) be the highest exterior power of V , and (W ) be the highest exterior power of W . Show that the restriction (A) : (V ) → (W ) gives the determinant of A. Example 1.4.6 (Grassmann algebra) On the polynomial algebra A = ∂f . Then with k[t1 . . . ,tn ], introduce the usual partial derivatives ∂j by ∂j f = ∂t j ej = dtj we let V = spank {dt1, . . . ,dtn } and form the Grassmann algebra ∧V ; then let R = k[t1, . . . ,tn ] ⊗ ∧V , which is an algebra and a left A-module. There is a k-linear map d : A → A ⊗ ∧1 V given by n  df = (∂j f ) dtj j =1

16

Background Results

such that d(f g) = f dg + gdf . We extend d to d : A ⊗ ∧V → A ⊗ ∧V by d(f ⊗ ω) = (df ) ∧ ω, so that d is a derivation on R as a left A-module. One proves that d 2 = 0.

1.5 Real Clifford Algebras In this section, we take k = R, and consider the Clifford algebras that arise when the quadratic form q is negative definite. The case of the quaternions provides the motivating example. Example 1.5.1 (Clifford matrices) Suppose that k = R, and the quadratic form on V satisfies q(v) < 0 for all v ∈ V \ {0}. Then q(v) = −v2 for some norm  ·  on V , and β is given by the corresponding inner product u,v = 2−1 (u2 + v2 − u − v2 ), so one can use the Gram–Schmidt process to produce a basis {ej } of V such that −β(ej ,em ) = ej ,em = δj m,

(1.5.1)

which is known as an orthonormal basis for V . (The Gram–Schmidt process requires us to take square roots of positive numbers, hence we have restricted attention to k = R.) In particular, let σj be the 2 × 2 Pauli matrices of Example 1.3.2. Then the 4 × 4 matrices, made up of 2 × 2 blocks

0 σ0 0 σj γ0 = , γj = (1.5.2) σ0 0 −σj 0 satisfy γ02 = I,

γ12 = γ22 = γ32 = −I

(1.5.3)

and anti-commute so that γj γk = −γk γj

(j = k).

(1.5.4)

Generally, let (ej )nj=1 be a family of p × p matrices, say with p = 2n , which satisfy the Clifford relations e em + em e = −2δm Ip ; then

We write v =



n

n 



2 am em

=−

m=1

m=1 am em ,

n 

(1.5.5)

 2 am

Ip .

m=1

and q(v) = −

n

2 m=1 am .

(1.5.6)

1.5 Real Clifford Algebras

17

Corollary 1.5.2 For each n = 1,2, . . ., there exists a real Clifford algebra Cln of dimension 2n with basis satisfying (1.5.5), which is unique up to automorphism. Proof This follows from the preceding discussion and Proposition 1.4.3. Example 1.5.3 (i) We have Cl0 = R, Cl1 = C and Cl2 = H. (ii) (Spin groups) On Cln , there is an anti-homomorphism x → x¯ such that ¯ One can define this by taking (λx + μy)− = λx¯ + μy¯ and xy = y¯ x. ei1 . . . eik → (−1)k eik . . . ei1

(1.5.7)

¯ = ε(x), and for i1 < · · · < ik and extending linearly. We observe that ε(x) − that both ε and are linear involutions. On V , we have α(v) = v¯ = −v, so α(v)v = vv ¯ = −v 2 = v2 . We proceed to construct some groups which have natural representations on this subspace. The set {v ∈ V : v = 1} is simply the sphere S n−1 in R of unit radius and centre 0, and the orthogonal group O(n) = {U ∈ Mn (R) : U U ∗ = I } is the group of linear transformations of V that stabilize S n−1 as a set. The discussion follows the approach taken to the quaternions in Example 1.3.2. First, the set of invertible elements of the algebra Cln forms a multiplicative group, denoted Cln× . This has a natural representation on Cln by conjugation, so u → ρu , where ρn (a) = ε(u)au−1 for a ∈ Cln . Next we introduce the Clifford group n , which is the subgroup of Cln× that stabilizes V as a subspace, namely n = {u ∈ Cln× : ε(u)vu−1 ∈ V , ∀v ∈ V }.

(1.5.8)

The Clifford group has a subgroup Spin(n) = {u ∈ n : ε(u) = u; uu ¯ = 1}; one checks readily that Spin(n) is a subgroup since ε is a homomorphism and − an anti-homomorphism. Clearly, ρ restricts to an action of Spin(n) on V ; moreover, one shows that ρu (v)2 = −ρu (v)ρu (v) = −uv uuv ¯ u¯ = uv2 u¯ = v2

(u ∈ Spin(n),v ∈ V ) (1.5.9)

so each ρu is a linear isometry on V , hence there is a group homomorphism Spin(n) → O(n). In [9], it is shown that there is a short exact sequence of multiplicative matrix groups {I } → {±I } → Spin(n) → SO(n) → {I }.

(1.5.10)

18

Background Results

1.6 Lie Bracket Definition 1.6.1 (Lie algebra) Let k be a field of characteristic zero and A an associative algebra over k. A Lie bracket on A is a map [ , ] : A × A → A that satisfies: (i) [λf + μg,h] = λ[f ,h] + μ[g,h]; (ii) [f ,g] = −[f ,g]; (iii) Jacobi’s identity       [f ,g],h + [h,f ],g + [g,h],f = 0 (1.6.1) for all f ,g,h ∈ A and λ,μ ∈ k, Given such a bracket, A is a Lie algebra. The centre of (A,[ , ]) is Z = {f ∈ A : [f ,g] = 0, ∀g ∈ A}. Except in trivial cases, the algebra (A,[ , ]) is not commutative and not associative. Exercise 1.6.2 (Lie algebras) (i) Let A = C ∞ (R). Verify that [f ,g] = f g  − f  g gives a Lie bracket. (ii) For V = R3 , let [f ,g] = f × g be the usual cross product of vectors. Verify Jacobi’s identity. Let g be an n-dimensional Lie algebra over k. Each X ∈ g determines a linear transformation adX on g by adX : Y → [X,Y ]. From Jacobi’s identity, we have adX ([Y,Z]) = [adX (Y ),Z] + [Y,adX (Z)],

(1.6.2)

so adX is a derivation of (g,[ , ]). Let trace be the trace on Endk (g), which we can obtain from the usual trace on Mn (k) with respect to some basis of g. Definition 1.6.3 (Killing form) The Killing form on g is the symmetric bilinear form K : g × g → k given by K(X,Y ) = trace(adX adY )

(X,Y ∈ g),

(1.6.3)

where adX adY is the composition of the endomorphisms, or equivalently the product of their matrix representations. Associated with K, there is a quadratic form q(X) = K(X,X) = trace((adX )2 ). Cartan introduced K to classify Lie algebras. When k = R and g is isomorphic to Rn , we can use Lagrange’s method to reduce the quadratic form by change of basis to a diagonal quadratic form. This involves replacing the matrix K0 that represents K as a bilinear form by S t K0 S, where S is a real and invertible matrix with transpose S t . Then K and K0 are said to be congruent.

1.6 Lie Bracket

19

Lemma 1.6.4 (Sylvester’s law of inertia) Let the rank of K be the dimension of the image of K; let the signature of K be the number of positive eigenvalues of K minus the number of negative eigenvalues of K, counted according to algebraic multiplicity. Then the rank and signature of K do not change under congruence. (1) Suppose that k = R and that K(X,Y ) = 0 for all X if and only if Y = 0. Then K is said to be non-degenerate, and the associated q has full rank. Proposition 1.6.5 (Cartan’s criterion) A finite-dimensional real Lie algebra g has a non-degenerate Killing form, if and only if g is semisimple and can be expressed as a direct sum of simple Lie algebras. Example 1.6.6 (Lie algebras) The basic examples of real Lie algebras include the following. (i) Let gl(n,C) be the space Mn (C) of n×n complex matrices with the usual product and the Lie bracket [A,B] = AB − BA. Let sl(n,C) be the subspace of X ∈ Mn (C) such that trace(X) = 0. Then sl(n,C) is a semisimple Lie algebra. (ii) For p,q,r,s ∈ Cn with p = (pj )nj=1 , etc., we introduce a bilinear form  : C2n × C2n → C by     n   qj sj  q s    , = (1.6.4) pj rj  , p r j =1

where | | denotes the determinant. Then we let sp(n,C) be the space of X ∈ M2n (C) such that     q s q s  X , + ,X = 0. (1.6.5) p r p r One easily checks that for all X,Y ∈ sp(n,C), the commutator [X,Y ] also satisfies [X,Y ] ∈ sp(n,C), so sp(n,C) is also a Lie algebra. (2) Suppose that k = R and that K(X,X) < 0 for all X ∈ g \ {0}, so q is negative definite. Then g is said to be of compact type. If G is a compact, connected and finite-dimensional Lie group, then its Lie algebra is of compact type. We encountered examples such as the spin group Sp(n) in Section 1.5. Example 1.6.7 (Infinite-dimensional Lie algebras) There are also infinitedimensional examples of spaces with Lie brackets. (3) Let M be an n-dimensional differentiable manifold, and T (M) the space of smooth vector fields, or tangent fields, so that locally F,G ∈ T (M) have the form

20

Background Results

F =

n 

fj (x)

j =1

∂ , ∂xj

G=

n 

gj (x)

j =1

∂ ∂xj

where x ∈ M and fj ∈ C ∞ (M);R). Then T (M) has a Lie bracket ⎛ ⎞  n n    ∂g ∂f i i ⎝ ⎠ ∂ . fj (x) [F,G] = − gj (x) ∂xj ∂xj ∂xi i=1

(1.6.6)

(1.6.7)

j =1

The Lie bracket is intended to be consistent with the usual composition rule n ∂ for differential operators. Given F = j =1 fj (x) ∂xj and a smooth scalar  ∂h function h, we write DF h = nj=1 fj (x) ∂x , for the derivative of h in the j n direction of the vector field (fj (x))j =1 at x, and one checks that D[F,G] h = DF (DG h) − DG (DF h), so D[F,G] = DF DG − DF DG .

(1.6.8)

There is a linear map C ∞ (M) → T (M) : f → Xf , where Xf = n ∂f j =1 ∂xj dxj , which is associated with the gradient of f . Example 1.6.8 (Divergence free fields) gence by div F =

On T (M), we define the diver-

n  ∂fj . ∂xj

(1.6.9)

j =1

Let T (M)0 = {F ∈ T (M) : div F = 0}. By a simple calculation, one checks that for all F,G ∈ T (M)0 , the commutator [F,G] is also in T (M)0 , so T (M)0 is a Lie subalgebra of T (M). In the special case of M = R3 , divergence free vector fields are important in fluid mechanics. With V = span{dxj ;j = 1, . . . ,n} we can form the skew-symmetric algebra ∧V and then the algebra C ∞ (M) ⊗ ∧∗ V ; in particular, we can let ⎧ ⎫ n ⎨ ⎬ fj dxj : fj ∈ C ∞ (M) , (1.6.10) T ∗M = ⎩ ⎭ j =1

and generally let k M be the space of k-forms with coefficients in C ∞ (M).  ∂f dxj , and d(f g) = Then we define d : C ∞ (M) → 1 M by df = nj=1 ∂x j g(df ) + f dg. Note the close analogy between df ∈ 1 M and the gradient Xf ∈ T (M). Also, there is d : 1 → 2 , given by  n n     ∂fj ∂fk dxj dxk . (1.6.11) fj dxj = (dfj )dxj = − d ∂xk ∂xj j =1

j =1

This satisfies d 2 = 0.

1≤j 0, so V is of polynomial growth. Then the Liouville measure e−βH (q,p) dq1 · · · dqn dp1 · · · dpn

(1.7.7)

is invariant under the flow generated by the canonical equations of motion, so the classical partition function  Z(β) = e−βH (q,p) dq1 · · · dqn dp1 · · · dpn (β > 0) (1.7.8) R2n

is constant with respect to time. The partition function reduces to an integral over Rn  √ n  2π Z(β) = e−βV (q) dq1 · · · dqn . n β R  Canonical forms 1.7.4 (i) On R2n , the canonical 1 form is ω = nj=1 pj dqj . The motivation for this choice is given by the calculus of variations.

24

Background Results

 (ii) The canonical 2 form is ω2 = nj=1 dqj ∧ dpj , which determines the Poisson bracket as follows. As in the preceding Section 1.6.5, f ,g ∈ C ∞ are associated with vector fields   n  n    ∂f ∂g ∂f ∂g Xg = dqj + dpj , dqj + dpj ; Xf = ∂qj ∂pj ∂qj ∂pj j =1

j =1

(1.7.9) since dqj ∧ dpj = −dpj ∧ dqj , we obtain  n   ∂f ∂g ∂f ∂g = {f ,g}, − ω2 (Xf ,Xg ) = ∂qj ∂pj ∂pj ∂qj

(1.7.10)

j =1

and we recover the Poisson bracket. One can take the existence of a nondegenerate 2 form as the starting point and deduce the existence of the Poisson bracket. An even-dimensional manifold with a non-degenerate 2 form satisfying various other axioms is called a symplectic manifold. Example 1.7.5 (A cyclic cocycle from Jacobians) In particular, we consider A = Cc∞ (R2 ;C) with the above Poisson bracket and consider  ϕ(f ,g,h) = f {g,h}dqdp. (1.7.11) R2

Suppose that the original variables (q,p) are written as differentiable functions of new variables (u,v) so that q = q(u,v) and p = p(u,v) under a bijective correspondence. Then  ∂(g,h) dqdp f ϕ(f ,g,h) = R2 ∂(q,p)  ∂(g,h) ∂(q,p) dudv = f R2 ∂(q,p) ∂(u,v)  ∂(g,h) dudv (1.7.12) = f 2 ∂(u,v) R by the Jacobian change of variables formula. This expression appears in multivariable calculus and in operator theory. Also ϕ : A3 → C is linear in each variable and we introduce bϕ : A4 → C as the following linear expression, in which we multiply successive pairs of variables and introduce a sign according to the position of the product in a cyclical fashion, bϕ(f0,f1,f2,f3 ) = ϕ(f0 f1,f2,f3 ) − ϕ(f0,f1 f2,f3 ) + ϕ(f0,f1,f2 f3 ) − ϕ(f3 f0,f1,f2 ).

(1.7.13)

1.8 Extensions of Algebras via Modules

25

We now show that bϕ(f0,f1,f2,f3 ) = 0.

(1.7.14)

Using axiom (iv), we rearrange the left-hand side as     f0 f1 {f2,f3 } − f0 {f1 f2,f3 } + f0 {f1,f2 f3 } − f3 f0 {f1,f2 }    = f0 f1 {f2,f3 } − f0 f1 {f2,f3 } − f0 f2 {f1,f3 }    + f0 {f1,f2 }f3 + f0 {f1,f3 }f2 − f3 f0 {f1,f2 } = 0.

(1.7.15)

1.8 Extensions of Algebras via Modules Let A be an algebra over k and M a left A-module. Then we form the k-vector space R = A ⊕ M and endow it with the multiplication (a,m) · (b,n) = (ab,an + mb)

(1.8.1)

so that R becomes an algebra over k. We identify M with the subspace {(0,m) : m ∈ M} of R and observe that (0,m) · (b,n) = (0,mb) and (b,n) · (0,m) = (0,bm), so M is an ideal of R. Furthermore, M has the trivial multiplication (0,m) · (0,n) = (0,0) for all m,n ∈ M, so M 2 = 0. We can realize M as the kernel of the canonical homomorphism π : R → A (a,m) → a, so we have an exact sequence 0 −→ M −→ R −→ A −→ 0.

(1.8.2)

We call this the extension of A by the square-zero ideal M. Conversely, given such an exact sequence with π : R → R/M, the canonical homomorphism, a linear lifting is a k-linear map ρ : A → R such that π ◦ ρ = I d. Then e = ρ ◦ π is a linear map on R such that e2 = e, so R = A ⊕ M, and we can write ρ(a) = (a,δ(a)) for some k-linear δ : A → M. The extension is called trivial if ρ is a homomorphism. Proposition 1.8.1 Let M be an A-bimodule and δ : A → M a derivation. Then there exists a trivial extension of A with M a square-zero ideal, a homomorphism ρ : A → E such that the homomorphism π : E → A satisfies π ◦ ρ = I dA . In particular, any inner derivation of the form δm (a) = am − ma for a ∈ A and m ∈ M gives a trivial extension.

26

Background Results

Proof We write  E=

a 0

 m : a ∈ A;m ∈ M a

so E is an algebra and consider the homomorphism

a m π: E → A : → a 0 a

(1.8.3)

(1.8.4)

to have nullspace M and

a ρ: A → E: a →  0

δ(a) a

(1.8.5)

which is a homomorphism with π ◦ ρ = I d and which is an algebra isomorphism of A to a subalgebra of E. Indeed, one checks that ρ is a homomorphism if and only if δ is a derivation. The map e : E → E given by ρ ◦ π = e is an algebra homomorphism and satisfies e2 = e and e(E) is isomorphic to A as an A-bimodule. We can regard E as having the multiplication of A deformed by the derivation δ. Definition 1.8.2 (i) (Homomorphism modulo an ideal) Let A and R be algebras over k, and I an ideal of R with canonical quotient homomorphism π : R → R/I . We say that a k-linear map ρ : A → R is a homomorphism modulo I if the map π ◦ ρ : A → R → R/I is a homomorphism; equivalently, −ρ(a0 a1 ) + ρ(a0 )ρ(a1 ) ∈ I for all a0,a1 ∈ R. (ii) (Curvature) Let ρ : A → R be a k-linear map. Then the curvature of ρ is ω : A2 → R ω(a0,a1 ) = ρ(a0 a1 ) − ρ(a0 )ρ(a0 ).

(1.8.6)

So ρ is a homomorphism if and only if its curvature is zero, so that ω = 0. Also, ρ is a homomorphism modulo I if and only if the range of ω is in I . In Sections 5.4.3 and 8.1.3 we see how this relates to curvature from differential geometry. In operator theory, the expression ω(a0,a1 ) is sometimes called the semi-commutator of ρ(a0 ) and ρ(a1 ), and the following result is used for Toeplitz operators, as in Section 3.6. Lemma 1.8.3 Let A and R be algebras over k, and let ρ : A → R be a homomorphism, modulo the ideal I of R. Then ρ induces a natural map A/[A,A] → (R/I )/[R/I,R/I ], with dual mapping from the traces on (R/I ) to the traces on A.

1.9 Deformations of the Standard Product

27

Proof We have ρ([a,b]) = ρ(ab) − ρ(a)ρ(b) + ρ(b)ρ(a) − ρ(ba) + ρ(a)ρ(b) − ρ(b)ρ(a) = ω(a,b) − ω(b,a) + [ρ(a),ρ(b)]

(1.8.7)

so the composition of ρ with the canonical quotient homomorphism π gives π ◦ ρ : A → R → R/I such that π ◦ ρ[a,b] = [π ◦ ρ(a),π ◦ ρ(b)], so π ◦ρ[A,A] ⊆ [R/I,R/I ]. The dual map takes a trace φ ∈ (R/I ) to φ ◦π ◦ρ ∈ A , where φ ◦ π ◦ ρ|[A,A] = 0 since φ|[R/I,R/I ] = 0. Example 1.8.4 Let A = C ∞ (Rn,R) and R = A[[h]] ¯ be the algebra of formal power series in the indeterminate h¯ with coefficients in A with the standard multiplication. Let ρ : A → R have the form ρ(a) = a +

∞  h¯ j j =1

j!

δj (a)

(1.8.8)

where δj : A → A is k-linear. Let I = (h), ¯ so I 2 = (h¯ 2 ), etc. Then ρ is 2 a homomorphism modulo I , provided that ρ(f g) − ρ(f )ρ(g) ∈ I 2 , so by comparing coefficients of h, ¯ we have δ1 (f g) = δ1 (f )g + f δ1 (g); that is, δ1 : A → A is a derivation. Given this, ρ is a homomorphism modulo I 3 , provided that f δ2 (g) − δ2 (f g) + δ2 (f )g = δ1 (f )δ1 (g).

(1.8.9)

To satisfy the first of these conditions, we can choose a smooth vector field v : Rn → Rn , and ∇ : C ∞ (Rn,R) → C ∞ (Rn,Rn ) the usual gradient operator ∂f ∂f , . . . , ∂x ). Then on Rn , so ∇f = ( ∂x n 1 δ1 (f ) =

n  j =1

vj (x)

∂f ∂xj

(1.8.10)

is a derivation on A.

1.9 Deformations of the Standard Product Let A be a commutative algebra over k, and for indeterminate h, ¯ let k[[h]] ¯ be the algebra of formal power series in h. ¯ Then supposing h¯ a = a h, ¯ we can introduce the k-vector space A[[h]] ¯ of formal power series in h¯ with coefficients from A and made it into an k[[h]]-bimodule with the obvious ¯

28

Background Results

operations. Likewise, given any k-bilinear map P : A×A → A, we can extend it to a k[[h]]-bimodule map P : A[[h]] ¯ ¯ × A[[h]] ¯ → A[[h]]. ¯ The space A[[h]] ¯ has an obvious multiplication arising from the formal product of power series in h¯ with coefficients in h¯ , and we write f g for this product of f ,g ∈ [[h]]. ¯ There are other multiplications one could define on A[[h]], ¯ and they satisfy the following uniqueness theorem, as in [67]. Proposition 1.9.1 such that

Suppose that ∗ is an associative multiplication on A[[h]] ¯ f  g = fg +

∞ 

h¯ n Pn (f ,g)

(1.9.1)

n=1

where Pn : A × A → A are k-bilinear maps. Then {f ,g} = P1 (f ,g) − P1 (g,f )

(1.9.2)

defines a Poisson bracket {·,·} : A × A → A. Proof The associative rule gives (a  b)  c = a  (b  c)

(a,b,c ∈ A)

(1.9.3)

so when we expand up to terms in h¯ 2 , we obtain (ab + h¯ P1 (a,b) + h¯ 2 P2 (a,b))  c = a  (bc + h¯ P1 (b,c) + h¯ 2 P2 (b,c)) (1.9.4) so 2 2 2 abc+ hP ¯ 1 (a,b)c + h¯ P2 (a,b)c + hP ¯ 1 (ab,c) + h¯ P1 (P1 (a,b),c)+ h¯ P2 (ab,c) 2 2 = abc + haP ¯ 1 (b,c) + h¯ aP2 (b,c) + h¯ P1 (a,bc) + h¯ P1 (a,P1 (b,c))

+ h¯ 2 P2 (a,bc) + O(h¯ 3 )

(1.9.5)

where the coefficients of h¯ are P1 (a,b)c + P1 (ab,c) = aP1 (b,c) + P1 (a,bc)

(1.9.6)

while the coefficients of h¯ 2 are P2 (a,b)c + P1 (P1 (a,b),c) + P2 (ab,c) = aP1 (b,c) + P1 (a,P1 (b,c)) + P2 (a,bc). Now we verify the axioms of the Poisson bracket in turn. (i) P1 is bilinear by hypothesis. (ii) Clearly the bracket is anti-symmetric.

(1.9.7)

1.9 Deformations of the Standard Product

29

(iii) We expand the bracket in terms of P1 , obtaining {ab,c} − a{b,c} − b{a,c} = P1 (ab,c) − P1 (c,ab) − aP1 (b,c) + aP1 (c,b) − bP1 (a,c) + bP1 (c,a) = 0,

(1.9.8)

after some reduction. (iv) We take one triple product and express it in terms of P1 and then P2 , obtaining {{a,b},c} = {P1 (a,b) − P1 (b,a)},c} = P1 (P1 (a,b),c)− P1 (P1 (b,a),c)−P1 (c,P1 (a,b))+P1 (c,P1 (b,a)) = −P2 (a,b)c + aP2 (b,c) − P2 (ab,c) + P2 (a,bc) + P2 (b,a)c − bP2 (a,c) + P2 (ab,c) − P2 (a,bc);

(1.9.9)

similar expansions are obtained for the other brackets by cyclically permuting a,b,c. The required identity follows by cancelling the terms. This is a uniqueness theorem and does not of itself indicate that the bracket { , } is useful or even that it is non-zero. However, we can take A = C ∞ (R2n ;R) with the Poisson bracket of Hamiltonian mechanics, introduce A[[h¯ ]] and ask: Does there exist an associative ∗ product on A[[h]] ¯ such that f  g = f g + h¯ {f ,g} + O(h¯ 2 )?

(1.9.10)

If so, can we classify all such  products? In Section 6.5, we consider the Fedosov product from [35], which gives algebras that to a first-order approximation reproduce the dynamical systems of classical mechanics.

2 Cyclic Cocycles and Basic Operators

Cyclic theory involves graded vector spaces which are linked by differentials to form chain complexes and hence produce homology groups by standard constructions from homological algebra, which we review here. The special feature of cyclic theory is the role of cyclic permutation operators on these chain complexes. In this chapter, we introduce the basic operators of cyclic theory and the notion of a cyclic cocycle for an associative noncommutative algebra. We also introduce one of the motivating examples of a cyclic cocycle on a compact differentiable manifold. The other notion is that of homomorphism modulo, an ideal of an algebra, which leads to an algebraic notion of curvature. One of the fundamental aspects of cyclic theory is the pairing with K-theory. For a commutative algebra A, we introduce the group K0 (A) for finitely generated projective A modules.

2.1 The Chain Complex Definition 2.1.1 (Chain complex) (i) A chain complex is a sequence of k-vector spaces Kj , linked by k-linear maps dj : Kj → Kj −1 such that dj −1 dj = 0, which we write: ...

dj +2 −→

Kj +1

dj +1 −→

dj −→

Kj

Kj −1

dj −1 ... −→

(2.1.1)

We abbreviate this by (K,d) = (Kj ,dj ), where d has degree (−1), and we write d 2 = 0 as shorthand, even when d 2 does not make sense as a composition of operators. If σj ∈ Kj has dj σj = 0, then we say that σj is a cycle. If βj ∈ Kj has the form βj = dj +1 ωj +1 for some ωj +1 ∈ Kj +1 , then we say that βj is a boundary. By hypothesis, every boundary is a cycle. By analogy with the case of simplicial homology, we call d a differential or boundary map. 30

2.1 The Chain Complex

31

Let Zj = Ker(dj ) and Bj = Im(dj +1 ), so that Bj ⊆ Zj , and define the homology group Hj = Hj (Mj ) = Zj /Bj to be the quotient vector space. When Zj = Bj , the sequence is said to be exact at Kj ; otherwise, Hj measures the discrepancy of the complex from exactness. A short exact sequence consists of 0 −→ K2

d2 d1 K1 K0 −→ 0 −→ −→

(2.1.2)

with d1 d2 = 0, d2 injective, d1 surjective and ker(d1 ) = im(d2 ). We think of K2 as the kernel of d1 and K0 as the cokernel of d1 .   N   (ii) Let (M,d) = (Mj ,dj )N j =0 and (M ,d ) = (Mj ,dj )j =0 be complexes,  and suppose that αj : Mj → Mj are k-linear maps such that dj αj = αj −1 dj so the maps round each basic cell Mj αj ↓ Mj

dj −→ −→ dj

Mj −1 ↓ αj −1 Mj −1

(2.1.3)

commute. Then we say that α : M  → M is a map of complexes. Evidently αj : im(dj +1 ) → im(dj +1 ) and αj : ker(dj ) → ker(dj ), so there is an induced map on the homology groups α˜ j : Hj → Hj . If α˜ j is an isomorphism for all j , then we say that α is a quasi-isomorphism.   N     (iii) Let (M,d) = (Mj ,dj )N j =0 , (M ,d ) = (Mj ,dj )j =0 and (M ,d ) =   (Mj,dj )N j =0 be complexes, with α : M → M and β : M → M maps of complexes such that βj αj = 0 for all j = 0, . . . ,N. Then we write 0 −→ M 

α β M  −→ 0. M −→ −→

(2.1.4)

In particular, if all of the sequences 0 −→ Mj

αj βj Mj M  −→ 0 −→ −→ j

(2.1.5)

are short exact, then we say that the map of complexes is exact. Lemma 2.1.2 Let (2.1.5) be a short exact sequence of complexes. Then there are connecting homomorphisms δj +1 : Hj +1 (Mj+1 ) → Hj (Mj ) such that Hj +1 (Mj+1 )

δj +1 α˜ j β˜j δj Hj (Mj ) Hj (Mj ) Hj (Mj ) Hj −1 (Mj ) −→ −→ −→ −→ (2.1.6)

is a long exact sequence of the homology groups.

32

Cyclic Cocycles and Basic Operators

Proof This is a standard result of algebraic topology. Consider the following diagram, in which the downward maps are differentials d. Mj +1 ↓ Mj ↓ Mj −1

αj +1 −→ αj −→ αj −1 −→

βj +1 −→

Mj +1 ↓ Mj ↓ Mj −1

βj −→ βj −1 −→

Mj+1 ↓ Mj ↓ Mj−1

(2.1.7)

Given x ∈ Mj+1 such that dx = 0, we need to trace back to z ∈ Mj such that dz = 0. See [33, p. 638] for discussion. Definition 2.1.3 (Cochain complex) Given a chain complex, namely a sequence of k-vector spaces Kj that are linked by k-linear maps dj : Kj → Kj −1 such that dj −1 dj = 0, we introduce C j = Homk (Kj ,k) and call the elements of C j cochains. We introduce the dual map dj : C j → C j +1 so that for a suitable linear pairing cj ,dj +1 ωj +1 = dj cj ,ωj +1

(ωj +1 ∈ Kj +1,cj ∈ C j )

(2.1.8)

we introduce the following array, called a cochain complex ...

dj −1 −→

Cj

dj −→

C j +1

dj +1 −→

C j +2

dj +2 ... −→

(2.1.9)

where dj +1 dj = 0. We abbreviate this as (C,d) = (C j ,dj ), where d has degree (+1), and we write d 2 = 0 as shorthand, even when d 2 does not make sense as a composition of operators. If σj ∈ C j has dj σj = 0, then we say that σj is closed. If βj ∈ C j has the form βj = dj −1 ωj −1 for some ωj −1 ∈ C j −1 , then we say that βj is exact, as in the particular case of de Rham cohomology. By construction, every exact form is closed. Let Zj = Ker(dj ) and Bj = Im(dj −1 ), so that Bj ⊆ Zj , and we define the cohomology group H j = H j (C j ) = Zj /Bj to be the quotient vector space. n

: fj ∈ C ∞ (Rn ;C)}  ∂f and let d0 : C ∞ (Rn ;C) → 1 be the differential df = nj=1 ∂x dxj . Then j p we let  be the space of differential p-forms with coefficients in C ∞ (Rn ;C) which is spanned by ω = f0 df1 ∧ · · · ∧ dfp where fj ∈ C ∞ (Rn ;C). There is a pairing betweendifferential p-forms and simplices of dimension p via integration ω,σ = σ ω. As in Lemma 1.4.4 and Example 1.4.6, n+1 = 0. Then we define dp : p → p+1 by dω = df0 ∧ df1 ∧ · · · ∧ dfp . Then Exercise 2.1.4 (Vector calculus) Let 1 = {

j =1 fj (x)dxj

2.1 The Chain Complex

33

dp+1 dp = 0, so here d is a differential of degree (+1). So we can introduce a cochain complex 0 −→ C −→ C ∞ (Rn ;C) −→ 1 −→ · · · −→ n −→ 0.

(2.1.10)

(i) Let n = 3, and show by vector calculus that the chain complex (2.1.10) is exact. Exactness at 1 is a standard result of vector calculus which states that a vector field is irrotational if and only if it is the gradient field of some function. Thus ∇ × (∇f ) = 0 for all smooth functions f , and for all smooth vector fields F such that ∇ × F = 0, there exists some smooth scalar field f such that F = ∇f . Likewise, show that η = F1 dx2 dx3 + F2 dx3 dx1 + F3 dx1 dx2 has dη = div(F)dx1 dx2 dx3 where F = (F1,F2,F3 ), and that div(F) = 0 if and only if F = ∇ × G for some smooth vector field G. See [71, p. 558] and [97]. (ii) One can show that (2.1.10) is exact for all n ≥ 1. The key point is that for ω ∈ p such that dω = 0, there exists σ ∈ p−1 such that dσ = ω by Poincar´e’s Lemma [97, p. 145]. Hence the cohomology of this cochain complex is trivial. (iii) We can introduce C ∞ (S 1,C) and 1 over the circle S 1 . The differential form ω = dθ has dω = 0, but ω is not an exact differential. In fact, we have an exact sequence 0 −→ C −→ C ∞ (S 1 ;C) −→ 1 −→ C −→ 0,

(2.1.11)

so H 1 (• ) = C. Exercise 2.1.5 (Exactness by formal integration) (i) Let k be a field of characteristic zero, and on the polynomial algebra A = k[x1, . . . ,xn ] introduce n ∂f p xj dxj = the differential df = j =1 ∂xj dxj and the formal integral p+1

xj

/(p + 1). Prove that

d d d n −→ 0 ··· (2.1.12) 1 −→ −→ −→ is an exact sequence. One approach, as in [48], is to use induction on n and eliminate variables one at a time by integration. In applications [98], it is important that one can use formal integration, without introducing the analytical definition of the indefinite integral. (ii) Show that H 0 (kn ) = H2n (kn ) = k, and that all the other cohomology and homology groups are zero. See [48, p. 53]. (iii) Extend these results to the algebra k[[x1, . . . ,xn ]] of formal power series. 0 −→ k −→ A

34

Cyclic Cocycles and Basic Operators

2.2 The λ and b Operators Let A be an associative algebra over k which is generally non-unital and noncommutative. In this context, the qualifier ‘non-’ means not necessarily. Let [a,b] = ab−ba be the commutator of a,b ∈ A and [A,A] = span{[a,b] : a,b ∈ A} the commutator subspace. A trace on A is a linear functional τ : A → k such that τ (ab) = τ (ba)

(a,b ∈ A);

(2.2.1)

equivalently, τ ∈ (A/[A,A])∗ . See Proposition 1.3.3. Elements of A⊗n will often be called chains, whereas elements of (A⊗n )∗ will be called cochains. The latter appear more directly in geometry, as in the next section. Consider the cyclic permutation (1,2, . . . ,p) → (p,1,2, . . . ,p − 1) on p symbols. The set of powers of this permutation gives the cyclic group of order p. Associated to this is the linear operator λ : (A⊗p )∗ → (A⊗p )∗ , namely the cyclic permutation with sign, so ϕ ∈ (A⊗p )∗ has λϕ(a1, . . . ,ap ) = (−1)p−1 ϕ(ap,a1, . . . ,ap−1 );

(2.2.2)

clearly λp = id. Taking linear combinations, we define b : (A⊗p )∗ → (A⊗p+1 )∗ by bϕ(a0,a1, . . . ,ap ) =

p−1 

(−1)i ϕ(a0, . . . ,ai ai+1, . . . ,ap )

i=0

+ (−1)p ϕ(ap a0,a1, . . . ,ap−1 ).

(2.2.3)

The final summand is called the cross-over term and it warns us of curious features in calculations (compare ‘crossover music’). Definition 2.2.1 (Cyclic cocycle) A cyclic n-cocycle on A is a cochain ϕ ∈ (A⊗n+1 )∗ such that bϕ = 0 and λϕ = ϕ. In particular, a cyclic 0-cocycle on A is precisely a trace on A, since bϕ(a0,a1 ) = ϕ(a0 a1 ) − ϕ(a1 a0 ).

(2.2.4)

2.3 Cyclic Cocycles on a Manifold Let V be a smooth differentiable manifold and A = C ∞ (V ;C) the space of infinitely differentiable complex functions on V , with topology defined by a suitable family of seminorms. The multiplication is defined pointwise, so A is

2.3 Cyclic Cocycles on a Manifold

35

commutative and [A,A] = {0}. A continuous trace on A is a distribution on V , in the sense of Schwartz; namely a linear functional that is continuous with respect to the family of seminorms that determine the topology on V . Examples include f → f (x0 ) for x0 ∈ V , which is given by a point mass measure at x0 . Let ∧n A be the subspace of A⊗n consisting of alternating tensors; see Lemma 1.4.4. Then an n-dimensional current γ on A is a continuous linear functional on the space of n-forms over V . Let ϕγ : ∧n A → C be  ω → ω (2.3.1) γ

and an n-dimensional current, which we write as an integral, and associate it with  a0 da1 . . . dan . (2.3.2) ϕγ (a0, . . . ,an ) = γ

A current is said to be closed if result relates to Example 1.7.5.

 γ

dω = 0 for all ω ∈ ∧n−1 A. The following

Proposition 2.3.1 Let ϕγ be a closed n-current. Then (i) bϕγ = 0; (ii) λϕγ = ϕγ . So every closed n-current defines a cyclic cocycle on A. Proof (i) For notational convenience we take p = 3 and look at   bϕ(a0, . . . ,a3 ) = a0 a1 da2 da3 − a0 d(a1 a2 )da3 γ γ   + a0 da1 d(a2 a3 ) − a3 a0 da1 da2 γ γ    = a0 a1 da2 da3 − a0 a1 da2 da3 − a0 da1 a2 da3 γ γ γ    + a0 da1 a2 da3 + a0 a1 da2 da3 − a3 a0 da1 da2 γ

γ

γ

= 0.

(2.3.3)

(ii) Note that λϕγ (a0, . . . ,ap ) = (−1)p ϕγ (ap,a0, . . . ,ap−1 )  p ap da0 . . . dap−1 = (−1) γ

(2.3.4)

36

Cyclic Cocycles and Basic Operators

where   ap da0 . . . dap−1 = d a0 da1 . . . dap−1 dap − (−1)p−1 a0 da1 . . . dap, (2.3.5) so when the current is closed   p ap da0 . . . dap−1 = a0 da1 . . . dap, (−1) γ

(2.3.6)

γ

hence we have cyclic permutations with sign λϕγ (a0, . . . ,ap ) = ϕγ (a0, . . . ,ap ).

(2.3.7)

Corollary 2.3.2 Let V be a compact C ∞ differentiable manifold of dimension n which is orientable. Then V has a canonical cyclic n-cocycle  ϕ(a0, . . . ,an ) = a0 da1 . . . dan . (2.3.8) V

Proof The space of alternating tensors of order n has dimension one. Suppose that V admits a continuous exterior differentiable form of degree n that is nowhere zero, and choose the form that has a particular orientation. Then the integral gives a cyclic cocycle by Proposition 2.3.1. Remarks 2.3.3 The term cyclic theory refers to the special status of the operator λ. We consider the noncommutative algebra C ∞ (V ) ⊗ MN , which we can identify with the algebra of smooth functions f : V → MN (C). We can extend ϕ : A⊗(n+1) → C to ϕ(N ) : (A ⊗ MN )⊗(n+1) → C by  a0 da1 . . . dan trace(T0 . . . Tn ), (2.3.9) ϕ(N ) (a0 ⊗ T0, . . . ,an ⊗ Tn ) = V

where trace : MN → C is the usual matrix trace. Note that ϕ(N ) (aσ (0) ⊗ Tσ (0), . . . ,aσ (n) ⊗ Tσ n = ε(σ )ϕ(N ) (a0 ⊗ T0, . . . ,an ⊗ Tn ) (2.3.10) for any cyclic permutation σ of {0, . . . ,n} with sign ε(σ ). Whereas ϕ is symmetric up to the change of sign with respect to all permutations, the extension ϕ(N ) to a noncommutative algebra is symmetric up to the change of sign only for cyclic permutations. This is the origin of the term cyclic theory. A cyclic cocycle is not the same as an element of ∧(A)∗ .

2.4 Double Complexes

37

2.4 Double Complexes Let Cn,m be a doubly indexed array of k-vector spaces which are connected by k-linear maps such that the vertical differentials satisfy djv+1,p djv+1,p = 0, so the columns give chain complexes and the horizontal differentials satisfy h h dj,p = 0, so the rows give chain complexes. We also impose the dj,p+1 condition that the maps round each basic cell Cj,p+1 v ↑ dj,p Cj,p

djh+1,p −→ −→ h dj,p

Cj +1,p+1 v ↑ dj,p+1 Cj,p+1

(2.4.1)

anti-commute so h v h dj,p+1 dj,p + djv+1,p dj,p = 0.

(2.4.2)

We sometimes add operators that have the same domain Cj,k but different codomains. So d = d h + d v is to be interpreted with this in mind. Under these conditions, d 2 = 0, since for example v h v v h h + d1,1 )d1,0 + (d2,0 + d2,0 )d1,0 (d1,1 v v h h h v v h = d1,1 d1,0 + d2,0 d1,0 + (d1,1 d1,0 + d2,0 d1,0 )

=0

(2.4.3)

and likewise at other entries in the array. Hence the array is a double complex, and we can form sums over the leading diagonals Ck = ⊕ ∞ j =−∞ Ck−j,j

(2.4.4)

h v so that dk−j,j + dk−j,j : Ck−j,j → Ck−j,j +1 + Ck+1−j,j gives the total differential:

C0,2 v ↑ d0,1 C0,1 v ↑ d0,0 C0,0

h d0,2 −→ h d0,1 −→ h d0,0 −→

C1,2 v ↑ d1,1 C1,1 v ↑ d1,0 C1,0

h d1,2 −→ h d1,1 −→ h d1,0 −→

C2,2 v ↑ d2,1 C2,1 v ↑ d2,0 C2,0

h d2,2 −→ h d2,1 −→

(2.4.5)

h d2,0 −→

In examples, we often suppress the subscripts and superscripts on the differentials.

38

Cyclic Cocycles and Basic Operators

2.5 The b and N Operators Let V be a k-vector space and let the V -valued cochains of order p be C p (A,V ) = Homk (A⊗p,V ) = {f : Ap → V

multilinear}

(2.5.1)

where we identify multilinear maps on Ap with linear maps on A⊗p . Let p C(A,V ) = ⊕∞ p=0 C (A,V )

(2.5.2)

be the space of sequences of cochains which are eventually zero. The basic operations on C(A,V ) are λ and b from Section 2.3, extended in the obvious way to cochains with values in V , and b : C p (A,V ) → C p+1 (A,V ) given by 

b f (a0, . . . ,ap ) =

p−1 

(−1)i f (a0, . . . ,ai ai+1, . . . ,ap );

(2.5.3)

i=0

and N : C p (A,V ) → C p (A,V ) by Nf (a1, . . . ,ap ) =

p−1 

λi f (a1, . . . ,ap ).

(2.5.4)

i=0

Note that b and b differ only in the cross-over term, which is missing from b . Clearly λN = N, so N produces cyclically symmetric cochains. Proposition 2.5.1 b = 0, 2

The operators on C(A,V ) satisfy the basic identities  2

(b ) = 0,

(1 − λ)b = b (1 − λ),

N b = bN

(2.5.5)

so that there is a double complex: −b ↑

b↑ C p+1 (A,V )

1−λ −→

b↑ C p (A,V )

1−λ −→

b↑ C p−1 (A,V )

1−λ −→

C p+1 (A,V ) −b ↑ C p (A,V ) −b

b↑ N −→ N −→

↑

C p−1 (A,V )

C p+1 A,C) b↑ C p A,C)

1−λ −→ 1−λ −→

(2.5.6)

b↑ N −→

C p−1 A,C)

1−λ −→

Proof This follows by direct calculation. Now let R be another k-algebra and consider the (unsigned) cup product C p (A,R) ⊗ C q (A,R) → C p+q (A,R)

(2.5.7)

given by taking the product of f ∈ C p (A,R) and g ∈ C q (A,R) in R, (f · g)(a1, . . . ,ap+q ) = f (a1, . . . ,ap )g(ap+1, . . . ,ap+q ).

(2.5.8)

2.6 Hochschild Cohomology

39

Example 2.5.2 (b and curvature) Let δ = −b . Then δ satisfies the following variant of Leibniz’s rule δ(f · g) = (δf ) · g + (−1)p f · (δg).

(2.5.9)

For g,f ∈ C 1 (A,R), we have δf (a0,a1 ) = −b f (a0,a1 ) = −f (a0 a1 ),

(2.5.10)

so (δf ) · g(a0,a1,a2 ) − f · (δg)(a0,a1,a2 ) = −f (a0 a1 )g(a2 ) + f (a0 )g(a1 a2 ) = δ(f · g)(a0,a1,a2 ).

(2.5.11)

In later discussion, the following example will be important. For f ∈ C 1 (A,R), δf + f · f = 0 if and only if f : A → R is an algebra homomorphism. Indeed (δf + f · f )(a0,a1 ) = −f (a0 a1 ) + f (a0 )f (a1 ).

(2.5.12)

2.6 Hochschild Cohomology First we review Hochschild cohomology for a unital associative algebra A over C. We write C = C1 and A¯ = A/C. An A-bimodule M as in Section 1.2 is equivalent to a left A ⊗ Aop module M with (a ⊗ c,m) → amc. Let P be the resolution b

−→

A ⊗ A¯ ⊗2 ⊗ A

b

−→

A ⊗ A¯ ⊗ A

b

−→

A⊗A

b

A

b

−→ −→

0

(2.6.1)

with ¯ ⊗n,M) = HomA⊗Aop (A ⊗ (A) ¯ ⊗n ⊗ A,M) C n (A,M) = Hom((A)

(2.6.2)

consisting of multilinear f : A×n → M such that f (a1, . . . ,an ) = 0 if aj = 1 for some j . The differential is δ : C n (A,M) → C n+1 (A,M) (δf )(a0, . . . ,an ) = a0 f (a1, . . . ,an ) +

n−1 

(−1)i+1 f (a0, . . . ,ai ai+1, . . . ,an )

i=0

+ (−1)n+1 f (a0, . . . ,an−1 )an .

(2.6.3)

Now we interpret H j (A,M) for j = 0,1,2. (0) First, C 0 (A,M) = M and δm ∈ C 1 (A,M) is the inner derivation δm (a) = am − ma. It follows that

40

Cyclic Cocycles and Basic Operators H 0 (A,M) = Ker(δ : C 0 (A,M) → C 1 (A,M)) = {m ∈ M : am − ma = 0,∀a ∈ A}

(2.6.4)

is the centre of M as an A-bimodule. (1) Next we have Z 1 (A,M) = Ker(δ : C 1 (A,M) → C 2 (A,M)) = {f : A¯ → M : δf (a1,a2 ) = 0}

(2.6.5)

is the set of all derivations D : A → M. Hence H 1 (A,M) = {Derivations : A → M}/{Inner derivations : A → M}. (2.6.6) Computing H 1 (A,M) can be difficult, even for specific examples. We consider a significant example in Proposition 4.7.7, and in Section 6.4 we consider conditions that ensure that H 1 (A,M) = 0 for all M. We recall from Proposition 1.8.1 that any derivation D and linear lifting map gives a trivial extension of A by a square-zero ideal M. Next we consider non-trivial extensions. (2) As in Section 1.8, we consider an algebra extension π 0 −→ M −→ E A −→ 0 (2.6.7) −→ with M 2 = 0. In particular, we can choose a linear lifting map ρ : A → E with π ◦ ρ = I d and ρ(a) = (a,ψ(a)), so E = A ⊕ M as k-vector spaces. Then we consider ϕ : A2 → M and (a,x) · (b,y) = (ab,ay + xb + ϕ(a,b)). Then the associative law holds for this multiplication, provided that ϕ : A2 → M satisfies δϕ(a0,a1,a2 ) = 0, or more explicitly a0 ϕ(a1,a2 ) − ϕ(a0 a1,a2 ) + ϕ(a0,a1 a2 ) − ϕ(a0,a1 )a2 = 0.

(2.6.8)

Then ω(a0,a1 ) = ρ(a0 a1 ) − ρ(a0 )ρ(a1 ) = (0,ψ(a0 a1 ) − a0 ψ(a1 ) − ψ(a0 )a1 ) (2.6.9) where σ (a0,a1 ) = ψ(a0 a1 ) − a0 ψ(a1 ) − ψ(a0 )a1 satisfies σ = −δψ and δσ (a0,a1,a2 ) = 0; so σ is a coboundary. A Hochschild 2-cocycle ϕ : A×A → M can be identified with an algebra extension with M 2 = 0 and a linear lifting map ρ : A → E such that π ◦ ρ = id. We call such an extension with square zero. Changing the lifting map by a linear map alters ϕ by a coboundary. Hence H 2 (A,M) is the set of isomorphism classes of square-zero algebra extensions of A by M. (3) The H n (A,M) with n ≥ 3 are harder to interpret. Loday [68, p. 43] describes H 3 (A,M) in terms of crossed bimodules.

2.7 Vector Traces

41

In this discussion, we have said little about M. In Section 2.9, we consider some specific classes of modules for specific A.

2.7 Vector Traces Let V be a k-vector space. Let τ be a vector trace; that is τ : R → V is a k-linear function such that τ |[R,R] = 0, so τ induces a linear map p R/[R,R] → V . Let Cλ (A,V ) be the space of f ∈ Homk (A⊗p,V ) such that λf = f ; we call this the cyclic cochains of degree p. Define trτ : C p (A,R) → p−1 Cλ (A,V ) by trτ f = N τf , or more explicitly trτ f (a1, . . . ,ap ) =

p−1 

(−1)i(p−1) τf (ai+1, . . . ,ap,a1, . . . ,ai ).

(2.7.1)

i=1

In algebraic topology, there is an operation called the cup product defined on singular cohomology and a coboundary operator which is a derivation for this multiplication; see [42, p. 196]. The following result is in this spirit. Proposition 2.7.1 (i) With differential δ and the cup product ·, the cochains C(A,R) form a differential graded algebra. (ii) Let r = degree(f ) and s = degree(g). Then trτ (f · g) = (−1)rs trτ (g · f ), trτ (δf ) = −btrτ (f ).

(2.7.2) (2.7.3)

Proof (ii) Note that τ (g · f ) = λp τ (f · g), so N τ (g · f ) = N λp τ (f · g).

(2.7.4)

For r + s = p, we have (−1)i(p−1) = (−1)i(r−1)(s−1) (−1)rs . Definition 2.7.2 (Homomorphism modulo an ideal) We say that a k-linear map ρ : A → R is a homomorphism modulo I if the map π ◦ ρ : A → R → R/I is a homomorphism; equivalently, −ρ(a0 a1 ) + ρ(a0 )ρ(a1 ) ∈ I for all a0,a1 ∈ R. Alternatively, one can say that ω = δρ + ρ · ρ has ω ∈ C 2 (A,I ). Write I 2 = span{rs : r,s ∈ R}. Then we have a descending sequence of ideals of R, R ⊇ I ⊇ I 2 ⊇ I 3 ⊇ . . . . Let τ : I n → V be a k-linear map such that τ |[R,I n ] = {0}, and define   p−1 p (2.7.5) f ∈ Cλ (A,I n ; trτ (f ) = N τ (f ) ∈ Cλ (A,V ) this gives a map trτ : C(A,I n ) → Cλ (A,V ).

42

Cyclic Cocycles and Basic Operators

Proposition 2.7.3 (i) C(A,I n ) is an ideal in the differential graded algebra (C(A,R), · ,δ) which is closed under δ. (ii) For f ∈ C r (A,I n ) and g ∈ C s (A,R), trτ (f · g) = (−1)rs trτ (g · f ).

(2.7.6)

Proof This is largely a formal consequence of Proposition 2.7.1.

2.8 Bianchi’s Identity Let A and R be algebras over k, and recall that a k-linear map θ : A → R is an (algebra) homomorphism if θ (ab) = θ (a)θ (b) for all a,b, ∈ A. Now let I be an ideal in R, so R/I is also an algebra; let π : R → R/I be the canonical quotient homomorphism as in Section 1.2. Definition 2.8.1 (Curvature)

The curvature of ρ ∈ C 1 (A,R) is

ω = δρ + ρ · ρ ∈ C 2 (A,R).

(2.8.1)

Let ωn ∈ C 2n (A,R) be ωn (a1,a2, . . . ,a2n ) = ω(a1,a2 )ω(a3,a4 ) · · · ω(a2n−1, a2n ), which has n factors of curvatures in the order of paired indices. Proposition 2.8.2

Let ρ be a homomorphism modulo I n . Then

(i) δ(ωn ) = [ωn,ρ]; (ii) trτ (ωn ) is a cyclic 2n − 1 cocycle on A. Proof (i) With ω = δρ + ρ · ρ, we have δω = (δρ) · ρ − ρ · (δρ) = (δρ + ρ · ρ) · ρ − ρ · (δρ + ρ · ρ) = ω · ρ − ρ · ω.

(2.8.2)

Suppressing the · in the cup product, we use the derivation rule to obtain δ(ωn ) =

n 

ωi−1 δ(ω)ωn−i

i=1

=

n 

ωi−1 (ωρ − ρω))ωn−i

i=1 n

= [ω ,ρ].

(2.8.3)

(ii) Then btrτ (ωn ) = −trτ (δ(ωn )) = trτ (ρωn − ωn ρ) = 0,

(2.8.4)

2.9 Projective Modules

43

since ωn ∈ C 2n (A,I n ) and τ is a trace on I n . Hence trτ (ωn ) is a 2n−1 cocycle on the algebra A. Also, trτ (ωn )(a1, . . . ,a2n ) = τ (ω(a1,a2 ) · · · ω(a2n−1,a2n )) + cyclic permutations with sign

(2.8.5)

is a cyclic (2n − 1)-cocycle.

2.9 Projective Modules Let A be a ring, which is unital but not necessarily commutative, and consider left modules over A. The basic idea is that projective modules over A share many of the properties of vector spaces over fields. Mainly we formulate definitions for arbitrary rings and then give specific results for commutative unital algebras. Definition 2.9.1 (1) (Free module) Let An = ⊕nj=1 A be the direct sum of A as left A-module. Then a left A-module F is free and finitely generated if F is isomorphic to An as a left A-module. (2) (Projective) A left A-module P is said to be projective if for all left A-modules M and all surjective left A-module maps α : M → P , there exists a left A-module map β : P → M such that α ◦ β = id. (3) (Finitely generated) A left A-module P is finitely generated if there exist n ∈ N and a surjective A-module map An → P . Lemma 2.9.2 A left A-module P is finitely generated and projective if and only if there exist n and a left A-module P  such that An ∼ = P ⊕ P  as left A-modules. Such a P  is called a complementary module, and both P and P  are the images of a k-linear projection on An that is also an A-module map. Proposition 4.6 of [33] shows that projectivity of P is equivalent to the condition that R → S → 0 exact implies HomA (P ,R) → HomA (P ,S) → 0 exact. This means that any A-module map ψ : P → S lifts to an A-module map ψ˜ : P → R, as in: ψ˜ P

The map ψ˜ is called a lifting of A.

 → ψ

R ↓ S ↓ 0

44

Cyclic Cocycles and Basic Operators

Example 2.9.3 (Projective modules) (i) Let V be a finite-dimensional k-vector space. Then V is projective and finitely generated as a k-module. This is essentially the rank-plus-nullity theorem of linear algebra; indeed, for any finite-dimensional W and surjective k-linear map π : W → V , one can select the bases of V and W so that W ∼ V ⊕ ker(π ) and π may be represented as the block matrix   π∼ I 0 . (2.9.1) (ii) We can take H = Cn and write the free module as An = A ⊗ H . Note that for all A-modules N, we have N ⊗A An ∼ = N n . For any finite-dimensional vector space V , the left A-module A⊗V is projective and finitely generated. In Section 4.4, we discuss A ⊗ W as a projective module for any vector space W . We pursue this idea in later sections. Lemma 2.9.4 For a finitely generated module M over a principal ideal domain A, let the torsion submodule of M be ! T = m ∈ M : xm = 0 for some x ∈ A \ {0} . (2.9.2) Then the following conditions are equivalent: (1) M is free; (2) M is projective; (3) M is torsion free, that is, T = {0}. Proof By [47, p. 125], T is a submodule of M, and there exists a finitely generated free submodule F of M such that M ∼ = T ⊕ F . All the listed conditions are equivalent to T = 0. (iii) Lemma 2.9.4 applies to the principal ideal domains A = C[X] and A = Z. See [47]. (iv) A commutative and unital ring R is said to be a local ring if R has only one maximal ideal. A finitely generated module over a local ring is projective if and only if it is free; by [64]. For example, R = C[X]/(X2 ) = {a1 + bX + (X2 ) : a,b ∈ C} is a local ring, where (X + (X 2 )) is the maximal ideal. (v) Let R = C(X) be the field of rational complex functions, which contains the polynomial algebra C[X] as a subalgebra. Indeed, C[X] is the subalgebra of f ∈ R such that f has no poles in C, only possible poles at ∞. Then by Lemma 2.9.4, all finitely generated modules over R and all torsion-free finitely generated modules over C[X] are projective. More generally, let S be a finite subset of the Riemann sphere C ∪ {∞} and RS the algebra of rational functions on C ∪ {∞} with possible poles in S, where we regard ∞ as a pole of a monic non-constant polynomial.

2.9 Projective Modules

45

Proposition 2.9.5 All finitely generated projective modules over RS are free. Proof Let T be the multiplicatively closed subset of C[X] that is generated by 1, (X − a) for a ∈ S, and let the ring of fractions be ! T −1 C[X] = p(X)/q(X) : p(X) ∈ C[X];q(X) ∈ T as in [8, p. 36]. There is a canonical homomorphism ϕ : C[X] → T −1 C[X] : p(X) → p(X), and any ideal I in T −1 C[X] is pulled back to a contractive ideal ϕ −1 (I ) in C[X], which is a principal ideal domain, so ϕ −1 (I ) = (p(X)) for some p(X) ∈ C[X]. Hence T −1 C[X] is also a PID, which we can identify with RS . By Lemma 2.9.4, all finitely generated projective modules over RS are free. (vi) Let M be a finitely generated projective module over the polynomial ring A = C[X1, . . . ,Xn ]. Then M is free, by the Quillen–Suslin theorem. This is much more difficult to prove than the preceding examples; see [64]. Definition 2.9.6 (Projective resolution) (1) A module M has a finite projective resolution if there exists an exact sequence 0 −→ Pn −→ · · · −→ P1 −→ P0 −→ M −→ 0

(2.9.3)

where the Pj are projective A-modules. The projective dimension of M is the minimum value of n for which such a sequence exists. Thus M is projective if and only if it has projective dimension zero. In Section 6.4, we consider modules of projective dimension 0 and 1. (In [55], the authors show how the projective dimension is related to other notations of dimension for modules over commutative algebras related to algebraic varieties.) (2) Let Proj(A) be the set of isomorphism classes [P ] of finitely generated projective left A-modules P , and introduce the commutative addition rule [P ] + [Q] = [P ⊕ Q]. On Proj(A)2 we introduce an equivalence relation ∼ by ([P ],[Q]) ∼ ([P  ],[Q ]) ⇔ [P ⊕ Q ⊕ V ] = [Q ⊕ P  ⊕ V ]

(2.9.4)

for some projective finitely generated left A-module V . Definition 2.9.7 (K0 )

Let K0 (A) be Proj(A)×2 / ∼, with

([P ],[Q]) + ([P  ],[Q ]) = ([P ⊕ P  ],[Q ⊕ Q ])

(mod ∼).

Note that ([P ],[Q]) + ([P  ],[Q ]) ∼ ([P  ],[Q ]) + ([P ],[Q])

(2.9.5)

46

Cyclic Cocycles and Basic Operators

and ([P ],[Q]) + ([Q],[P ]) ∼ ([P ⊕ Q],[Q ⊕ P ]) ∼ (0,0), so K0 (A) is an abelian group. The map Proj(A) → K0 (A) : [P ] → ([P ],0)/ ∼ is a homomorphism of additive semigroups, but is not necessarily injective. Definition 2.9.8 (Augmented algebra) Let A be a commutative algebra over k, not necessarily with a unit. Then the augmented algebra of A is A+ = {(f ,c) : f ∈ A,c ∈ k} with the multiplication ((f ,c),(g,d)) → (f ,c) · (g,d) = (f g + df + cg,cd). We regard A+ as A with a unit adjoined, and think of A+ as an abbreviation for A⊕k1. (In the literature, the notation A˜ is also used for augmented algebra, ¯ but we use A¯ = a/k1 and seek to avoid confusion between A˜ and A.) + There is a surjective homomorphism ψ : A → A such that A+ /k is isomorphic to A, and ι : k → A+ : c → (0,c) is a surjection onto the nullspace of ψ, so that A+ has a unit (0,1), and A+ contains A as an ideal, so A+ = A ⊕ k, and [A+,A+ ] = [A,A]; hence A+ /[A+,A+ ] = (A/[A,A]) ⊕ k. This ψ induces a surjective homomorphism ψ ∗ : K0 (A+ ) → K0 (k), while ι induces ι∗ : K0 (k) → K0 (A+ ) such that ψ ∗ ◦ ι∗ = id : K0 (k) → K0 (k). Thus we can write K0 (A+ ) = K˜ 0 (A)⊕K0 (k) for some group K˜ 0 (A). Observe that if A already has a unit, then K˜ 0 (A) = K0 (A), so K˜ 0 gives us nothing new; however, K˜ 0 extends the definition of K0 from unital to non-unital algebras. Exercise 2.9.9 (Grothendieck groups) (1) Let A be a commutative algebra and M an A-module; let [M] be the isomorphism class. Let F A be the free abelian group that is generated by the classes of finitely generated A-modules. Let EA be the subgroup of FA that is generated by [M2 ] − [M1 ] − [M3 ], where 0 −→ M1 −→ M2 −→ M3 −→ 0 is an exact sequence of A-modules. Then the Grothendieck group of A is the abelian group GA = FA /EA . (2) Suppose that all finitely generated A-modules have finite projective dimension. Show how one can compute GA using the projective resolutions. See [64, p. 210] for details. We now give a description of K0 (A) which emphasizes the analogy with linear algebra. Let A be a unital algebra over the field k of characteristic zero and Mn (A) = Mn (k) ⊗k A be the algebra of n × n matrices with entries in A,

2.9 Projective Modules

47

with the usual matrix multiplication. We introduce ιn : Mn (A) → Mn+1 (A) via the non-unital homomorphism

T 0 T → . (2.9.6) 0 0 We write M∞ (A) = ∪∞ n=1 Mn (A) for the algebra of matrices with entries in A that have only finitely many non-zero entries. Now M∞ (A) is non-unital, so we form the augmented algebra M∞ (A)+ = M∞ (A) + k1 which is obtained by adjoining a unit; then M∞ (A) is a subalgebra of M∞ (A)+ . Let Idemn (A) = {e ∈ Mn (A) : e2 = e} be the set of idempotents in Mn (A), and Idem(A) = ∪∞ n=1 Idemn (A). Under ιn , the identity of Mn (A) is mapped to an idempotent e = ιn (1), where e is not the identity of Mn+1 (A). Let GLn (A) be the group of invertible n × n matrices with values in A under multiplication, and ι : GLn (A) → GLn+1 (A) the group homomorphism (different from (2.9.6))

T 0 T → . (2.9.7) 0 1 + Then GL(A) = ∪∞ n=1 GLn (A) is a subset of M∞ (A) and a multiplicative group. One operates on Idem(A) with GL(A) via conjugation, so e → geg −1 .

Lemma 2.9.10 Proj(A) is canonically isomorphic to the orbit space of Idem(A) under the action of GL(A) by conjugation. The addition rule corresponds to

[P ] 0 [P ] + [Q] → . (2.9.8) 0 [Q] Proof Each p ∈ Idemn (A) is associated with the projective left A-module An p, where An = An p ⊕ An (1 − p) is a free left A-module. We check that if such projective left A-modules are isomorphic, then the isomorphism can be implemented by multiplying on the right by an invertible matrix with entries from A. Let p ∈ Idemn (A) and q ∈ Idemm (A), and suppose that An p and Am q are isomorphic as left A-modules. Then we can introduce a ∈ Mn×m (A) and b ∈ Mm×n (A) such that ab = p and ba = q; we also impose the conditions a = pa; a(1 − q) = 0 and likewise bp = b and b(1 − p) = 0; then the isomorphism S : An p ⊕ An (1 − p) → Am (1 − q) ⊕ Am q is represented by right multiplication by

1−p a T → T S : [An p ⊕ An (1 − p)] → [An (1 − q) ⊕ An q] S = b 1−q (2.9.9)

48

Cyclic Cocycles and Basic Operators

such that S 2 = I d, so S is invertible and conjugation by S implements an isomorphism on Idemn+m (A) such that



p 0 0 0 S S= . (2.9.10) 0 0 0 q We can also think of S as an intertwining operator



p 0 0 0 S = S. 0 0 0 q

(2.9.11)

Let tr be the standard trace on Mn . Note that tr gives an isomorphism of Mn (C)/[Mn (C),Mn (C)] with C by Proposition 1.3.3. Given an A-bimodule P , we can regard P as a left A⊗Aop module with operation (a⊗b) : p → apb. Then [A,P ] = {ap − pa : a ∈ A,p ∈ P } is the space of commutators. We let P = P /[A,P ] be the commutator quotient space, and write  : P → P for the quotient map. We obtain the first glimpse of the Chern character, which involves the vector trace Tr taking values in the vector space V = A/[A,A]. Proposition 2.9.11 (Chern character) There is a natural map Tr = (tr⊗1) : K0 (A) → A . Proof Let e be an idempotent in Mn (A) = Mn ⊗ A, and g ∈ GLn (A) so that f = geg −1 is also an idempotent. We check that Tr(e) = Tr(f ), so that Tr(e) depends only on the isomorphism class of e. Consider h = eg −1 , so  hij gj i (2.9.12) (tr ⊗ 1)(e) = (tr ⊗ 1)(hg) = ij

while (tr ⊗ 1)(f ) = (tr ⊗ 1)(gh) =



gj i hij ,

(2.9.13)

 [hij ,gij ] ∈ [A,A].

(2.9.14)

ij

and so (tr ⊗ 1)(e) − (tr ⊗ 1)(f ) =

ij

By considering (2.3.3), we obtain the map K0 (A) → A/[A,A]. Proposition 2.9.12 Let A be a commutative and unital algebra. Then K0 (A) is a commutative and unital ring. Proof Let P and Q be finitely generated projective A-modules; then P ⊗ Q is also a finitely generated and projective A-module, as we now check. Since P is

2.9 Projective Modules

49

an A-module, and A is commutative, we can regard P as a right A-module by pa = ap so we form P ⊗A Q in which pa ⊗ q = p ⊗ aq. Then we introduce complementary A-modules such that An = P ⊕ P  and Am = Q ⊕ Q , then observe that P ⊗ Q is a direct summand of Anm = An ⊗ Am , and P ⊗ Q is isomorphic to Q ⊗ P . By [33, p. 573], there is a natural isomorphism HomA (P ,HomA (Q,R)) ∼ HomA (P ⊗A Q,R) for all A-modules R, and we deduce that P ⊗A Q is projective as a left A-module. Furthermore, P ⊗A A may be identified with P , so tensor multiplication on A-modules by A acts as a multiplicative unit. If R is also a finitely generated projective left A-module, then R ⊗ (P ⊕ Q) is isomorphic to (R ⊗P )⊕(R ⊗Q). Hence we can introduce binary operations ⊕ and ⊗ on Proj(A) such that ⊗ is distributive over ⊕. Whereas some proofs in this section easily extend to noncommutative algebras, the proof of Proposition 2.9.12 soon runs into problems that can only be remedied by reformulating the statement. Exercise 2.9.13 (Ranks of modules) (1) Let p ∈ Idem(k), so p is an n × n idempotent matrix p which represents a linear projection on kn for some n. Then Tr(p) = rank (p), since k = k and tr(p) = trace(p) = rank(p). Then Tr(p) takes values in Z+ . (2) For X, a compact metric space, let A = C(X;R). Then p ∈ Idem(A) is represented by a matrix with entries in A such that p(x) is an idempotent matrix for each x ∈ X. (One can say that p(x) is a projection, but it may not be self-adjoint.) Hence Tr(p) = rank(p(x)) gives the rank of the vector bundle An p, hence is a continuous function with values in Z+ . When X is connected, the rank is constant on X. (3) Let T be a real m × m matrix of rank one. Then P = T ∗ T /trace (T ∗ T ) is a self-adjoint projection such that ker(P ) = ker(T ). Likewise Q = T T ∗ / trace (T T ∗ ) is a self-adjoint projection such that ker(Q) = ker(T ∗ ), so im(Q) = im(T ). (4) Deduce that T may be expressed as T = T ξ ⊗ η, where ξ,η are unit vectors. Discuss the matrix T −1 η ⊗ ξ . Exercise 2.9.14 (K0 (P I D)) Let A be a principal ideal domain. Use Lemma 2.9.4 and Exercise 2.9.13 to show that K0 (A) ∼ = Z. Definition 2.9.15 (Vector bundle) Let X be a compact metric space. Then the trivial real vector bundle of dimension n is the metric space X × Rn with the projection p : X × Rn → X : (x,v) → x. Generally we say that a metric space E is a continuous real vector bundle over X, when there exist:

50

Cyclic Cocycles and Basic Operators

(1) a continuous surjection p : E → X such that the fibre Ex = {v ∈ E : p(v) = x} is a real finite-dimensional vector space for all x ∈ X; (2) for all x ∈ X there exist an open U ⊂ X such that x ∈ X, a finite-dimensional real vector space V and a homeomorphism ϕ : {v ∈ E : p(v) ∈ U } → U × V , such that (3) ϕ|Ex is a linear homeomorphism from Ex to V . Then the dimension of Ex as a real vector space is a locally constant function on X, hence bounded and constant on each connected component, and supx∈X dimR Ex is called the dimension of E as a vector bundle (which is possibly smaller than the dimension of E as a metric space). Let Vect(X) be the set of isomorphism classes [E] of continuous real vector bundles E over X; also let Vectn (X) be the set of isomorphism classes [E] of continuous, n-dimensional, real vector bundles E over X. A bundle E is trivial if [E] contains X × Rn . The direct sum of vector spaces leads to a direct sum operation on (the fibres of) vector bundles, and we define [E] ⊕ [F ] = [E ⊕ F ] for [E],[F ] ∈ Vect(X). Now let E be a locally trivial finite-dimensional real vector bundle over a compact metric space X, so p : E → X is a continuous and surjective map and Vx = {f ∈ E : p(f ) = x} is a family of finite-dimensional real vector spaces, such that the dimension is locally constant. The space of sections of E is (X,E) = {s : X → E : continuous

p ◦ s = id},

(2.9.15)

and the space of sections has a natural addition structure. Theorem 2.9.16 (Swan) (i) Let A = C(X;R). Then (X;E) is a finitely generated and projective A-module. (ii) Up to isomorphism, every finitely generated and projective A module arises as the spaces of sections of some continuous finite-dimensional real vector bundle over X. Proof (i) The details are presented in [91, p. 34; 101], so the following sketches the basic ideas. Observe that An is the typical finitely generated free left A-module, and that we can identify An ∼ = C(X;Rn ) so that each n s ∈ C(X;R ) gives a section of the trivial rank n vector bundle over X. Any n-dimensional vector bundle E over X has a complementary bundle E  such that E ⊕ E  is isomorphic to the trivial vector bundle C(X;Rn ). A pair of sections s of E and s  of E  corresponds to a pair P ,P  of C(X;R) modules so that P ⊕ P  ∼ = C(X;Rn ).

2.9 Projective Modules

51

(ii) Note also that any finitely generated projective left A-module P is a direct summand of An . Using the identification of sections of An with functions X → Rn , one can produce a locally trivial vector bundle E with section s to represent P . Thus one can define with A = C(X;C) an abelian K0 -group K0 (C(X;C)). In Chapter 3 we consider how to compute this group in specific cases. By Example 6.2.4(iii) relating to X = S 2 , we cannot replace projective by free in the statement of Theorem 2.9.19; local triviality is not the same as global triviality for vector bundles. Exercise 2.9.17 (Picard group) (See 5.4.2 of [45], and 6.12 of [49].) Let A be a unital commutative algebra so K0 (A) is a commutative unital ring by Proposition 2.9.12. Let the Picard group Pic (A) be the set of equivalence classes of finitely generated projective A-modules P that have a multiplicative inverse under ⊗. Show that Pic (A) defines an abelian group. See Example 8.2.5 for a geometrical example of this. (i) Let G(A) = {u ∈ A : ∃v ∈ A, uv = 1} be the multiplicative group of units in A and let P be a finitely generated projective A bimodule. Show that φ ∈ HomA (A,A) is determined by φ(1), and Aφ(1) = A if and only if φ(1) ∈ G(A). (ii) Show that there is an A-module map A → HomA (P ,P ) given by a → λa , where λa (p) = ap for all a ∈ A,p ∈ P . Describe the image of G(A). (iii) Let Pˇ = HomA (P ,A), known as the dual module of P . Show that Pˇ is an A-module and that there is an A-module map π : Pˇ ⊗A P → HomA (P ,P ), given by ⎛ ⎞   π⎝ pˇ j ⊗ pj ⎠ : r → pˇ j (r)pj . j

j

(iv) State a condition on P under which Pˇ ⊗A P is a projective module that is isomorphic to A via the maps π and λ. (See lemma 3.17, p. 118 of [64].) In this circumstance, one writes P = L, since L has rank one and is some sort of line bundle, and L−1 = Lˇ = HomA (L;A), since HomA (L;A) ⊗A L ∼ A, and A is the multiplicative unit of Pic (A). (v) Find Pic (A) where A = C(X;R), and X is a connected and compact Hausdorff space. In [42, 5.4.2], Pic (A) is identified with a group of line bundles on X. See also [63] page 35.

52

Cyclic Cocycles and Basic Operators

Definition 2.9.18 (Group algebra) Let G be a group, and introduce the  complex group algebra C[G] = { g∈G ag g : ag ∈ C} with multiplication  g∈G

ag g

 

   bh h = akh−1 bh k.

b∈G

k∈G

(2.9.16)

h∈G

Proposition 2.9.19 (Maschke’s theorem) Let G be a finite group, and P a submodule of a finitely generated C[G]-module M. Then P is projective. Proof Since M is a finite-dimensional C-vector space, we can choose a basis {p1, . . . ,pk } of P and extend to a basis {p1, . . . ,pk ,mk+1, . . . ,mn } of M. Then we select a linear map e : M → M such that e(pj ) = pj and e(m ) = 0 for all j, ; evidently e is a projection with range P . Now let eG : M → M be the linear map eG =

1  geg −1, G g∈G

so that eG h = heG and range(eG ) ⊆ range(e) = P since P is a submodule, so heh−1 eG = heeG h−1 = heG h−1 = eG , hence by averaging over h ∈ G we deduce that eG eG = eG . Now eG M and (1 − eG )M are complementary C[G]modules. Finally, dim eG M = trace(eG ) = trace(e) = dim P , so eG M = P , and P has a complementary module. Maschke’s theorem does not necessarily hold if one replaces C by a field of characteristic p dividing the order of G. Also, examples in [59] show that one cannot necessarily replace G by an infinite group. Nevertheless, in Corollary 4.7.3, we obtain a result about Hilbert modules over C[Z]. In the representation theory of finite groups, complemented modules are sometimes called completely reducible.

2.10 Singular Homology In the context of arbitrary compact topological spaces, the most appropriate ˇ topological theory is Cech cohomology. Rather than develop this theory, we prefer to impose special assumptions on X so that we can work with the elementary theory of simplicial homology. Let X be a compact topological space which is a simplicial complex. (0) Let r be the number of path components of X. Then H0 (X,Z) ∼ = Zr is a free module.

2.10 Singular Homology

53

(1) A loop is a continuous function : [0,1] → X such that (0) = (1). Suppose that x0 ∈ X and let π1 (X,x0 ) be the set of homotopy classes of loops in X that are based at x0 . Then π1 (X,x0 ) forms a group, namely the fundamental group for the space X with point x0 , as described in [50]. The derived group π1 (X,x0 ) is the normal subgroup of π1 (X,x0 ) that is generated by the commutators XY X−1 Y −1 . Let H1 (X,Z) be the group of singular 1-simplices in X. Any loop : [0,1] → X with (0) = (1) = x0 gives a singular 1-simplex. Proposition 2.10.1 (i) There is a natural homomorphism χ : π1 (X,x0 ) → H1 (X,Z) formed by taking the homotopy class of a loop to the homology class of the singular 1-simplex . (ii) If X is path connected, then there is an exact sequence of groups {1} −→ π1 (X,x0 ) −→ π1 (X,x0 ) −→ H1 (X,Z) −→ {1}. Proof See [92, Theorems 4.2.7 and 4.2.9]. Example 2.10.2 (Punctured sphere) (a) One can express the fundamental group of a connected simplicial complex with base point in terms of generators and relations. One can deduce that (i) the circle has fundamental group Z; (ii) the torus has fundamental group Z ⊕ Z; (iii) a figure of eight has fundamental group F2 , the free group on two generators; see [92, Ex. 7.17]. (b) Let S 2 = C ∪ {0} be the Riemann sphere. Then (i) S 2 \ {0} has fundamental group {1}; (ii) S 2 \ {0,1} has fundamental group Z; (iii) X = S 2 \ {0,1,∞} has fundamental group F2 and H1 (X,Z) = Z ⊕ Z. Definition 2.10.3 (Smooth triangulation) Let K˜ be a simplicial complex, and ˜ so K is a compact set. Given a C ∞ manifold K the point set associated with K, X, a smooth triangulation consists of a homeomorphism h : K → X such that ˜ h|σ : σ → X has an extension hσ to a neighbourhood for each simplex σ˜ of K, U of σ in the plane of σ such that hσ : U → X is a smooth submanifold. By [97, p. 146], every smooth compact manifold has a smooth triangulation. In particular, for algebraic curves, one can produce smooth triangulations via Theorem 6.1.3; see [62] and [34]. The significance of this result for cyclic theory is that it provides a means for relating the homology theories in topological and smooth categories. For a smooth compact manifold M, we have K0 (C(M;C)) = K0 (C ∞ (M;C)) by 8.2.6 of [68]. Exercise 2.10.4 (Euler characteristic) Let X be a compact topological space that has a triangulation. Then the simplicial homology groups Hq (X;Z) are finitely generated abelian groups, namely finitely generated Z-modules. Any

54

Cyclic Cocycles and Basic Operators

finitely generated abelian group A may be written as A = T ⊕ F where T is the subgroup of torsion elements, and F = A/T is a free abelian group isomorphic to Zn , so we define the rank of A to be n. See [47, p. 125]. In particular, we define the qth Betti number βq to be the rank of Hq (X;Z). Then the Euler characteristic is  (−1)q βq , (2.10.1) χ (X) = q

and the genus g satisfies 2g = β1 . For the triangulation with V vertices, E edges and F faces, we have χ = V − E + F . Show that χ (S 2 ) = 2, and that for X the complex torus of Example 6.1.5, we have χ (X) = 0. See [42, p. 120].

3 Algebras of Operators

Cyclic theory incorporates the index theory of elliptic operators as a particular application. In this chapter, we consider the simplest interesting case of index theory, namely Toeplitz operators on Hardy space over the circle. The chapter begins by introducing the basic facts about operators on Hilbert space that are required subsequently. In Section 2.9, we imposed the condition that modules over algebras are finitely generated. To enable us to use such results, we consider Fredholm operators that have finite-dimensional kernels and cokernels, so that an index can be defined. The basic idea is to consider an algebra R over C with an ideal I and a trace on I , namely a linear map τ : I /[R,I ] → C. Then we introduce an algebra A over C and a linear map A → R which is a homomorphism modulo I , so that the curvature ω takes values in I . This enables us to consider ϕ(a0,a1 ) = τ ω(a0,a1 ) for all a0,a1 ∈ A. However, the algebras are infinite dimensional, so I and τ require careful definition.

3.1 The Gelfand Transform (i) Let A be a complex, unital and commutative Banach algebra. Then A has a norm such that SA ≥ 0 for all with SA = 0 only if S = 0; S + T A ≤ SA + T A ; ST A ≤ SA T A

λSA = |λ|SA ;

(3.1.1)

(S,T ∈ A,λ ∈ C),

(3.1.2)

and A is complete for the metric S − T . The space of bounded linear functionals ψ : A → C is denoted A , and known as the dual space. 55

56

Algebras of Operators

Exercise 3.1.1 (Connectedness) Let Y be a compact Hausdorff space, and C(Y ;C) the space of continuous functions f : Y → C with pointwise multiplication and the norm f ∞ = sup{|f (y)| : y ∈ Y }. Verify that C(Y ;C) is a unital and commutative Banach algebra. (1) Show that the space Y is disconnected, if and only if C(Y,C) has non-trivial idempotent elements e2 = e, namely e ∈ C(Y,C) such that e(x) ∈ {0,1} for all x ∈ Y corresponding to the indicator functions of connected components of Y . (2) Let G(C(Y ;C)) be the set of f ∈ C(Y ;C) that have gf = 1 for some g ∈ C(Y ;C). By elementary results, if f (y) = 0 for all y ∈ y, then there exists δ > 0 such that |f (y)| ≥ δ for all y ∈ Y , and 1/f is continuous. Deduce that G(C(Y ;C)) = C(Y ;C∗ ) where C∗ = C \ {0}. (3) Let f ∈ C(Y ;C). Deduce that σ = {f (y) : y ∈ Y } is a compact subset of C, and λ − f is invertible for all λ ∈ C \ σ . Let the spectrum of f be the set of λ ∈ C such that f does not have a multiplicative inverse in C(Y ;C). Deduce that the spectrum of f is equal to the range of f . There is a systematic way of mapping a complex, unital and commutative Banach algebra into a space of continuous functions, known as the Gelfand transform. (ii) We introduce the unit ball of A by BA = {S ∈ A : SA ≤ 1} and the unit ball of the dual space by BA = {φ ∈ A : |φ(S)| ≤ 1,∀S ∈ BA }. We have a weak∗ topology on BA given by the basic open sets U (T1, . . . ,Tm ;z1, . . . ,zm ;ε1, . . . ,εm ) = {φ ∈ BA : |φ(Tj ) − zj | < εj ,j = 1 . . . ,m}

(3.1.3)

for all m ∈ N, Tj ∈ BA, zj ∈ C and εj > 0. One shows using Tychonov’s theorem that BA with the weak∗ topology is compact and Hausdorff. Furthermore, if A is separable for the norm topology, then BA with the weak∗ topology is metrisable. (iii) Let G(A) be the set of invertible elements of A. Then I ∈ G(A), and G(A) is a multiplicative group. By considering perturbation series, one can show that G(A) is an open subset of A for the norm topology. For all T ∈ A, the series exp(T ) = 1 + T + T 2 /2! + · · · converges, and one checks that exp(S + T ) = exp(S) exp(T ) and exp(tS) → 1 as t → 0. Hence exp(A) = {exp(T ) : T ∈ A} forms a connected subgroup of G(A) that contains 1, and we can form the quotient group G(A)/ exp(A).

(3.1.4)

3.1 The Gelfand Transform

57

(iv) An ideal M of A is said to be maximal if (1) M is proper and (2) if M  is any proper ideal such that M ⊆ M  ⊂ A, then M = M  . Suppose that M is a maximal ideal. Then all the elements of M are not invertible and the closure M¯ is an ideal containing M such that all the elements of M¯ are limits of elements in M, hence are not invertible. We deduce that M¯ is proper, and hence M¯ = M. It follows that M is closed and that the quotient map φ : A → A/M is a homomorphism and A/M is a field containing {φ(λI ) : λ ∈ C}. Definition (Spectrum) Let A be a unital Banach algebra. Then the spectrum σ (T ) of T ∈ A consists of those λ ∈ C such that λI − T does not have an inverse in A. Theorem 3.1.2 (Gelfand) empty subset of C.

The spectrum of T ∈ A is a compact and non-

Proof This follows from Liouville’s theorem. See [2]. Corollary 3.1.3 (Gelfand–Mazur) Let A be a commutative and unital Banach algebra in which every non-zero element has an inverse. Then A is isomorphic to C. (v) (Multiplicative linear functional) A linear functional φ : A → C is multiplicative if φ(ST ) = φ(S)φ(T ) for all S,T ∈ A and φ(I ) = 1. By (iv), all multiplicative linear functionals are continuous, and one can easily check that |φ(T )| ≤ T A for all T ∈ A. We write X = {φ ∈ BA : φ(ST ) = φ(S)φ(T ),∀S,T ∈ A}

(3.1.5)

which is a closed subspace of BA for the weak ∗ topology, hence compact and Hausdorff. Furthermore, if A is separable for the norm topology, then X with the weak∗ topology is metrisable. The kernel of φ is ker φ = {S ∈ A : φ(S) = 0}. One can show that ker φ is a maximal ideal in A such that A/ker φ is isomorphic to C and there is a bijective correspondence between multiplicative linear functionals and maximal ideals HomC (A;C) ↔ X : φ ↔ ker φ.

(3.1.6)

We associate with each T ∈ A the continuous function Tˆ : X → C by Tˆ (φ) = φ(T ) on the maximal ideal space X. This map T → Tˆ is known as the Gelfand transform. Theorem 3.1.4 (Gelfand) The maximal ideal space of a unital complex commutative Banach algebra is a non-empty compact Hausdorff space X. (i) Then map T → Tˆ is an algebra homomorphism A → C(X,C) such that ˆ " = Tˆ S; ST

58

Algebras of Operators

(ii) Tˆ ∞ ≤ T A ; (iii) If Tˆ (φ) = 0 for all φ ∈ X, then T is invertible. Proof (iii) Suppose that T is not invertible, and let (T ) = {ST : S ∈ A}. Then (T ) is an ideal of A that contains T , but does not contain I . So by Zorn’s Lemma, (T ) is contained in some maximal ideal M; then φ : A → A/M is a multiplicative linear functional such that φ(T ) = 0. By Corollary 3.1.3, A/M is isomorphic to C, so φ ∈ X, and Tˆ (φ) = 0. Corollary 3.1.5

The Gelfand transform gives a group homomorphism

G(A)/ exp(A) → G(C(X;C))/ exp(C(X;C)).

(3.1.7)

By the Arens–Royden theorem, this is actually a group isomorphism. Proving surjectivity needs the theory of several complex variables, as in [2]. Example 3.1.6 (exp on continuous functions) We return to the example of C(X;C), as in Exercise 3.1.1. Let C∗ = C \ {0}. Then we observe that G(C(X,C)) = C(X,C∗ ). Also, exp(C(X;C)) is the connected component of C(X,C∗ ) that contains the identity. Using this result, it is easy to see that C(X;C∗ )/ exp(C(X;C)) is the group of homotopy classes of continuous functions X → C∗ . Hence the cohomology is H 1 (X,Z) ∼ = G(C(X;C))/ exp(C(X;C)).

(3.1.8)

Let A be a commutative and unital Banach algebra such that the maximal ideal space X is homeomorphic to a simplicial complex. Then one can prove the following. (0) H 0 (X,Z) is isomorphic to the group generated by the idempotents in A; (1) H 1 (X,Z) is isomorphic to G(A)/ exp(A); (2) H 2 (X,Z) is isomorphic to the Picard group Pic (A) of Exercise 2.9.17. This is a result of Forster; see [103, p. 175]. In Example 8.2.7 we obtain a similiar result when X is a compact Riemann surface. Exercise 3.1.7 (Bounded derivations) (i) Let C 1 (S 1,C) be the Banach algebra of continuously differentiable functions on the circle with pointwise multiplication and the norm f C 1 = f ∞ + f  ∞ .

(3.1.9)

Show that C(S 1,C) is a C 1 (S 1,C) module for the pointwise multiplication and δ : C 1 (S 1,C) → C(S 1,C) : f → f  is a module map.

3.2 Ideals of Compact Operators on Hilbert Space

59

(ii) (Disc algebra) Let A be the disc algebra of continuous complex ¯ = {z ∈ C : |z| ≤ 1} such that f is functions f on the closed disc D holomorphic on the open disc D = {z ∈ C : |z| < 1} with pointwise multiplication and f  = supz {|f (z)| : |z| ≤ 1}. For 0 < r < 1, let δf (z) = f  (rz). Show that f (z) → f (rz) is a homomorphism A → A and δ : A → A is a continuous derivation with respect to this homomorphism. (iii) Let A be a commutative semisimple Banach algebra. Then there are no non-zero continuous derivations δ : A → A; see [2, 5.8]. Discuss how this does not contradict (i) or (ii). Exercise 3.1.8 (Traces)

Let A be a Banach algebra.

(i) Show that every continuous multiplicative functional φ : A → C gives a continuous trace τ : A → C such that τ |[A,A] = 0. (ii) Show that M2 (C) has a non-trivial trace, but the only continuous multiplicative linear functional is zero. (iii) For A commutative, show that the continuous traces are the continuous linear functionals φ : A → C. (iv) For X a compact metric space, and A = C(X,C), show that the traces on A are given by bounded measures μ on X via τ (f ) = X f (x)μ(dx). (v) Let A = C0 (R;C) be the space of continuous functions f : R → C such that f (x) → 0 as x → ±∞. By considering the one-point compactification R ∪ {∞} as in [101], show that A ⊕ C is isomorphic to C(R ∪ {∞};C). Hence find the traces on A. (vi) In Exercise 3.5.4, we introduce the Toeplitz algebra T such that T /[T ,T ] is isomorphic to C(S 1,C). Find the traces on T . Exercise 3.1.9 (Banach limits) Let ∞ (Z) be the space of bounded complex sequences (an )∞ n=−∞ with coordinate-wise addition and the norm (an ) = supn∈Z |an |.  that ∞ (i) Let (pn )∞ n=−∞ be a complex sequence such n=−∞ |pn | converges.  p Show this gives a trace on ∞ (Z) via (an ) → ∞ n=−∞ n an . (ii) Let LI M be the Banach limit as in [66, p. 31]. Show that LI M defines a trace on ∞ (Z) which is not given by any sequence in (i).

3.2 Ideals of Compact Operators on Hilbert Space We express the essential facts as succinctly as possible in this section and refer the reader to texts such as [31] and [32] for explanation.

60

Algebras of Operators

Definition 3.2.1 (1) (Hilbert space) Let H be a complex separable Hilbert space with inner product · | · : H × H → C, so that for all f ,g,h ∈ H and μ,λ ∈ C, (i) f | g = g | f ; (ii) f | μg + λh = μ f | g + λ f | h , so the inner product is linear in the second variable; (iii) h | h ≥ 0, and h | h = 0 ⇒ h = 0; (iv) H is complete for the metric associated with the norm f − g = f − g | f − g 1/2 ; (v) H is infinite dimensional, but separable for the norm of (iv). One can prove that there exists a complete orthonormal basis (zj )∞ j =0 , such  a that any f ∈ H may be expressed as a convergent series f = ∞ j =0 j zj in the ∞ 2 2 norm of H , where f  = j =0 |aj | , and the aj are uniquely determined by aj = zj | f . Another basic result is the Cauchy–Schwarz inequality, which gives | f | g | ≤ f g for all f ,g ∈ H . Example 3.2.2 (Fourier coefficients) Consider the circle S 1 = {z ∈ C : |z| = 1} with normalized Lebesgue measure dθ/(2π ). With z = eiθ , let H = L2 (S 1 ;dθ/ (2π )) be the Hilbert space of square integrable functions with the inner product  g | f = 0

2π

f (θ )g(θ ¯ )

dθ . 2π  2π

(3.2.1)

The Fourier coefficients of f ∈ H are fˆ(n) = 0 f (θ )e−inθ dθ/(2π ) for n ∈ Z. By the Riesz–Fischer theorem, H has a complete orthonormal basis (zn )∞ n=−∞ . Definition 3.2.3 (Bounded linear operators) Let T : H → H be a C-linear map. Then T is said to be bounded with operator norm T L(H ) where T L(H ) = sup{Tf  : f ∈ H ;f  ≤ 1}

(3.2.2)

f

is finite; the set of all such f is L(H ). Then L(H ) forms an algebra under multiplication given by composition, and ST L(H ) ≤ SL(H ) T L(H ) . We often write T  for T L(H ) . Definition 3.2.4 (Finite-rank operators) The space F of finite-rank operators N  on H is {T = N j =1 ξj ⊗ ηj ;ξj ,ηj ∈ H } so T ζ = j =1 ηj | ζ H ξj . Each T ∈ F determines a bounded linear operator on H . Also, F determines an ideal in L(H ) since ST ∈ F and T S ∈ F for all T ∈ F and S ∈ L(H ).

3.2 Ideals of Compact Operators on Hilbert Space

61

(4) Again let T : H → H be a C-linear map. Then T is said to be compact if {Tf : f ∈ H ;f  ≤ 1} has compact closure in H for the norm topology; in particular, such a T is bounded. Lemma 3.2.5 (Compact operators) Let K(H ) be the space of compact operators. Then K(H ) is an ideal in L(H ) for the usual multiplication operation, so that (i) (ii) (iii) (iv) (v)

ST ∈ K(H ) and T S ∈ K(H ) for all T ∈ K(H ) and S ∈ L(H ); F ⊂ K(H ); T ∈ K(H ) if and only if T ∗ ∈ K(H ); T ∈ K(H ) if and only if T ∗ T ∈ K(H ); if Tn ∈ K(H ) and T ∈ L(H ) has T − Tn  → 0 as n → ∞ then T ∈ K(H ).

Also K(H ) is uniquely determined as the smallest set that satisfies (i)–(v). Proof See [29]. Definition 3.2.6 (Hilbert–Schmidt operators) A linear operator T : H → H  2 is Hilbert–Schmidt if T 2L2 = ∞ j =0 T zj  is finite. Then T is compact, and the set L2 (H ) of all Hilbert–Schmidt operators forms an ideal in L(H ) under the usual addition and multiplication; indeed, T  ≤ T L2 ;

U T V L2 ≤ U T L2 V .

(3.2.3)

Example 3.2.7 (Hilbert–Schmidt integral operators) When H = L2 (S 1, dθ/(2π )), we can express any T ∈ L2 (H ) as an integral operator  dy K(x,y)f (y) (3.2.4) Tf (x) = 1 2π S where K(x,y) is confusingly called the kernel and K ∈ L2 (S 1 × S 1 ; dxdy/(2π )2 ); indeed  dx dy T 2L2 = |K(x,y)|2 (3.2.5) 2π 2π S 1 ×S 1 by the Hilbert–Schmidt theorem. This criterion is easy to check in applications. Definition 3.2.8 (Trace class operators) Let I = L2 (H ), then L1 (H ) = I 2 = {ST : S,T ∈ L2 (H )} is the ideal of trace class operators. We can take the norm to be T L1 = inf{U L2 V L2 : T = U V ;U,V ∈ L2 }. Then L1 (H ) forms an ideal in L(H ) for the usual multiplication T  ≤ T L2 ≤ T L1 ;

U T V L1 ≤ U T L1 V .

(3.2.6)

62

Algebras of Operators

Let (zj ) be any complete orthonormal basis of H . Then on L1 (H ), there is a continuous trace : L1 (H ) → C defined by trace(T ) =

∞ 

zj | T zj ,

(3.2.7)

j =0

such that |trace(T )| ≤ T L1 ; also trace(ST ) = trace(T S)

(S,T ∈ L2 (H ));

trace(U V ) = trace(V U )

(U ∈ L1 (H ),V ∈ L(H )). (3.2.8) ∞ ∞ Given a linear operator T = j =1 ξj ⊗ ηj with j =1 ξj ηj  < ∞, then T ∈ L1 (H ) and we can analogously define τ (T ) =

∞  ξj | ηj H . j =1

Proposition 3.2.9 (Traces) The trace class operators I with τ : I → C given by τ (T ) = trace(T ) satisfy the following: (i) I is a self-adjoint ideal in L(H ) that contains all the finite-rank operators ⎧ ⎫ N ⎨ ⎬  ξj ⊗ ηj ;ξj ,ηj ∈ H ; T = ⎩ ⎭ j =1

(ii) I is a Banach space for some suitable norm so that U T V I ≤ U T I V 

(3.2.9)

for all T ∈ I and U,V ∈ L(H ), so multiplication is norm continuous; (iii) τ : I → C is a continuous linear functional such that τ |[I,L(H )] = 0, so τ |[I,I ] = 0, and τ is a trace on I . Further, up to scalar multiples, τ is the unique trace with the properties (i), (ii) and (iii). Proof We can use (3.2.6) and the choice L1 (H ) = L2 (H )L2 (H ) to prove existence. To see uniqueness, we consider τ satisfy on (i), (ii) and (iii), and observe that by continuity, we need only determine the value of τ on finiterank T . Then for any finite-rank operator T ∈ I , we can write T = (1/2) (T + T ∗ ) + (1/2i)(iT − iT ∗ ), where T + T ∗ and iT − iT ∗ are self-adjoint and finite-rank operators in I . Also, for any unitary U ∈ L(H ), we have τ (U T U ∗ − T ) = τ (U (T U ∗ ) − (T U ∗ )U ) = 0,

(3.2.10)

3.2 Ideals of Compact Operators on Hilbert Space

63

so τ is invariant under unitary conjugation. Given a finite-rank and selfadjoint S ∈ I , we introduce the eigenvalues λ1, . . . ,λm of S and the orthogonal projection e of rank m onto the range of S, so we can present S as an m × m complex matrix on eH with respect to some orthonormal basis, and the trace on eH is uniquely determined as in Proposition 1.3.3. (In some contexts, discontinuous traces are used on L1 (H ), but we do not use them here.) When H = L2 (S 1,dθ/(2π )), and T ∈ L1 (H ), then T also belongs to L2 (H ) and hence may be expressed as an integral operator with kernel K(x,y). Suppose that K is continuous. Then by [66, p. 344].  dx K(x,x) . trace(T ) = 1 2π S

(3.2.11)

(6) We can regard the operator ideals as contained L(H ) ⊃ K(H ) ⊃ L2 (H ) ⊃ L1 (H ) ⊃ F

(3.2.12)

with the trace defined on the smallest two of them. The ideals I here are all self-adjoint in the sense that A ∈ I ⇒ A∗ ∈ I . Note that F and L1 (H ) are dense linear subspaces of K; the trace τ is densely defined but unbounded on K. Exercise 3.2.10 (Homomorphisms and traces) Let B,C be algebras over a field k, and let Hom(B,C) be the space of k-algebra homomorphisms B → C. (i) Show that there is a natural map Hom(C,k)×Hom(B,C) → Hom(B,k) (φ,T ) → φ ◦ T . (ii) Let Tr(C) = {τ : C → k : τ (ab) = τ (ba);a,b ∈ C} be the space of k-valued traces on C. Show that Hom(C,k) ⊆ Tr(C) and that there is a natural map Tr(C) × Hom(B,C) → Tr(B) (φ,T ) → φ ◦ T . (iii) One can show that the only norm continuous trace on the algebra B = K(H ) is the zero trace. Deduce that the spaces of continuous homomorphisms Hom(B,Mn ) and Hom(B,C(X,C)) are also trivial. Exercise 3.2.11 (Logarithmic kernel) − log 4 sin2

θ −φ = 2

(i) Show that ∞  n=0;n=−∞

ein(θ−φ) . |n|

(ii) Show that the integral operator T on L2 (S 1,dθ/2π ) with this kernel function is Hilbert–Schmidt, but not trace class. Indeed, T ∈ Lp (L2 (S 1,dθ/2π )) for all p > 1. (iii) Describe the operator F = (−id/dθ )T in terms of Fourier series; prove that F ∈ L(L2 (S 1,dθ/2π )).

64

Algebras of Operators

3.3 Algebras of Operators on Hilbert Space Definition 3.3.1 (C ∗ -algebra) Let A be a complex Banach ∗-algebra so that A is equipped with an adjoint operation ∗ such that a ∗ ∈ A for all a ∈ A, (a ∗ )∗ = a;

(ab)∗ = b∗ a ∗

¯ ∗ (λa)∗ = λa

(a,b ∈ A,λ ∈ C); (3.3.1)

and a norm  ·  : A → [0,∞) that satisfies the usual Banach norm axioms a + b ≤ a + b;λa = |λ|a;

(3.3.2)

a = 0 ⇒ a = 0;

(3.3.3)

A is complete for the norm; and A satisfies the special assumptions a ∗  = a;

a ∗ a = a2 ;

ab ≤ ab.

(3.3.4)

Example 3.3.2 (C ∗ -algebras) (i) Let A be a closed ∗-subalgebra of L(H ) where H is a Hilbert space. Then A is a C ∗ algebra, as one can easily check. Remarkably, the converse is also true, as described in [81] and Theorem 4.2.3. The first step towards the converse is Theorem 3.3.3. (ii) There is at most one norm on a ∗-algebra that makes it a C ∗ -algebra; see [80]. (iii) Let (X,d) be a compact metric space, μ a probability measure on X such that μ(U ) > 0 for all non-empty open subsets of X. Let H = L2 (X,μ;C) be the Hilbert space of square integrable functions f : X → C with inner ¯ μ(dx). Let A = C(X;C) the space of product g | f H = f (x)g(x) continuous functions on X with the usual pointwise multiplication, conjugation f ∗ (x) = f¯(x) and f  = supx {|f (x)| : x ∈ X}. Then each f ∈ A is associated with a multiplication operator Mf : g → f g, so Mf L(H ) = f  and Mf∗ = Mf¯ . Thus A is realized as a norm closed ∗ subalgebra of L(H ), so A is a commutative unital C ∗ -algebra. The continuous traces τ : A → C on A are precisely the bounded linear  functionals on A, and by Riesz’s theorem have the form τ (f ) = X f (x)ν(dx) where ν is a measure on X of bounded total variation. Each point x ∈ X determines a linear functional φx : f → f (x)  such that φx (f g) = φx (f )φx (g) for all f ,g ∈ A; one can express φx (f ) = X f (y)δx (dy) where δx is the unit point mass at x ∈ X. One can establish an equivalence between multiplicative linear functionals on A, maximal ∗-ideals in A and points of X, as follows. Theorem 3.3.3 (Gelfand–Naimark) Let A be a commutative and unital C ∗ algebra with maximal ideal space X. Then the Gelfand transform T → Tˆ

3.3 Algebras of Operators on Hilbert Space

65

is a bijective unital ∗ homomorphism from A onto C(X,C) such that T A = Tˆ ∞ . Proof We already know that the Gelfand transform is a homomorphism. One checks that the map is a ∗-homomorphism, that T A = Tˆ ∞ for all T ∈ A and that the image separates the points of X. By the Stone–Weierstrass theorem, a self-adjoint closed subalgebra of C(X,C) that separates the points of X and contains the constant functions must be all of C(X,C); see [66]. Exercise 3.3.4 (Normal operators) Let N be a bounded normal operator on Hilbert space such that N N ∗ = N ∗ N. Let A be the closed algebra generated by I , N and N ∗ . Show that A is isomorphic as a C ∗ -algebra to C(X,C), where X is the spectrum of N . Define a linear functional τ : A → C such that τ (I ) = 1 and |τ (A)| ≤ A for all A ∈ A. Definition 3.3.5 (i) (Compactification) First we observe that the one point compactification of the line R is homeomorphic to the circle S 1 , while the one-point compactification of the plane R2 is homeomorphic to the sphere S 2 . Let C0 (R;C) be the space of continuous f sR → C such that f (x) → 0 as x → ±∞; such f are bounded so we use the  · ∞ norm. There is an exact sequence 0 −→ C0 (R;C) −→ C(S 1 ;C) −→ C −→ 0,

(3.3.5)

and we deduce that we can adjoin a unit to C0 (R : C) and obtain a unital C ∗ algebra C0 (R : C)+ ∼ = C(S 1 ;C). (ii) (Suspension) Let A be a C ∗ algebra. Then for any topological space X, the space Cb (X;A) of bounded and continuous functions f : X → A gives a C ∗ algebra for pointwise multiplication and addition, f ∗ (x) = (f (x))∗ and f  = sup{f (x)A : x ∈ X}. This may be identified with Cb (X) ⊗ A, where the tensor norm is completed for the injective tensor product norm, and hence an exact sequence 0 −→ C0 (R;C) ⊗ A −→ C(S 1 ;C) ⊗ A −→ A −→ 0, where C(S 1 ;A) is the space of loops in A, and {f ∈ C(S 1 ;A) : f (1) = 0} is the subalgebra of loops that vanish at 1. The latter algebra, or equivalently C0 (R;C)⊗A, is called the suspension of A. Hence C0 (R2 ;C)⊗A is the double suspension of A, denoted S 2 A. Definition 3.3.6 Let A be a commutative C ∗ -algebra. Then we define K˜ 0 (A) as in Section 2.9, and then K1 (A) = K˜ 0 (SA). If A is a commutative and unital C ∗ -algebra, then by Exercises 2.9.9 and 2.9.13, K0 (A) is a commutative and unital ring. This does not extend easily

66

Algebras of Operators

to noncommutative C ∗ -algebras. One of the main achievements of Connes’s noncommutative geometry is the notion of the Chen character, which is defined for noncommutative C ∗ -algebras and extends the ring structure of topological K-homology. There are more natural and illuminating ways of defining K1 , as discussed in Exercise 4.4.4, [90] and [103]. One shows that K1 (C) = 0. Continuing the process of suspension, we can progress to define K˜ 2 (A) = K0 (S 2 A), and so on. This process does not produce new groups indefinitely, due to the following fundamental result. Theorem 3.3.7 (Bott’s periodicity theorem) There is a natural isomorphism between K˜ 0 (A) and K˜ 0 (S 2 A). The Gelfand–Naimark theory does not readily extend to noncommutative C ∗ -algebras, for reasons evidenced in the following examples. (i) A unital C ∗ -algebra has no nilpotent ideals other than 0. For if J is an n ideal such that J n = 0, and T ∈ J , then (T ∗ T )2n = 0, so (T ∗ T )2  = 0. We deduce that T ∗ T  = 0, so T = 0. This is important when we come to consider extensions of C ∗ -algebras later in this chapter, since the discussion in Section 1.8 is not directly applicable. (ii) Let Mm (C) be the space of linear transformations on Cm with the usual matrix norm, matrix multiplication and adjoint [aj k ]∗ = [a¯ kj ]. Then Mn (C) is a C ∗ -algebra. The only traces on Mm (C) have the form trace([aj k ]) =  We temporarily write τm for the usual c m j =1 ajj ; see Proposition 1.3.3. m matrix trace with τm ([aj k ]) = j =1 ajj . (iii) Any C ∗ -algebra A that is finite dimensional over C has the form A = ⊕N j =1 Mnj (C)

(3.3.6)

for some N,nj ∈ N, by Wedderburn’s theorem as in Complement 1.3.5. The traces are given by Proposition 1.3.3, and have the form τ=

N 

cj τnj .

(3.3.7)

j =1

 (iv) The space F of finite-rank operators on H is {T = N j =1 ξj ⊗ ηj ;ξj , N  ηj ∈ H } so T ζ = j =1 ζ,ηj H ξj . Then T has a trace τ (T ) = nj=1 ηj | ξj H which matches with the notion of trace used for matrices in linear algebra. The space F is a ∗ -subalgebra of L(H ) and an ideal in L(H ), but F is not closed for the operator norm, hence is not a C ∗ -algebra.

3.3 Algebras of Operators on Hilbert Space

67

(v) The space K(H ) of compact operators on H is a C ∗ -algebra. Indeed, K(H ) is an ideal in K(H ), and is the smallest norm-closed ideal that contains F. This C ∗ -algebra does not have a unit, so we consider also K(H )+ = CI + K(H ) = {λI + K : K ∈ K(H ),λ ∈ C},

(3.3.8)

which is now a unital C ∗ -algebra. However, K(H ) does not have a non-trivial trace that is continuous for the norm topology. (vi) The compact operators form a norm closed subspace of L(H ), so the quotient C(H ) = L(H )/K(H ) is a Banach algebra called the Calkin algebra, which is also a C ∗ -algebra, and there is an exact sequence of C ∗ -algebras and ∗-homomorphisms π 0 "→ K(H ) −→ L(H ) C(H ) −→ 0. (3.3.9) −→ Exercise 3.3.8 (Reducing subspaces) (i) Let E be a closed linear subspace of a Hilbert space H , then let E ⊥ be the orthogonal complement E ⊥ = {ξ ∈ H : η | ξ = 0, ∀η ∈ E}. The orthogonal projection e ∈ L(H ) with range E and nullspace E ⊥ is represented as

I 0 E . (3.3.10) e= 0 0 E⊥ Show that there is an exact sequence of Hilbert spaces 0 −→ E ⊥ −→ H −→ E −→ 0.

(3.3.11)

(ii) Let A ∈ L(H ). Say that E is invariant for A if Ae = eAe. Show that this is equivalent to A having the block form

E A1 A2 . (3.3.12) A= 0 A4 E⊥ Let A(E) = {A ∈ L(H ) : Ae = eAe}. Show that A(E) forms an algebra under operator multiplication. (iii) Show that E and E ⊥ are invariant for A, if and only if [A,e] = 0, or equivalently A has the block form

E A1 0 . (3.3.13) A= 0 A4 E⊥ In this case, we say that E is reducing for A. (iv) Let E = {A ∈ L(H ) : [A,e] = 0}. Show that E is a C ∗ -algebra. Exercise 3.3.9 (Dirichlet space) Let A be the space off ∈ C(S 1 ;C) that  inθ , where a = 2π f (eiθ )e−inθ dθ/ have Fourier series f (θ ) = ∞ n n=−∞ an e 0

68

Algebras of Operators

 2 (2π ), and ∞ n=∞ |n||an | converges. This is sometimes known as the Dirichlet space, especially when written in the style of Exercise 3.7.7. See also [85, p. 82]. (i) Show that K(θ,φ) =

f (θ ) − f (φ) eiθ − eiφ

(3.3.14)

defines the kernel of a Hilbert–Schmidt integral operator on L2 (S 1 ;C) with K2L2 =

∞ 

|n||an |2 .

n=−∞

(ii) Let F : L2 (S 1 ;C) → L2 (S 1 ;C) be the operator  2π φ dφ F g(θ ) = PV g(θ − φ) cot , 2 2π 0

(3.3.15)

and Mf the multiplication operator Mf g = f g. Find the kernel of [Mf ,F ] and show that [Mf ,F ] is Hilbert–Schmidt. (iii) By considering δ(f ) = [Mf ,F ], show that A is an algebra. (iv) For f0,f1,f2 ∈ A, show that  f1 (θ1 ) − f1 (θ2 ) f2 (θ2 ) − f2 (θ1 ) dθ1 dθ2 f0 (θ1 ) ϕ(f0,f1,f2 ) = tan(θ1 − θ2 )/2 tan(θ2 − θ2 )/2 2π 2π S 1 ×S 1 (3.3.16) converges and   ϕ(f0,f1,f2 ) = trace Mf0 [Mf1 ,F ][Mf2 ,F ] .

(3.3.17)

Exercise 3.3.10 (Restricted general linear group) Let A be a subalgebra of L(H ) and δ : A → L(H ) a derivation of the form δ(A) = [A,e] for some orthogonal projection e. Show that the following give subalgebras of A: (i) A0 = {A ∈ D : δ(A) = 0}; (ii) A∞ = {A ∈ D : δ(A) ∈ K(H )}. (iii) Deduce that there is an algebra



 L(H ) K(H ) R S = : S,T ∈ K(H );R,U ∈ L(H ) . (3.3.18) K(H ) L(H ) T U (iv) Likewise, show that

 L(H ) L2 (H ) R = A2 = T L2 (H ) L(H )



S 2 : S,T ∈ L (H );R,U ∈ L(H ) U

3.4 Fredholm Operators

69

gives a unital algebra A2 , and the set G(A2 ) of invertible elements in A2 gives a group, so that δ(U −1 ) = −U −1 δ(U )U −1 for all U ∈ G(A2 ). This is the restricted general linear group, as in [85, p. 80]. Exercise 3.3.11 (Idempotents) Let E ∈ L(H ) satisfy E 2 = E. Show that E ∈ K if and only if E has finite rank, and trace(E) = rank(E).

3.4 Fredholm Operators Theorem 3.4.1 (Atkinson)

For T ∈ L(H ), the following are equivalent:

(i) there exists S ∈ L(H ) such that 1 − ST and 1 − T S are compact; (ii) the image of T is closed in H , dim Ker(T ) is finite and dim Ker(T ∗ ) is finite; (iii) π(T ) is invertible in the Calkin algebra C(H ) = L(H )/K(H ). See [90] for discussion. An operator satisfying these equivalent conditions is called Fredholm. By condition (iii), the space of Fredholm operators forms a multiplicative semigroup under composition. Also, the set of invertible elements in C(H ) is open in the norm topology on C(H ), and the inverse image of any open subset of C(H ) is open in L(H ). All separable infinitedimensional complex Hilbert spaces are unitarily equivalent. However, it is sometimes necessary to deal with Hilbert spaces that have a fixed interpretation as function spaces, so we formulate the following definition for pairs of Hilbert spaces. Definition 3.4.2 (Fredholm operator) A bounded linear operator P : H1 → H2 between separable and infinite-dimensional complex Hilbert spaces is said to be Fredholm if there exists a bounded linear operator Q : H2 → H1 such that QP −1 and P Q−1 are compact; then Q is called a parametrix. The index of a Fredholm operator is index(P ) = dim Ker(P ) − dim Coker(P ).

(3.4.1)

Here Coker(P ) = (P H1 )⊥ = ker(P ∗ ). The index does not depend upon the particular choice of Q, so we can select any convenient Q. This notion of index is consistent with the terminology of Section 2.9. One can introduce a functor K0 on C ∗ -algebras such that K0 (K) = Z. Theorem 3.4.1 shows that the set F(H ) of Fredholm operators on H is open, the index is well-defined on F(H ) and constant on connected components of F(H ). In the next section, we show how to compute the index. For a more analytical presentation, see appendix 1 of [18].

70

Algebras of Operators

Exercise 3.4.3 (Cokernels) (i) Let Aj be finite-dimensional complex vector spaces with dimension aj such that 0 −→ A0 −→ A1 −→ A2 −→ A3 −→ 0 is an exact sequence. Show that a0 + a2 = a1 + a3 , hence that A0 ⊕ A2 ∼ = A1 ⊕ A3 . (ii) Let T : H0 → H1 be a Fredholm operator and F a finite-dimensional subspace of H1 such that im(T ) + F = H1 , where the sum is not necessarily direct. Show that there is an exact sequence 0 −→ Ker(T ) −→ T −1 (F ) −→ F −→ Coker(T ) −→ 0,

(3.4.2)

and deduce that Ker(T ) ⊕ F ∼ = T −1 (F ) ⊕ Coker(T ). (iii) Deduce that F ∼ = T −1 (F ) if and only if the index of T is zero. As in Section 2.1, a complex is a sequence of k-vector spaces Kj , linked by k-linear maps dj : Kj → Kj +1 such that dj +1 dj = 0, which we write in the following array: dj −1 dj dj −2 (3.4.3) Kj −1 Kj Kj +1 . . . −→ −→ −→ Let Zj = Ker(dj ) and Bj = Im(dj −1 ), so that Bj ⊆ Zj , and we define the homology group H j = Zj /Bj to be the quotient vector space. By an endomorphism T of a complex, we mean a sequence T = (Tj ) of k-linear maps Tj : Kj → Kj such that Tj +1 dj = dj +1 Tj , and so the following diagram commutes: ...

Kj Tj ↓ Kj

dj −→ dj −→

Kj +1 Tj +1 ↓ Kj +1

(3.4.4)

Such an endomorphism induces an endomorphism Hˆ i (T ) of the homology groups H i . See [5]. When the Kj are complex separable Hilbert spaces, we suppose that the dj are Fredholm operators. Then the H j are finite dimensional, so we can express the endomorphism as a finite matrix, and then take the trace. Definition 3.4.4 (Lefschetz number) Suppose that dim Kj = 0 for all j < 0 and all j > n, and that dim H j is finite for all j = 0, . . . ,n so the complex reduces to 0 −→ K0 −→ K1 −→ · · · −→ Kn−1 −→ Kn −→ 0.

3.5 Index Theory on the Circle via Toeplitz Operators

Then the Lefschetz number of T is n  (−1)j trace Hˆ j (T ). (T ) =

71

(3.4.5)

j =0

The following result gives a means for computing the index in terms of parametrices. We use this in specific examples in 3.5.2, 3.6.2, 3.8.6 and 3.8.8. Proposition 3.4.5 Suppose that P is Fredholm with parametrix Q, so QP −1 and P Q − 1 are both trace class. Then index(P ) = −trace(1 − P Q) + trace(1 − QP ).

(3.4.6)

Proof The proof that follows is not the most direct, but is intended to link with other theories. Given that we have a parametrix Q, it is natural to use the trace formula (3.2.18) to compute the index. There is a diagram 0

0

−→

H1

1−QP

↓

−→

H1

P −→ Q

H2

−→

↓

1−P Q

H2

−→

'

−→ P

0 (3.4.7) 0

with induced maps on the homology groups Ker(P ) −→ Coker(P ) ↓ ↓ Ker(P ) −→ Coker(P )

(3.4.8)

The Lefschetz number is  = trace(1 − QP ) − trace(1 − P Q),

(3.4.9)

and when 1 − QP and 1 − P Q are trace class, this is  = dim Ker(P ) − dim Coker(P ).

(3.4.10)

The analytical details are described in [56, theorem 19.1.15].

3.5 Index Theory on the Circle via Toeplitz Operators In the context of Example 3.2.2 with H = L2 (S 1 ;dθ/(2π )), let the Hardy space H+ be #∞ $ ∞   n 2 H+ = an z : |an | < ∞ (3.5.1) n=0

n=0

72

Algebras of Operators

and # H− =

−1 

an zn :

n=−∞

−1 

$ |an |2 < ∞

(3.5.2)

n=−∞

so that H = H+ ⊕ H− where the complementary subspaces H+ and H− are orthogonal. Now H+ is the image of the orthogonal projection e : H → H+ , known as the Hardy projection # 0, for n < 0; n (3.5.3) e(z ) = n z , for n = 0,1,2, . . .. The image of 1 − e is H−, the orthogonal complement of H+ in L2 . There are equivalent descriptions of H+ as the space of square summable power series as in (3.5.2), the space of f ∈ L2 (S 1 ;dθ/(2π )) such that fˆ(n) = 0 for n < 0; and the space of holomorphic functions g : D → C such that 2π sup0 1, it is important to note that  is a special lattice such that Cg / inherits a complex structure; see [76]. In particular, if X is a hyperelliptic curve of genus g, Mumford constructs a special meromorphic function ℘ on the Jacobian J ac(X) and an invariant vector field D∞ on J ac(X). Then ℘ (2g−1) ℘ has poles on ' ⊂ X and derivatives ℘ (1) = D∞ ℘, . . . ,℘ (2g−1) = D∞ (1) (2g−1) generate the affine ring of J ac(X) \ '. As in such that ℘,℘ , . . . ,℘ theorem 10.3 of [79], this gives an embedding J ac(X) \ ' → C2g . (iv) We pursue this further by momentarily assuming some results from sheaf theory, as in [36, p. 123]. Now we loosen up and allow C ∞ sections, so we can patch together sections of line bundles on X by means of C ∞ partitions of unity on X. Let U be an open neighbourhood in X and f : U → C a nowhere vanishing C ∞ function. Then there exists a neighbourhood V ⊆ U and a C ∞ function g : V → C such that f = e2π ig . Hence there is a short exact sequence of groups 0 −→ Z

ι −→

θ

e2π i· −→

θ∗

−→ 0.

(8.2.14)

This gives rise to an exact sequence of cohomology groups H 1 (X;Z) −→ H 1 (X;θ ) −→ H 1 (X;θ ∗ )

δ H 2 (X;Z) −→ H 2 (X;θ ), −→ (8.2.15)

so the connecting map δ gives a group homomorphism δ : H 1 (X;θ ∗ ) → H 2 (X;Z) which gives a version of the first Chern class c1 (E) = δ(E) ∈ H 2 (X;Z) for E ∈ H 1 (X;θ ∗ ) by [63, p. 33]. Then by general results about fine sheaves, we have H 1 (X;θ ) = H 2 (X;θ ) = 0, so in the C ∞ category there is an isomorphism of groups δ : H 1 (X;θ ∗ ) → H 2 (X;Z). For a compact Riemann surface X of genus g, there are four interpretations of the Picard group: (i) Pic(X) is the multiplicative group H 1 (X;θ ∗ ) of complex line bundles over X; (ii) Pic(X) is the divisor class group, with subgroup Pic0 (X); (iii) Pic0 (X) is isomorphic to the Jacobi variety Cg / ∼ = Xg / ∼; (iv) Pic(X) is the group of first Chern classes c1 (E) in H 2 (X;Z). Example 8.2.8 (Second Chern class) Donaldson and Kronheimer [28] give a detailed discussion of the second Chern class c2 for a four-dimensional manifold.

8.3 Deforming Flat Connections

213

8.3 Deforming Flat Connections Suppose that V = E ⊕ E  is a linear direct sum of C-vector spaces with a specified inner product. Usually we take dim E = dim E  , so dim V is even. An involution is ε ∈ End(V ) such that ε2 = id. We have embeddings ι and j with corresponding projections ι∗ and j ∗, as in the following array: E

ι∗ ←− −→ ι

V

j∗ −→ ←− j

E ,

which may be expressed as a block matrix

∗

0 ι 0 ι I= 0 j∗ 0 j

(8.3.1)

E . E

(8.3.2)

We write d for the trivial connection for the bundle V × M → M with fibre V , and introduce connections ∇ = ι∗ dι,

∇  = j ∗ dj,

(8.3.3)

so we can express ∇ 2 and likewise (∇  )2 as the product of two tensors ∇ 2 = (ι∗ dι)(ι∗ dι) = ι∗ d(1 − jj ∗ )dι = −(ι∗ dj )(j dι∗ ),

(8.3.4)

where ι∗ dj ∈ 1 (M,Hom(E ,E)) and j ∗ dι ∈ 1 (M,Hom(E,E  )). We change notation and suppose that we have a (flat) connection D where



∗ 0 0 ι∗ Dj ι Dι + ∗ D= 0 0 j ∗ Dj j Dι := ∇ + α and ε=

(8.3.5)

∗ 0 ι ι , 0 −j ∗ j

so that ε2 = I , ε∇ = ∇ε and αε = −εα. Then D 2 = ∇ 2 + ∇α + α∇ + α 2 = ∇ 2 + α 2 + [∇,α],

(8.3.6)

where ∇ 2 + α 2 commutes with ε, whereas [∇,α] anti-commutes with ε. If D 2 = 0, then ∇ 2 + α 2 = 0 = [∇,α]. Suppose that (ut )t∈[0,1] is a continuously differentiable family of unitary operators on V . Then εt = ut εu∗t is a continuously differentiable family of involutions on V , so the complementary subspaces E and E  change as t varies. We wish to investigate the effect on the cohomology classes.

214

Connections

Proposition 8.3.1 Let D be a flat connection on a bundle V over m with fibre Cm and ε an endomorphism of V such that ε2 = id. Then D = ∇ + α,

(8.3.7)

where (i) ∇ = (1/2)(D + εDε) is a connection which commutes with ε; (ii) α = (1/2)(D − εDε) is a 1-form anti-commuting with ε; (iii) the forms tr(α 2n ) and tr(εα 2n ) are closed and their de Rham cohomology class does not change as ε is changed smoothly. Proof By flatness, we have 0 = D 2 = ∇ 2 + ∇α + α∇ + α 2 = ∇ 2 + α 2 + [∇,α],

(8.3.8)

where ∇ 2 + α 2 commutes with ε, whereas [∇,α] anti-commutes with ε. Hence ∇ 2 + α 2 = 0 = [∇,α], so   (8.3.9) dtr (∇ 2 )n = tr[∇,(∇ 2 )n ] = 0, so tr(∇ 2 )n is closed. Hence tr(∇ 2 )n = tr(−α 2 )n = (−1)n trα 2n = 0,

(8.3.10)

since tr α 2n = tr αα 2n−1 = −tr α 2n−1 α = −tr α 2n . Also dtr(εα 2n ) = tr d(εα 2n ) = tr([∇,εα 2n ]) = 0,

(8.3.11)

since [α,∇] = 0 = [∇,ε]. Now suppose that ε = εt is a smoothly varying family of involutions on the bundle V with connection D fixed. (We can easily produce these by taking a one-parameter subgroup of ut ∈ U (m) and letting εt = ut ε0 u∗t .) Then we have εε˙ + ε˙ ε = 0, so α˙ = (−1/2)(˙ε Dε + εD ε˙ ) = (−1/2)(˙ε (∇ + α)ε + ε(∇ + α)˙ε ) = (−1/2)(˙ε ∇ε + ε∇ ε˙ ) + (1/2)(˙ε αε + εα ε˙ ),

(8.3.12)

where the first summand anti-commutes with ε, whereas the final summand commutes with ε. The anti-commuting part is (−1/2)[∇,εε˙ ]. Now  ∂ tr(εα 2n ) = tr(˙ε α 2n ) + tr(εα i αα ˙ 2n−1−i ) ∂t n

i=1

˙ 2n−1 ). = tr(˙εα 2n ) + 2ntr(εαα

(8.3.13)

8.4 Universal Differentials

215

The first summand here vanishes, while the second depends only upon the part of α˙ that anti-commutes with ε. Hence ∂ tr(εα 2n ) = (−1/2)2ntr(ε[∇,εε˙ ]α 2n−1 ) ∂t = −ntr([∇, ε˙ α 2n−1 ]) = −ndtr(˙εα 2n−1 ),

(8.3.14)

since [α,∇] = 0 = [∇,ε], which gives an exact differential form, so  1 2n 2n τ (εα )t=1 = τ (εα )t=0 − d nτ (˙εα 2n−1 ) dt, (8.3.15) 0

and the cocycles

τ (εα 2n )

t=1

and

τ (εα 2n )

t=0

are cohomologous.

Example 8.3.2 Let A = C ∞ (M) and let e ∈ Mr (A) be an idempotent, so we can introduce a vector bundle over M as the range of e. Then CI + Ce is an algebra generated by one idempotent as in Example 7.4.3, and we consider cn ([e]) (2n)! = (−1)n−1 trace(ede . . . de) ∈ 2n A, n! 2(n! )

(8.3.16)

where the image depends on the class [e] of e in K0 (A). As in Corollary 8.2.4, ev (M). The choice of constants the sum of these terms gives an element in HDR is given by Example 7.4.3. See section 8.3 of [68] for further discussion.

8.4 Universal Differentials We start by considering universal properties of R, and then use this as a tool for studying connections over manifolds. Definition 8.4.1 (Derivation) With R an algebra and M an R-bimodule, a derivation D : R → M is a k-linear map such that D(xy) = x(Dy) + (Dx)y.

(8.4.1)

Example 8.4.2 (Inner derivations) Let m ∈ M and let D be the inner derivation D(x) = xm − mx = [x,m]. To clarify the various notations, we write Z 1 (R,M) = Der(R,M) = HomR⊗R op (1 R,M). We also let b : R ⊗n → R ⊗n+1 be n  b (x1, . . . ,xn ) = (−1)i−1 (x1, . . . ,xi xi+1, . . . ,xn ) i=1

(8.4.2)

(8.4.3)

216

Connections

and s : R ⊗n → R ⊗n+1 be s(x1, . . . ,xn ) = (1,x1, . . . ,xn ).

(8.4.4)

Proposition 8.4.3 (i) Then b is an R-bimodule map such that b2 = 0; (ii) s is a right R-module map such that b s + sb = id; (iii) the following sequence is exact: ...

b

−→

R ⊗4

b

−→

R ⊗3

b

−→

R ⊗2

b

−→

R −→ 0.

(8.4.5)

Let 1 R = Ker(b : R ⊗2 → R) which may also be expressed as Im(b : R ⊗3 → R ⊗2 ) or as Coker(b : R ⊗4 → R ⊗3 ). In the following result, we use the latter interpretation. Proposition 8.4.4 (1,y,1) satisfies

(i) 1 R is an R-bimodule, and ∂ : R → 1 R : ∂y = f ∂(gh)k = f (∂g)hk + f (g∂h)k.

(8.4.6)

(ii) The map HomR⊗R (1 R,M) → Der(R,M) : u → u ◦ ∂ is surjective. Proof (i) We have b (x,y,z,w) = (xy,z,w) − (x,yz,w) + (x,y,zw),

(8.4.7)

so ∂(yz) = (1,yz,1) = (y,z,1) + (1,y,z)

mod(b R ⊗4 ),

(8.4.8)

so ∂(yz) = y(∂z) + (∂y)z.

(8.4.9)

(ii) For any D there is a unique u such that R

D −→ ∂ /

M ↑u 1 R

(8.4.10)

We take the first-order differentials in 1 R = R ⊗3 /b R ⊗4 . The sequence 0 −→ 1 R

j b R −→ 0 R⊗R −→ −→

(8.4.11)

is exact, where j : x(∂y)z → (xy,z) − (x,yz) is the bimodule map which corresponds to the derivation y → y ⊗ 1 − 1 ⊗ y and j is induced by b : R ⊗3 → R ⊗2 .

8.5 Connections on Modules over an Algebra

217

8.5 Connections on Modules over an Algebra Having reviewed the concept of a connection in differential geometry in Section 8.1, we proceed to introduce a connection for noncommutative differential forms. Definition 8.5.1 (Right connection) Let E be an R-bimodule, where R is a unital algebra over C. A right connection is a linear map ∇ : E → E ⊗R 1 R that satisfies the Leibniz rule ∇(xξ ) = x∇ξ, ∇(ξ x) = (∇ξ )x + ξ dx

(ξ ∈ E,x ∈ R)

(8.5.1)

for the right module operation. Lemma 8.5.2 Let R be quasi-free. Then there exists a right connection ∇ : 1 R → 2 R. Proof We have an exact sequence of R-bimodules 0 −→ 2 R

j μ 1 1 R ⊗ R  R −→ 0, −→ −→

(8.5.2)

¯ where j (αdx) = αx⊗1−α⊗x and μ(α⊗x) = αx. Now 1 R⊗R ∼ = R⊗R⊗R 1 is a free R-bimodule, and  R is projective by Definition 6.4.7(iii), so there is a splitting map s : 1 R → 1 R ⊗ R such that j (∇ξ ) = ξ ⊗ 1 − s(ξ ).

(8.5.3)

Now   j ∇(ξ a) − (∇ξ )a = ξ a ⊗ 1 − ξ ⊗ a − s(ξ a) + s(ξ )a

(8.5.4)

or equivalently   j ∇(ξ a) − (∇ξ )a − ξ da = s(ξ a) − s(ξ )a,

(8.5.5)

so s is a right R-module map if and only if ∇ is a right connection. Example 8.5.3 (Noncommutative tangents) Connections on the A-bimodule E = 1 A are the noncommutative analogues of connections on the tangent bundle T M to a manifold M. With A = C ∞ (M), and V a vector bundle over M, there is SA (M,V ), the symmetric algebra of sections of the bundle V . Definition 8.5.4 (Cup product) Let L,M,N be A-bimodules and suppose that L ⊗A M → N : x ⊗ y → x · y is an A-bimodule map. Then the cup product is defined by C p (A,L) ⊗ C q (A,M) → C p+q (A,N )

218

Connections

f (a1, . . . ,ap ) ⊗ g(ap+1, . . . ,ap+q ) → f (a1, . . . ,ap ) · g(ap+1, . . . ,ap+q ), (8.5.6) so f ∪ g(a1, . . . ,ap+q ) = f (a1, . . . ,ap ) · g(ap+1, . . . ,ap+q ).

(8.5.7)

δ is a derivation when δ(f ∪ g) = (δf ) ∪ g + (−1)|f | f ∪ δg.

(8.5.8)

Example 8.5.5 (Universal n-cocycle) For d ∈ C 1 (A,A), the universal derivation, we have d ∪ d : (a1,a2 ) → da1 da2 . Also d ∪ . . . ∪ d to n factors is the universal n-cocycle. Recall from Lemma 6.4.8 that A is quasi-free if and only if 1 A is a ¯ is free and an A-bimodule. projective A-bimodule. Also, 1 A⊗A = A⊗A⊗A The following data exist for A if and only 1 A is projective as an A-bimodule, and hence give a further criterion for A to be quasi-free. This completes the proof of Proposition 6.7.4. Proposition 8.5.6 The following data are equivalent: (i) a one cochain φ : A¯ → 2 A such that −δφ = d ∪ d, that is a1 φ(a2 ) − φ(a1 a2 ) + φ(a1 )a2 + da1 da2 = 0;

(8.5.9)

(ii) a lifting homomorphism for A → RA/(I A)2 (see Corollary 6.7.3); (iii) a bimodule splitting of j mr 1 1 A ⊗ A 0,  A −→ −→ −→ −→ where j (ωda) = ωa ⊗ 1 − ω ⊗ a and mr (ω ⊗ a) = ωa; (iv) a right connection ∇r : 1 A → 2 A where 2 A = 1 A ⊗A 1 A. 0

2 A

Proof (i) ⇔ (ii) Recall from Corollary 6.7.3 that RA/(I A)2 = A ⊕ 2 A with the Fedosov product. A linear lifting φ : A → RA/(I A)2 sending 1 → 1 has the form a → a − φ(a), where φ : A → 2 A is linear. This lifting is a homomorphism for  when a1 a2 − φ(a1 a2 ) = (a1 − φ(a1 ))  (a2 − φ(a2 )) = a1 a2 − da1 da2 − φ(a1 )a2 + dφ(a1 )da2 − a1 φ(a2 ) (8.5.10) modulo forms of degree ≥ 4; hence we have a lifting homomorphism if and only if −φ(a1 a2 ) = −da1 da2 − φ(a1 )a2 − a1 φ(a2 ).

(8.5.11)

8.5 Connections on Modules over an Algebra

219

(i) ⇔ (iv) Let ∇r be linear and compatible with left multiplication, so ∇r (a0 da1 ) = a0 φ(a1 ) where φ : A¯ → 2 A is linear. Then ∇r satisfies the Leibniz rule for right multiplication if and only if ∇r (ωa) = ∇r (ω)a + ωda

(8.5.12)

∇r ((a0 da1 )a2 ) = ∇r (a0 da1 )a2 + a0 da1 da2,

(8.5.13)

if and only if

that is ∇r (a0 d(a1 a2 )) − ∇r (a0 a1 da2 ) = ∇r (a0 da1 )a2 + a0 da1 da2,

(8.5.14)

a0 φ(a1 a2 ) − a0 a1 φ(a2 ) = a0 φ(a1 )a2 + a0 da1 da2 .

(8.5.15)

so

(i) ⇔ (iii) See Lemma 8.5.2. Example 8.5.7 (Tensor algebras) (i) Consider ∇ : 1 R → 1 R ⊗R 1 R = 2 R satisfying ∇(xξ ) = x∇ξ and ∇(ξ x) = (∇ξ )x + ξ dx. Recall that R ⊗ R¯ ⊗ R is isomorphic to 1 R via x ⊗ y¯ → xdy. Then ∇ is determined by ϕ : R¯ → 2 R, satisfying ϕ = ∇d where ∇(xy) = x∇(y) + ϕ(x)y + dxdy.

(8.5.16)

(ii) Let R = T (V ) be the tensor algebra on V , so R ⊗ V ⊗ R is isomorphic to 1 R via x ⊗ v ⊗ y → x(dv)y. Then there is a canonical connection ∇ determined by the requirement that ∇dv = 0 for all v ∈ V . We have an exact sequence 0 −→ 2 R −→ 1 R ⊗ R −→ 1 R −→ 0

(8.5.17)

and maps vdy → v(y ⊗ 1 − 1 ⊗ y) and (dv)y ⊗ 1 → (dv)y, with lifting

((dv)y) = dv ⊗ y. Then the requirement ∇dv = 0 gives   ∇(x(dv)y) = x ∇(dv)y + dvdy = xdvdy, (8.5.18) so there is an extension to V ⊗n for n > 1 given by n    ∇d v1 . . . vn = ∇ v1 . . . vj −1 (dvj )vj +1 . . . vn j =1

=

n−1  j =1

v1 . . . vj −1 (dvj )d(vj +1 . . . vn ).

(8.5.19)

220

Connections

More generally, suppose that E is a projective R-bimodule. Then there are the following equivalent data: (i) a right connection ∇ : E → E ⊗R 1 R; (ii) a right module splitting of 0 −→ E ⊗R 1 R −→ E ⊗ R −→ E −→ 0;

(8.5.20)

(iii) a bimodule section s : E → E ⊗ R. Let E be a bimodule over R. A connection on E is a connection on E as a right module such that ∇(xξ ) = x∇ξ . These are equivalent to module liftings

: E → E ⊗ R of the right multiplication m : E ⊗ R → E : m(ξ ⊗ x) = ξ x such that m ◦ = I d. They exist if and only if E is projective as a bimodule. Proposition 8.5.8 Let ∇ be a connection on the bimodule E. (i) Then there is an extension of ∇ to E ⊗R R such that b∇ + ∇b = 1

in degree

j

(j = 1,2, . . .).

(8.5.21)

(ii) The Hochschild complex is contractible in degree j for j = 1,2, . . ., and Hj (R,E) = 0 for all j > 0. Proof (i) Given a connection, we can extend ∇ to a degree (+1) operator on E ⊗ R satisfying ∇(αω) = (∇α)ω + (−1)|α| αdω

(8.5.22)

for α ∈ E ⊗R R and ω ∈ R. Let n > 0, and consider E ⊗ n R, which is spanned by ξ dx1 . . . dxn with ξ ∈ E and x1, . . . ,xn , and on which b is defined by   (8.5.23) b(αdx) = (−1)deg(α) αx − xα . Then we compute

  ∇b(αdx) = (−1)|α| ∇ αx − xα   = (−1)|α| (∇α)x − (−1)|α| αdx − x∇α

(8.5.24)

and for comparison

so

  b∇(αdx) = b (∇α)dx + αd 2 x   = (−1)|α|+1 (∇α)x − x∇α ,

(8.5.25)

  ∇b + b∇ (αdx) = αdx.

(8.5.26)

(ii) This follows from (i).

8.6 Derivations and Automorphisms

Definition 8.5.9

221

(i) (Curvature) One has ∇ 2 (αω) = (∇ 2 α)ω,

so we define ∇ 2 : E → E ⊗R 2 R to be the curvature. (ii) (Chern character classes) Suppose further that E is finitely generated and projective as a right R-module, so R n = E ⊕ E  for some complementary module, and endomorphisms of E may be regarded as block matrices in Mn (R). Then one can define (Chern character classes)   traceE (∇ 2 )n ∈ 2n /[,]2n . (8.5.27) In our discussion Normalization 8.2.2 of the Chern character for connections in differential geometry, we introduced a normalizing constant so as to ensure that the Chern classes take integer values. In the noncommutative setting, there is also a natural normalization, chosen so as to fit with results from Kasparov’s KK theory. We refer readers to [18] for a statement of the result.

8.6 Derivations and Automorphisms First, we review the connection between derivations and semigroups of automorphisms of an algebra. Definition 8.6.1 (Semigroup of automorphisms) Let R be a unital Banach algebra, and (ρt )t≥0 a semigroup such that: (1) ρt ∈ L(R) for all t ≥ 0; (2) ρt (f ) → f in R as t → 0+ for all f ∈ R; (3) ρt ◦ ρs = ρs+t for all s,t ≥ 0; (4) ρt (fg) = ρt (f )ρt (g) for all f ,g ∈ R and t ≥ 0. In this case, we say that (ρt ) is a semigroup of automorphisms. Proposition 8.6.2 Suppose that (ρt ) satisfies (1)–(3). (i) Then there exists a dense linear subspace D on which the limit exists −Df = lim

t→0+

ρt (f ) − f t

(f ∈ D).

(8.6.1)

(ii) Then (ρt ) also satisfies (4), if and only if D is a dense subalgebra of R and D(f g) = D(f )g + f D(g)

(f ,g ∈ D),

so that D is a derivation, and there is a subalgebra of R given by ! ker(D) = f ∈ R : ρt (f ) = f ,∀t > 0 .

(8.6.2)

222

Connections

Proof (i) This is standard Hille–Yoshida theory of semigroups; see [31]. (ii) (⇒) Assume (4) holds. Then for f ,g ∈ D, we have ρt (g) − g ρt (f ) − f ρt (fg) − f g = ρt (g) + f t t t → −(Df )g − f D(g) (t → 0+);

(8.6.3)

so f g ∈ D and D(f g) = (Df )g + f (Dg).

(8.6.4)

By semigroup theory, D is dense in R, and f D = f R + Df R gives an algebra norm such that f gD = f gR + (Df )g + f (Dg)R ≤ f R gR + Df R gR + f R DgR ≤ f D gD ;

(8.6.5)

also D is complete for  · D since D is closed. (ii) (⇐) Conversely, suppose that −D generates (ρt ). Then for f ,g ∈ D and 0 ≤ s ≤ t we have 1   2  ∂ ρt−s ρs (f )ρs (g) = −ρt−s −D ρs (f )ρs (g) ∂s     + ρt−s −Dρs (f )ρs (g) + ρt−s ρs (f )(−D)ρs (g) 2 1  2 = ρt−s D ρs (f )ρs (g) −Dρs (f )ρs (g)−ρs (f )Dρs (g) = 0,

(8.6.6)

by (8.6.2). Hence ρt−s (ρs (f )ρs (g)) is constant, and by comparing the values at s = 0 with s = t, we obtain ρt (f g) = ρt (f )ρt (g).

(8.6.7)

Example 8.6.3 (Automorphisms of the continuous functions) Let X be a compact Hausdorff space. A ∗-homomorphism of C(X,C) is a linear map such that α(f¯) = α(f ) for all f ∈ C(X,C). A strongly continuous group of ∗-automorphisms has the form ρt (f ) = f ◦ θt , where (i) θt : X → X is a homeomorphism of X such that (ii) R × X → X : (t,x) → θt (x) is continuous, and (iii) θt ◦ θs (x) = θt+s (x) for all s,t ∈ R and x ∈ X. This follows from the Banach–Stone theorem.

8.7 Lifting and Automorphisms of QA

223

Exercise 8.6.4 (Automorphisms of the disc algebra) For the disc algebra A of Exercise 3.1.7, let the Poisson semigroup be defined by ρt (f )(z) = f (e−t z) for z ∈ C with |z| < 1. (i) Verify that (ρt )t>0 is a strongly continuous semigroup of homomorphisms ρt : A → A. (ii) Show that the generator is −D, where Df (z) = zf  (z). (iii) Show that ρt (f ) → Pf as t → ∞, where Pf (z) = f (0). Exercise 8.6.5 (Connections on tensor algebra) Let A and R be algebras, and M an A-bimodule. Let ρt : TA M → R be a differentiable one-parameter family of algebra homomorphisms, so that ρt corresponds to a pair (ut ,vt ) as in Proposition 6.4.10(i). Use Proposition 8.6.2 to differentiate this pair.

8.7 Lifting and Automorphisms of QA In this section we consider groups of automorphisms ρt = e−tD on the Cuntz algebra. The calculations are simpler to understand if one looks at the Example 5.8.1 first. In that example, we considered the harmonic oscillator semigroup acting on the Hermite polynomials. Here we have a graded algebra ev A and produce a semigroup of automorphisms on it. In the following proof, we consider a family of automorphisms e−tD of RA such that the limit limt→∞ e−tD gives an embedded copy of A in RA. So we consider a derivation on RA, and regard 2n A as playing a similar role to H e2n in Example 5.8.1. Recall that ! Homalg (RA,R) = ρ : A → R; ρ linear; ρ(1) = 1 , (8.7.1) where the linear unital map ρ gives rise to a homomorphism ρ∗ via ρ∗ (a) = ρ(a)

(a ∈ A);

(8.7.2)

ρ∗ (a0 da1 da2 . . . da2n ) = ρ(a0 )ω(a1,a2 ) . . . ω(a2n−1,a2n ),

(8.7.3)

where ω(a,b) = ρ(ab) − ρ(a)ρ(b) and ωn (a1, . . . ,a2n ) = ω(a1,a2 )ω(a3,a4 ) . . . ω(a2n−1,a2n ).

(8.7.4)

Now such elements span (I A)k for k ≤ n, and Homalg (RA/(I A)n+1,R) = ρ : A → R;

ρ

linear;

ρ(1) = 1;

! ωn+1 = 0 .

(8.7.5)

224

Connections

Theorem 8.7.1 Let A be quasi-free. Then: (i) there exists a linear map φ : A → 2 A such that −δφ = d ∪ d; (ii) there exists a unique derivation D on RA such that D(a) = φ(a); 9 2j (iii) D : RA → RA corresponds to H + L on ∞ j =0  A, where H is the 2n 2n+2 number operator, and L has degree 2, so L :  A →  A; −L L −L 2−k . Let Uk ∈ Cc∞ (Rn ;R) be a nonnegative function such that Uk (x) = 0 for all x ∈ k and Uk (x) = 2k x2 for all x ∈ ck+1 ; also Uk ≤ Uk+1 . It is clear from (9.4.6) that the kernel

of e−h¯ #/2−Uk is monotone decreasing in k. The limiting operator # is the Laplacian in L2 (;C) with Dirichlet boundary conditions. Hence n   2  1 vol() (h¯ → 0) (9.4.7) trace e−h¯ #/2 , √ h¯ 2π 2

by the monotone convergence theorem. In 17.5 of [56], H¨ormander uses wave equation methods to give a more precise version of this result, with sharp bounds on the error term. Of course, the terms in (9.4.3) become increasingly complicated and difficult to interpret geometrically. In the context of Riemannian manifolds, McKean and Singer [74] compute the first three terms in the asymptotic expansion of the heat kernel e−t# (x,y) in terms of local geometrical quantities such as the curvature tensor; however, they noted that the task of describing all the terms in the asymptotic expansion seemed pretty hopeless. Atiyah, Bott and Patodi subsequently overcame these difficulties [7, theorem E.III]. The following approach is more algebraic, and seeks to use cocycles defined by the commutator products to compute the traces of the leading order terms. Let W be the cotangent space of R2n and V = W ∗ the space of linear functions on W . Let ω : ∧2 V → C be the natural symplectic form on W , and write ω = ω+ − ω− , where ω± are rank n bilinear forms on complementary subspaces. We can take the south-west corner of the matrix for ω+ , and the north-east corner for ω− ; results of [57, p. 333] show one can always do this. Let S(W ) be the Schwartz class functions on W , which are C ∞ smooth and of rapid decay. For h¯ > 0, we introduce a group law on Hh¯ = R × V by (t1,v1 ) ◦ (t2,v2 ) = (t1 + t2 + hω(v ¯ 1,v2 )/2,v1 + v2 ).

(9.4.8)

Proposition 9.4.3 Given unitary groups (eip·D )p∈Rn and (eiq·X )q∈Rn on a Hilbert space such that ¯ + (p,q) eiq·X eip·D , eip·D eiq·X = ei hω

(9.4.9)

then with v = (q,p), there is a unitary representation of πh¯ : Hh¯ → U (H ) by ¯ + (q,p)/2+iq·X+ip·D . πh¯ (t,q,p) = eit−i hω

(9.4.10)

242

Cocycles for a Commutative Algebra over a Manifold

Then πh¯ extends to S(W ) → L(H ) by  πh¯ (f ) = fˆ(q,p)πh¯ (0,q,p) dqdp.

(9.4.11)

V

9.5 The Poisson Bracket and Symbols We begin by introducing pseudo-differential operators within the formalism of H¨ormander’s general theory. Definition 9.5.1 (Pseudo-differential operator) For u ∈ L1 ∩ L2 (Rn ), let  u(ξ ˆ )= e−i ξ,x u(x) dx Rn

be the Fourier transform of u. Let Dx = −i∇ be differentiation   ∂u ∂u Dx u = , ,..., i∂x1 i∂xn

(9.5.1)

which is represented by u(ξ ˆ ) → (ξj u(ξ ˆ )). Then for m ∈ R, let S m = S m (Rn × n ∞ n R ;C) be the set of all p ∈ C (R × Rn ;C) such that for all multi-indices α and β there exists Cα,β > 0 such that  α β  ∂ ∂  m−|α|   ; (9.5.2)  ∂ξ α ∂x β p(x,ξ ) ≤ Cα,β (1 + |ξ |) such a p is a symbol of order m. The corresponding integral operator is P = (p) where  1 P u(x) = ei x,ξ p(x,ξ )u(ξ ˆ ) dξ, (9.5.3) (2π )n Rn and we say that P has order m, and σ (P ) = p. The integrand involves both x and ξ variables and can be simplified using the formula  1 iDξ ·Dy e a(x,y,ξ )|y=x = ei x−y,η−ξ a(x,y,ξ ) dydη (9.5.4) (2π )n Rn ×Rn for a ∈ Cc∞ (Rn × Rn × Rn ;R). We write DO m = {(f ) : f ∈ S m (Rn × Rn )} for the pseudo-differential operators of order m on L2 (Rn ;C). The notion of a pseudo-differential operator combines two basic examples of operators from Fourier analysis, described in the following Lemma. Lemma 9.5.2 (i) Let p(x,ξ ) = U (x) where U : Rn → C is bounded and measurable. Then P is the bounded linear operator of multiplication by U (x).

9.5 The Poisson Bracket and Symbols

243

(ii) Let p(x,ξ ) = p0 (ξ ), where p0 ∈ L1 has pˆ 0 ∈ L1 (Rn ;C). Then P is convolution P u(x) =  a Fourier multiplier, which may be written as the ∂ pˆ 0 (x − y)u(y) dy, or equivalently as P = p0 (−i ∂x1 , . . . , − i ∂x∂ n ), and P is bounded on Lq (Rn ;C) for 1 ≤ q ≤ ∞. (iii) Let U ∈ L2 (Rn ;C), p0 ∈ L2 (Rn ;C) and p(x,ξ ) = U (x)p0 (ξ ). Then is the Hilbert–Schmidt operator on L2 (Rn ;C) with kernel U (x)p0 (−i h∂/∂x) ¯ U (x)pˆ 0 (x − y). (iv) In particular, (iii) applies to p0 (x) = (1 + |x|2 )−s/2 for s > n/2. Proof (ii) This is Young’s convolution inequality, as in u ∗ pˆ 0 Lq (Rn ;C) ≤ uLq (Rn ;C) pˆ 0 L1 (Rn ;C) . (iii) Here U (x)pˆ 0 (x − y) is square integrable over R2 since    2 2 2 |U (x)| |pˆ 0 (x − y)| dxdy = 2π |U (x)| dx |p0 (y)|2 dy R2

R

R

by Plancherel’s theorem, hence giving a Hilbert–Schmidt kernel. (iv) We verify the particular choice (1 + |x|2 )−s/2 ∈ L2 (Rn ;C) for s > n/2 by using polar coordinates. This example may be compared with Exercise 3.2.11. Proposition 9.5.3 The pseudo-differential operators on L2 (Rn ;C) give a graded algebra, in the following sense. (i) Let p ∈ S m (Rn × Rn ;C), where m ≤ 0. Then the corresponding P = (p) belongs to L(L2 (Rn )). (ii) Let p ∈ S m and q ∈ S μ for m,μ ∈ Z. Then the operators P = (p) and Q = (q) have product P Q = R, where R is a pseudo-differential operator of order m + μ, and the commutator [P ,Q] is a pseudo-differential operator of order m + μ − 1. (iii) The pseudo-differential operators of order zero form a subalgebra DO 0 of L(L2 (Rn )). (iv) Suppose that P is trace class and p is integrable where P = (p). Then  1 trace(P ) = p(x,ξ )dxdξ . (9.5.5) (2π )n Rn ×Rn Proof (i) This is a variant of Schur’s Lemma on integral operators [56, 18.8.17]. (ii) As in [56, 1.8.18]. The symbol of R is  1 e−i x−y,ξ −η p(x,η)q(y,ξ ) dydη, (9.5.6) r(x,η) = (2π )n Rn ×Rn

244

Cocycles for a Commutative Algebra over a Manifold

which reduces to r(x,η) = eiDξ ·Dy p(x,η)q(y,ξ )|y=x,η=ξ .

(9.5.7)

Hence DO m ◦DO μ ⊆ DO m+μ , where ◦ denotes operator composition. Let P1 and P2 be pseudo-differential operators of order zero. Then P1 and P2 define bounded linear operators on H = L2 (Rn ) such that [P1,P2 ] is a pseudo-differential operator with symbol  1 σ ([P1,P2 ])(x,η) = e−i x−y,ξ −η (p1 (x,η)p2 (y,ξ ) n n (2π )n R ×R − p2 (x,η)p1 (y,ξ )) dydη   = eiDξ ·Dy p1 (x,η)p2 (y,ξ ) − p1 (y,ξ )p2 (x,η) |y=x,η=ξ . (9.5.8) (iii) This combines (i) and (ii). (iv) This is a variant of Mercer’s trace formula. Next we link up the Poisson structure on R2n , as discussed in Section 9.1, with the group representation considered in Section 9.3. The first step is to extend representations of the Heisenberg group to the Schwartz class. The Schwartz class has pointwise multiplication and differentiation, hence we can consider the Poisson bracket {·,·} of Section 9.1. Also, where possible, we extend the definition to H¨ormander’s symbol classes S m via Proposition 9.5.3. We recall the Fourier transform  ˆ f (q,p) = f (x,ξ )e−i q,x −i p,ξ dxdξ Rn ×Rn

for smooth functions f that are of rapid decay, which enables us to define the pseudo-differential operator π1 (f ) where  1 fˆ(q,p)ei(q·X+p·D) u dqdp. (9.5.9) π1 (f )u = f (X,D)u = (2π )2n Rn ×Rn We use the subscript to indicate the implicit choice h¯ = 1. This formula differs from (3.7.12) in two respects: first, the operators X and D are unbounded on L2 (Rn ); second, we have the exponential of i(q · X + p · D) instead of a Wick-ordered product of exponentials. Proposition 9.5.4 (Weyl calculus) (i) Let f ∈ S m (Rn×Rn ), g ∈ S μ (Rn × Rn ). Then their usual Poisson bracket {f ,g} satisfies π1 ({f ,g}) = −i[π1 (f ),π1 (g)]

(9.5.10)

modulo, a pseudo-differential operator with symbol in S m+μ−2 (Rn × Rn ).

9.5 The Poisson Bracket and Symbols

245

(ii) H¨ormander’s class (S 0,{·,·}) is a Lie algebra, and π1 : (S 0,{·,·}) → (L(H ), − i[·,·]) is a Lie algebra homomorphism modulo pseudo-differential operators of order −2. (iii) Suppose that f ∈ S gives π1 (f ) ∈ L1 (H ). Then the trace satisfies  1 trace(π1 (f )) = f (x,ξ ) dxdξ . (9.5.11) (2π )n Rn ×Rn (iv) The adjoint on L(H ) satisfies π1 (f¯) = π1 (f )∗ , where f¯ denotes a complex conjugate. Proof (i) We introduce a special convolution for f ∈ S m and g ∈ S μ such that π1 (f )π1 (g) = π1 (f ∗1 g); then by page 62 of [104], (f ∗1 g)(x,ξ ) = f (x,ξ )g(x,ξ ) + (i/2){f ,g}

mod

S m+μ−2 .

(ii) Here π1 (f ) ∈ L(H ) for all f ∈ S 0 by theorem 18.1.11 of [56] and we can apply (i). In particular DO −2 forms an ideal in DO 0 , so E = DO 0 /DO −2 is an algebra and we have a natural map S 0 → E arising from π1 . We can pass to the commutator quotient spaces so S 0 /{S 0,S 0 } → E/[E,E]. (iii) The kernel is given by 

 π1 (f )(x,y) ↔ since π1 (f )u(x) =

1 (2π )n

f Rn



 Rn ×Rn

f

 x+y ,ξ ei(x−y)·ξ dξ 2

 x+y ,ξ ei(x−y)·ξ u(y)dydξ ; 2

(9.5.12)

(9.5.13)

so when we integrate along x = y, we obtain the stated trace formula. (iv) We can take the adjoint of (9.5.9). The results in this section have been formulated for Euclidean space. In Section 9.6, we give versions that hold for Riemannian manifolds. For a compact manifold M and H = L2 (M), the grading on DO matches better with the grading on the Lp (H ). Exercise 9.5.5 (i) Show that there is an increasing sequence of ideals in the algebra DO 0 . . . ⊆ DO −2 ⊆ DO −1 ⊆ DO 0 .

246

Cocycles for a Commutative Algebra over a Manifold

(ii) By reviewing Corollary 3.7.2, show that there is a commutative ∗-algebra C0 that makes the following sequence exact: 0 −→ DO −1 −→ DO 0 −→ C0 → 0. (iii) Using Section 3.7, describe the principal symbol of P ∈ DO 0 , namely the highest order term. Exercise 9.5.6 (i) By taking f (q,p) = e−t (p +q ) in (9.5.12), recover Mehler’s kernel of Proposition 5.8.2. (ii) With φn as in (5.8.8), compute π(f ) for f (p,q) = φn (p)φm (q). Let S(W ) be the space of Schwartz class functions on W = V ∗ , which has ? various products defined on it through the results of Section 9.2. Let + V ?+ V is be the alternating tensors on V that have even degree. Observe that commutative, since ω ∧ η = (−1)|η| η ∧ ω = η ∧ ω for tensors of even degree. When V has dimension 2n over C, we have an exact sequence of commutative algebras 2

2

2j n 2j 0 −→ ⊕n−1 j =0 ∧ V −→ ⊕j =0 ∧ V −→ C −→ 0,

(9.5.14)

via the determinant, so S(W ) ⊗ ∧+ V is a nilpotent extension of S(W ). By Proposition 9.5.4, we can realize S as pseudo-differential operators on Hilbert space via π1 . If the DO are trace class, then the trace is expressed as integration over the phase space. There are natural maps on the K-groups associated with the trace      K0 S(W ) ⊗ ∧+ V −→ K0 S(W ) C. (9.5.15) −→ We now return to the situation of Proposition 9.5.4 with phase space W ∼ = R2n , and recall the maps πh¯ : S(W ) → L(H ), which are a deformation family including the map π1 : S(W ) → L(H ) of Proposition 9.5.4, and π0 which corresponds to pointwise multiplication on S. We can attempt to introduce a new multiplication operation on S by using πh¯ , operator composition in L(H ), and the symbol map σ , as in the following array (9.5.16). We can regard Sh¯ as a quantized version of S, where the modified multiplication has the Poisson bracket as the first-order correction term. The trace is taken to be common to all the Sh¯ and given by the integration over phase space as in (9.5.5) and (9.5.11). Let h¯ −n C[[h]] ¯ be the space of formal Laurent series in h¯ that have a pole of order at most n at zero. As in Example 9.4.2 and [7, theorem E.III], Fedosov [35] observed that for a bounded open set  ⊂ R2n and H = L2 (;C), the 2 Dirichlet Laplacian # gives rise to a heat kernel e−h¯ #/2 ∈ L1 (H ), such that

9.5 The Poisson Bracket and Symbols

247

the trace has an asymptotic expansion trace e−h¯

2 #/2

,

∞ 

cj h¯ j

(h¯ → 0+),

j =−n

hence we should therefore interpret the trace as a linear map Sh¯ → h¯ −n C[[h]]. ¯ This set-up is the algebra of quantum observables on R2n .

πh¯ ⊗ πh¯

Sh¯ ⊗ Sh¯ ↓ 0 DO ⊗ DO 0 ↓ L(H ) ⊗ L(H )

∗h¯ −→

−→ −→

Sh¯ ↑ A0 ↑ σ L(H )

(9.5.16)

Claim 9.5.7 Suppose that σ is the symbol map from the pseudo-differential operators of order zero on L2 (W ) to A0 ⊂ S 0 such that: (i) σ ◦ πh¯ (f ) = f for all h¯ > 0 and f ∈ S 0 ; (ii) trace πh¯ (f ) = W f (w)dw for all f ∈ S such that πh¯ (f ) ∈ L1 (H )); (iii) f ∗h¯ g = σ (πh¯ (f )πh¯ (g)) defines an associative multiplication on S, giving a family of algebras (Sh¯ ,∗h¯ ) for h¯ > 0. Then the K-theory of Sh¯ is a homotopy invariant of algebras as h¯ → 0. This outline of a theorem has been realized in certain specific circumstances, including the following. (i) In Section 9.6, we consider the Poisson structure {f1,f2 } and associated commutators [ρ(f1 ),ρ(f2 )] of Toeplitz operators, and recall some results of Helton and Howe. By passing to the case of compact manifolds M, we ensure that the grading on products of the pseudo-differential operators in −1 matches the grading on L(L2 (M)) associated with the Schatten ideal DOM m I = L (L2 (M)) and its powers I,I 2, . . . ,I m , so we can define traces and generalized determinants for the Toeplitz operators. (ii) By Proposition 5.2.2, the heat kernel e−h¯ #/2 over a compact manifold has smoothing properties, which enable us to define Fredholm modules with cyclic cocyles as in (5.6.7). In Section 11.12, we use these to state a more sophisticated type of index theorem. (iii) In Theorem 8.7.1, we showed that a quasi-free algebra A is a subalgebra n + 1 and Proposition 9.6.1 applies to give cocycles on A. Proof Let U be an open connected subset of M. By Proposition 9.5.4, [P1,P2 ] has an asymptotic expansion in which the leading symbol is the multiple of the Poisson bracket −i{p1,p2 }, which has order −1. Hence the range of [P1,P2 ] is contained in {f ∈ L2 (U ) : ∇f ∈ L2 (U )}. The map ρ : f → Pf , where σ (Pf ) = f , satisfies σ ◦ ρ = id. Also, T = ρ(f g) − ρ(f )ρ(g) is a pseudo-differential operator of order −1, hence (T ∗ T )n is pseudo-differential of order −2n, hence Hilbert–Schmidt by Lemma 9.5.2(iii), so T gives a compact operator on H by Lemma 3.2.5(iv). We prove that [ρ(f1 ), . . . ,ρ(f2n+4 )] ∈ L1 (H ) for all fj ∈ C ∞ (U ;C), and refer the reader to [51, 52] for the rest of the proof. Given an even n, let n = 2 , and form the product [ρ(f1 ),ρ(f2 )][ρ(f3 ),ρ(f4 )] . . . [ρ(f2 +1 ),ρ(f2 +2 )],

(9.6.7)

which is a pseudo-differential operator of order −( + 1) < −n/2, hence Hilbert–Schmidt by Lemma 9.5.2(iii), and likewise [ρ(f2 +3 ),ρ(f2 +4 )][ρ(f2 +5 ),ρ(f2 +6 )] . . . [ρ(f2n+3 ),ρ(f2n+4 )] (9.6.8) is a pseudo-differential operator of order −n + 2 + ( + 1) < −n/2, hence Hilbert–Schmidt by Lemma 9.5.2(iii), so [ρ(f1 ), . . . ,ρ(f2n+4 )]

(9.6.9)

250

Cocycles for a Commutative Algebra over a Manifold

is a sum of products of Hilbert–Schmidt operators, hence is trace class. A similar result holds for odd n when we take n = 2 + 1. S0 ⊗ S0 −i{·,·}

π1 ⊗π1 −→

−→

↓ S −1

L(H ) ⊗ L(H ) ↓

−→ π1

Ln+2 (H )

[·,·]

(9.6.10)

⊂ L(H )

Helton and Howe [52] proceed to evaluate the traces introduced in Corollary 9.6.2. The cotangent bundle T ∗ M has fibres linearly homeomorphic to Rn , so we introduce the unit co-sphere bundle S ∗ M with fibres Sx∗ M = {ξ ∈ Tx∗ M : ξ  = 1} and the disc bundle D ∗ M with fibres Dx∗ M = {ξ ∈ Tx∗ M : ξ  ≤ 1}, so D ∗ M is a compact 2n-dimensional manifold; also the boundary of D ∗ M is S ∗ M. Given a continuous function f : Sx∗ M → C, we extend f to f : Dx∗ M → C by solving the Dirichlet problem on Dx∗ M with boundary condition f |Sx∗ M. In particular, let U ⊂ Rn be open and relatively compact. A pseudo-differential operator can be defined by patching together operators defined locally as in (9.5.3). Given a coordinate patch u, a symbol p restricts to σ (P )(x,ξ ) = p(x,ξ ) for x ∈ U and |ξ | = 1, so we extend the symbol to σ (P )(x,ξ ) for ξ  ≤ 1; by patching together such symbols we can introduce σ (P ) on D ∗ (M). Let H = L2 (U,dx). The following result concerns the linear functional on the K¨ahler differentials 2m A →C  f0 df1 . . . df2m → f0 df1 ∧ df2 ∧ · · · ∧ df2n (9.6.11) D ∗ (M)

and gives a trace formula extending the case of Toeplitz operators over the circle, in the style of Corollary 3.7.6. Theorem 9.6.3

There is an extension 0 −→ K(H )

ι σ C ∞ (D ∗ M) −→ 0 P −→ −→

(9.6.12)

and a linear map ρ : C ∞ (D ∗ M;C) → L(H ) such that (i) σ ◦ ρ = id and ρ is a homomorphism modulo K(H ), so that ρ(f ) is a pseudo-differential operator with symbol f ; (ii) [ρ(f1 ), . . . ,ρ(f2n )] ∈ L1 (H ) for all fj ∈ C ∞ (D ∗ M;C),  n! trace[ρ(f1 ), . . . ,ρ(f2n )] = df1 ∧ df2 ∧ · · · ∧ df2n . (9.6.13) (2π i)n D ∗ (M)

9.6 Cocycles Generated by Commutator Products

251

Proof The proof requires a refinement of Proposition 9.6.1. Let A0 be a subalgebra of L(H ), and let A1 be the ideal in A0 that is generated by the commutator subspace [A0,A0 ]. Then we let A2 be ideal in A0 that is generated by A21 and the commutator subspace [A0,A1 ]. Generally we let Aj +1 be the j +1 ideal generated by A1 and the commutator subspace [A0,Aj ]. Thus we produce a decreasing filtration L(H ) ⊃ A0 ⊃ A1 ⊇ A2 ⊇ · · ·

(9.6.14)

as in Exercise 1.6.9. If A1 ⊆ then Am ⊆ and the trace is defined on Am , and we can apply Proposition 9.6.1 and Corollary 9.6.2. We refer the reader to [52] for the details of how this trace in (9.6.13) is computed in terms of the symbols of the pseudo-differential operators. Lm ,

L1 (H )

10 Cyclic Cochains

Suppose that we have an exact sequence of algebra homomorphisms 0 −→ B −→ E

π R −→ 0 −→

(10.0.1)

and a linear map ρ : R → E that splits the sequence, so π ◦ ρ = id, as vector spaces. When can we split the sequence as algebras? Now the deviation of ρ from being a homomorphism is measured by the curvature ω(x,y) = −ρ(xy) + ρ(x)ρ(y). If R is a free algebra R = T (V ), then we can define a canonical homomorphism h : R → E by h(v1, . . . vn ) = ρ(v1 ) . . . ρ(vn ). In this chapter, we make use of this basic idea. This chapter considers the tensor algebra and coalgebra associated with a noncommutative associative algebra. The idea is to introduce universal models for derivations, especially the Hochschild b operator, which is associated with the bar construction. There are some subtleties as to how unital and non-unital algebras are treated. Ultimately, we derive the exact sequences that are needed for the Connes’s bicomplex, which is considered in the next chapter.

10.1 Traces Modulo Powers of an Ideal Let A and R be algebras, and I an ideal in R. Recall that a cyclic 0-cocycle on A is precisely a trace on A. Let ρ : A → R be a k-linear map that is an algebra homomorphism modulo I , so the curvature ω = δρ + ρ 2 has ω ∈ C 2 (A,I ), where ω(a1,a2 ) = ρ(a1 )ρ(a2 ) − ρ(a1 a2 ). Observe that I n is an R-bimodule, so [R,I n ] ⊆ I . Let τ : I n → V be a k-linear map to a vector space V such that p−1 τ |[R,I n ] = 0. For f ∈ C p (A,I n ), we define trτ (f ) = N τ (f ) ∈ Cλ (A,V ). When n = 1 and V = k, we have the situation of Lemma 1.8.3. However, we 252

10.1 Traces Modulo Powers of an Ideal

253

wish to produce interesting cyclic cocycles on A in the case in which I n has interesting traces for some n > 1, but I and R/I have no interesting traces. Proposition 10.1.1 (i) trτ (ωn ) is a cyclic 2n − 1-cocycle on A. (ii) The cyclic cohomology class depends only upon the induced homomorphism ρ˜ : A → R/I . Proof (i) By Bianchi’s identity, we have btrτ (ωn ) = −trτ (δωn ) = −trτ ([ρ,ωn ]) = 0;

(10.1.1)

also trτ (ωn )(a1, . . . ,a2n ) = τ (ω(a1,a2 ) . . . ω(a2n−1,a2n )) + cyclic permutations with sign.

(10.1.2)

(ii) Let ρ be a family of such maps, depending differentiably upon a parameter t ∈ [0,1]. Then writing ˙ for d/dt, we have ω˙ = δ ρ˙ + ρρ ˙ + ρ ρ˙ = δ ρ˙ + [ρ,ρ] ˙ for the superbracket. Hence  d trτ ωn = trτ (ωi ωω ˙ n−i−1 ) dt n

=

i=1 n 

trτ (ωi δ ρω ˙ n−i−1 ) +

i=1

˙ = ntrτ δ ρω

n−1

n 

n−i−1 trτ (ωi [ρ,ρ]ω ˙ )

i=1 n−1

+ ntrτ [ρ,ρ]ω ˙

.

(10.1.3)

Now δω = [ρ,ω], so ˙ n−1 ) = trτ (δ ρω ˙ n−1 ) + btrτ (ρω = trτ (δ ρω ˙ n−1 ) +

n  i=1 n 

trτ (ρω ˙ i−1 (δω)ωn−i−1 ) trτ (ρω ˙ i−1 [ρ,ω]ωn−1−i )

(10.1.4)

i=1

and 0 = trτ [ρ, ρω ˙

n−1

] = trτ ([ρ, ρ]ω ˙

n−1

)+

n−1 

trτ (ρω ˙ i−1 [ρ,ω]ωn−1−i ),

i=1

(10.1.5) so d trτ ωn = nbtrτ (ρω ˙ n−1 ). dt

(10.1.6)

254

Cyclic Cochains

Example 10.1.2 (Cocycles for powers of an ideal) Connes [18, proposition 3, p. 72] considered the case of R = L(H ) and I = Lp/2 (H ), so that I n ⊆ L1 (H ) for all n ∈ N such that n ≥ p/2. Then the usual trace τ on L1 (H ) satisfies τ |[I n,R] = 0. Given ρ : A → R so that ρ is a homomorphism modulo I , let A = {a ∈ A : ρ(a) ∈ I } and introduce A to make the sequence 0 −→ A −→ A −→ A −→ 0

(10.1.7)

exact. Then we have an exact sequence 0 −→ I −→ ρ(A) + I −→ A −→ 0.

(10.1.8)

We can proceed to define   trace ρ(a0 )ω(a1,a2 ) . . . ω(a2n−1,a2n ) .

(10.1.9)

In this discussion we do not assume the existence of traces on I or on R/I . This is an essentially different situation from Lemma 1.8.3, and resembles more closely the results of Section 9.6.

10.2 Coalgebra We introduce the notion of a coalgebra via a similar diagram by which we introduced a unital associative algebra; see [1]. Given a field k, a coalgebra is a triple (C,#,ε) consisting of a vector space C over k, a k-linear map # : C → C × C and a co-unit ε, which is a k-linear map ε : C → k with the following properties. Following a standard shorthand from the theory of Hopf algebras,  we write #(c) = c(1) ⊗ c(2) , for c(1),c(2) ∈ C, with notation alluding to some suitably chosen set of indices, and we require that the following diagrams commute: first to express co-associativity, C⊗C⊗C ↓ C⊗C 

←− C ⊗ C ↑ −→ C

 c(1)(1) ⊗ c(1)(2) ⊗ c(2) = c(1) ⊗ c(2)(1) ⊗ c(2)(2) ↓  c(1) ⊗ c(2)

←− =



c(1) ⊗ c(2) ↑ c (10.2.1)

10.2 Coalgebra

255

then to describe the operation of the co-unit on the leftmost tensor:   C ⊗ C ←− C ⊗ C c(1) ⊗ c(2) ←− c(1) ⊗ c(2) ↓ ↑ ↓ ↑  = c k ⊗ C −→ C ε(c(1) )c(2) Then to describe the operation of the co-unit on the right tensor:   c(1) ⊗ c(2) C ⊗ C ←− C ⊗ C c(1) ⊗ c(2) ←− ↓ ↑ ↓ ↑  = c ε(c(2) )c(1) C ⊗ k −→ C

(10.2.2)

(10.2.3)

Example 10.2.1 (Hopf algebras) (i) Let C = C[N ∪ {0}] be the space of complex-valued and finitely supported functions on the non-negative integers, (n) = 0 for n = j . with basis {cj }∞ j =1 , where cj (n) = 1 for n = j and cj  Then we define the co-multiplication by taking #(cn ) = nj=0 cj ⊗ cn−j and extending linearly; then we define ε by ε(cj ) = 1 for j = 0 and ε(cj ) = 0 for all j = 0, then extending linearly. Another notation for this Hopf algebra has C[x] where ε(x j ) = 1 for j = 0 and ε(x j ) = 0 for all j = 1,2, . . . ; then we write the coproduct as #(x n ) = n j n−j . We also take τ (x n ) = ε for n = 1 and τ (x n ) = 0 for all j =0 x ⊗ x n = 1. (ii) Let C = C[G] be the group algebra, as in Proposition 2.9.19. There is additional structure on C[G] given by a coproduct # : C[G] → C[G] ⊗ C[G]     #: ag g → ag g ⊗ g g∈G

and a co-unit ε : C[G] → C ε:

g∈G

 g∈G

ag g →



ag .

g∈G

(iii) A coalgebra C over k has a co-associative coproduct # : C → C ⊗ C and η : C → k. In particular, let A be a k-vector space with T (A) = ⊗n ⊕∞ n=0 A ; then C = T (A) is a coalgebra with # : C → C ⊗ C given by #(a0, . . . ,an ) =

n  (a0, . . . ,ai ) ⊗ (ai+1, . . . ,an ). i=0

Here we have (a0, . . . ,ai ) ⊗ (ai+1, . . . ,an ) = a0 ⊗ a1 ⊗ · · · ⊗ an,

(10.2.4)

256

Cyclic Cochains

but our notation makes the grouping of terms in the coproduct more clear. We have diagrams C

#

−→

C ⊗2

#⊗id−id⊗#

−→

C ⊗3

(10.2.5)

Let σ be the forward shift operator σ (a1, . . . ,an ) = (an,a1, . . . ,an−1 ); then let  (ak+1, . . . ,an,a0,a1, . . . ,aj ) ⊗ (aj +1, . . . ,ak ) I (a0 ⊗ (a1, . . . ,an )) = 0≤j ≤k≤n

(10.2.6) and define J = I ⊗ # − # ⊗ I + σ (I ⊗ #).

(10.2.7)

We use the mixture of tensor and bracket notation so that the arguments of the operators are more transparent.

10.3 Quotienting by the Commutator Subspace Lemma 10.3.1

The following sequence is exact:

I J C ⊗2 C ⊗3 (10.3.1) −→ −→ Proof We consider the case of a tensor algebra R = T (V ) on a k-vector space V , so 0 −→ A ⊗ C

R ⊗4

b ⊗3 p R R⊗V ⊗R −→ −→

(10.3.2)

is exact for maps b (x,y,z,w) = (xy,z,w) − (x,yz,w) + (x,y,zw)

(10.3.3)

and p(x ⊗ v1, . . . ,vn ⊗ y) =

n  (xv1, . . . ,vi−1 ) ⊗ vi ⊗ (vi+1, . . . ,vn y). i=1

(10.3.4) For any algebra R and R-bimodule M, we have HomR-bimod (Coker(b ),M) = Der(R,M)

(10.3.5)

10.3 Quotienting by the Commutator Subspace

257

and for R = T (V ), we have Der(R,M) = Homk (V ,M) by the universal property of T (V ). Also Homk (V ,M) = HomR-bimod (R ⊗ V ⊗ R,M),

(10.3.6)

with D ∈ Homk (V ,M) producing D(v1, . . . ,vn ) =

n 

v1 . . . vi−1 (Dvi )vi+1 . . . vn

(10.3.7)

i=1

on the subspace V n ⊂ R. Now we quotient each module M by [R,M] to produce M → M⊗R R ⊗4 ↓ R ⊗3

b −→

R ⊗3 ↓ R ⊗2

b −→

p −→ p¯ −→

R⊗V ⊗R ↓ R⊗V

−→

0 (10.3.8)

−→

0

so b(x,y,x) = (xy,z) − (x,yz) + (zx,y) and p(x ¯ ⊗ v1 . . . vn ) =

n 

vi+1 . . . vn xv1 . . . vi−1 ⊗ vi .

(10.3.9)

i=1

The next step uses the diagram R ⊗3 ↓σ R ⊗3

b −→ b˜ −→

R ⊗2 ↓σ

p −→

R⊗V ↓σ

−→ 0

R ⊗2

p˜ −→

R⊗V

−→ 0

(10.3.10)

where b˜ = σ bσ : (x,y,z) → (y,zx) − (xy,z) + (x,yz), p˜ : (v1 . . . vn ⊗ x) →

n 

vi ⊗ vi+1 . . . vn xv1 . . . vi−1,

(10.3.11)

(10.3.12)

i=1

hence the bottom row is exact. To prove the Lemma in the case where dim A < ∞, we let V = A∗ so Cn = A⊗n has dual Cn∗ = V ⊗n . The dual sequence is exact, and we check ˜ Indeed, with μ : R ⊗ R → R the that the duals map to I and J are p˜ and b. multiplication μ(x,y) = xy, we have b˜ = (1 ⊗ μ)σ −1 − μ ⊗ 1 + 1 ⊗ μ

(10.3.13)

J = σ (1 ⊗ #) − # ⊗ 1 + 1 ⊗ #

(10.3.14)

with dual

258

and

Cyclic Cochains

B  C  I a0 ⊗ (a1, . . . ,an ) ,(v0 . . . vp ) ⊗ (vp+1 . . . vn ) B C = a0 ⊗ (a1, . . . ,an ), p((v ˜ 1 . . . vp ) ⊗ (vp+1, . . . ,vn ) .

(10.3.15)

To see this, expand the left-hand side as & %  (ak+1, . . . ,an,a0, . . . ,aj )⊗(aj +1, . . . ak ),(v0 . . . vp )⊗(vp+1 . . . vn ) 0≤j ≤k≤n

(10.3.16) in which the summands contribute zero unless k − j = n − p. Meanwhile, the right-hand side is & % p  vi ⊗ vi+1 . . . vp (vp+1 . . . vn )v0 . . . vi−1 (10.3.17) a0 (a1, . . . ,an ), i=0

=

p 

a0,vi a1,vi+1 . . . ap−i ,vp . . . an,vi−1

(10.3.18)

i=0

=

B

(an−i−1, . . . ,an,a0, . . . ,ap−1 ) ⊗ (ap−i−1, . . . ,an ), C (v0, . . . ,vp ) ⊗ (vp+1, . . . ,vn ) .

(10.3.19)

To obtain the general statement of the Lemma for infinite-dimensional V , we take the inductive limits over finite-dimensional subspaces.

10.4 Bar Construction We carry out constructions with values in a tensor algebra since this makes the definitions simplest. Given the results for the tensor algebra, we subsequently use universal properties to deduce results for other algebras. Let A be a nonunital algebra over k and let A+ = k ⊕ A be the augmented algebra, so A+ is unital. We locate A as the degree one component A[1] of C = T (A), so C = ⊗∞ n=0 Cn with C0 = k and C1 = A[1]. Let #(a0, . . . ,an ) be as above, J = 1 ⊗ # − # ⊗ 1 + σ −1 (1 ⊗ #) (10.4.1)  (−1)n(k+1) (ak+1, . . . ,an,a0,a1, . . . ,aj ) I (a0 ⊗ (a1, . . . ,an )) = 0≤j ≤k≤n

⊗ (aj +1, . . . ,ak ).

(10.4.2)

10.4 Bar Construction

259

Definition 10.4.1 (Bar construction) The bar construction of A+ is the differential graded algebra C = T (A) with b . (We don’t need to quotient out by an ideal since A is non-unital.) Theorem 10.4.2

There is an exact sequence of complexes 0 −→ A[1] ⊗ C

I J C⊗C C ⊗3 −→ −→

(10.4.3)

where A[1] ⊗ C is equipped with the Hochschild differential b. Proof We use the lemma as follows. Let ν : C → A be the projection onto the component A[1]. Then I is injective since (ν ⊗ I )I = id; indeed I (a0 ⊗ (a1, . . . ,an )) =



(−1)n(k+1) (ak+1, . . . ,an,a0, . . . ,aj −1 )

0≤j ≤k≤n

⊗ (aj , . . . ,ak )

(10.4.4)

has I b = (b ⊗ id + id ⊗ b )I .

(10.4.5)

Indeed, (b ⊗ id + id ⊗ b )I (a0 ⊗ (a1, . . . ,an ))  = (−1)n(k+1) b (ak+1, . . . ,an,a0, . . . ,aj −1 ) ⊗ (aj , . . . ,ak ) + (ak+1, . . . ,an,a0, . . . ,aj ) ⊗ b (aj +1, . . . ,ak )

(10.4.6)

and so (ν ⊗ id)(b ⊗ id + id ⊗ b )(a0 ⊗ (a1, . . . ,an ))  (−1)n(k+1) νb (ak+1, . . . ,an,a0, . . . ,aj ) ⊗ (aj +1, . . . ,ak ) = 0≤j ≤k≤n

+ (−1)(j +1)nk ν(ak+1, . . . ,an,a0, . . . ,aj ) ⊗ b (aj +1, . . . ,ak ). (10.4.7) To give a non-zero contribution, ν should operate on a tensor in A[1]. The final term contributes only if j = 0 and k = n; the first term in the sum contributes when k = n − 1 and j = 0; or k = n and j = 1; so

260

Cyclic Cochains (ν ⊗ id)(b ⊗ id + id ⊗ b )I (a0 ⊗ (a1, . . . ,an )) = −a0 ⊗ b (a1, . . . ,an ) + (−1)n b (an,a0 ) ⊗ (a1, . . . ,an−1 ) + b (a0 ) ⊗ (a1, . . . ,an ) = −a0 ⊗

n  (−1)i−1 (a1, . . . ,ai ai+1, . . . ,an )

i=1 n + (−1) an a0

⊗ (a1, . . . ,an−1 ) + a0 a1 ⊗ (a2, . . . ,an )

= b(a0 ⊗ (a1, . . . ,an )).

(10.4.8)

This suffices to prove (10.4.3) since A ⊗ C has no terms in degree zero. Definition 10.4.3 (Hochschild complex) (A ⊗ C,b) is called a Hochschild complex. When A is unital, H∗ (A ⊗ C;b) = H∗ (A,A). Also # k, for n = 0; (10.4.9) Hn (C) = 0, for n = 0. The tensor algebra has a co-unit η : C → k, so η picks out the component in degree zero. A⊗C

I id ⊗ η C⊗C C ⊗ k = C. −→ −→

(10.4.10)

Recall the cyclic permutation with sign λ and the symmetrizing operator N = n i ⊗n+1 . Then on A⊗n+1 in A[1] ⊗ C, i=0 λ on A N(a0, . . . ,an ) =

n 

(−1)j (n+1) (aj +1, . . . ,an,a0, . . . ,aj )

j =0

= (1 ⊗ η)I (a0, . . . ,an ).

(10.4.11)

¯ To avoid confusion, we write this as N = ∂¯ and conclude that b ∂¯ = ∂b. The map # − σ # : C → C ⊗ C is dual to R ⊗ R → R : (x,y) → [x,y]. Also J (# − σ #) = 0 since b2 = 0. With u = (ν ⊗ id)(# − σ #) has u : C → A[1] ⊗ C : u(a1, . . . ,an ) = (a1, . . . ,an ) − (−1)n−1 (an,a1, . . . ,an−1 ) so u = 1 − λ on A⊗n , hence b(1 − λ) = (1 − λ)b . We have a commuting diagram: C u↓ A[1] ⊗ C

/ #−σ # →

(10.4.12) C⊗C

10.5 Cochains with Values in an Algebra

261

Thus we have the chain version of Connes’s bicomplex: A⊗3

N ←−

A⊗2

N ←−

A

N ←−

A⊗3 ↓b A⊗2 ↓b A

Definition 10.4.4 (Cyclic complex) Im{∂¯ : A ⊗ C → C}. From the exact sequence 0 ←− CC(A) ←− A ⊗ C

1−λ

←− 1−λ ←− 1−λ ←−

A⊗3 ↓ b A⊗2 ↓ b A

(10.4.13)

The cyclic complex of A is CC(A) =

1−λ C ←− CC(A) ←− 0 ←−

(10.4.14)

we have a long exact sequence H Cn (A)

I B S B Hn+1 (A,A) H Cn+1 (A) H Cn−1 (A) .... −→ −→ −→ −→ (10.4.15)

In Lemma 2.1.2, we asserted that the connecting map exists via a diagram chase that involves selecting representatives for homology classes. In Section 11.2, we will define the connecting map S by specifying how these representatives can found explicitly.

10.5 Cochains with Values in an Algebra So far we have carried out computations with cochains that take values in the tensor algebra T (A). To bring the discussion down to earth, we wish to follow through similar calculations for cochains with values in an algebra such as L(H ). Let A and its tensor algebra C = T (A) be as above, with A[1] in degree one, and let L be an algebra over k with multiplication μ : L × L → L. Let C(A,L) = Homk (C,L), which is differential graded algebra with the product defined by f g = μ(f ⊗ g)#

C

# f ⊗g μ C⊗C L⊗L L. −→ −→ −→

(10.5.1)

262

Cyclic Cochains

More explicitly, we have for f ∈ C p (A,L) and g ∈ C q (A,L), the signed cup product f g(a1, . . . ,ap+q ) = (−1)pq f (a1, . . . ,ap )g(ap+1, . . . ,ap+q ).

(10.5.2)

The differential is δf = −(−1)deg(f ) f ◦ b so δf (a1, . . . ,ap+1 ) = (−1)p+1 f (b (a1, . . . ,ap+1 )) = (−1)p+1

p−1 

f (a1, . . . ,ai+1 ai+2, . . . ,ap+1 ). (10.5.3)

i=0

We define a pairing on the bar cochains with values in the Hochschild cochains C((A ⊗ [1] ⊗ B,k) → C(A) : (f ,h) → τ ((∂f )h).

(10.5.4)

Let M be an L-bimodule with μ : L × M → M the left multiplication; also let τ : M → k be a trace, namely a linear functional such that τ |[L,M] = 0. More generally, one can consider k-linear maps from M into a vector space. A Hochschild cochain is an element of Homk (A ⊗ C,k) of the form τ ((δf )h) as defined for f ∈ C(A,L) and h ∈ C(A,M) by A⊗C

I f ⊗h μ τ C⊗C L⊗M M k. −→ −→ −→ −→

Proposition 10.5.1

(10.5.5)

Homk (A ⊗ L,k) is a complex with differential δ where δϕ = −(−1)deg(ϕ) ϕ ◦ b.

The differential satisfies (i) δτ ((∂f )h) = τ ((∂(δf ))h) + (−1)deg(f ) τ ((∂f )δh); (ii) for f ,g ∈ C(A,L) and h ∈ C(A,M), τ (∂(f g)h) = τ ((∂f )gh) + (−1)deg(f )(deg(g)+deg(h)) τ ((∂g)hf ).

(10.5.6)

10.5 Cochains with Values in an Algebra

263

Proof (i) When f ∈ C p (A,L) and h ∈ C q (A,M) with n = p + q, the right-hand side is   τ ∂(δf )h)(a0 ⊗ (a1, . . . ,ap+q ) + (−1)p+1 τ (∂f δh)(a0 ⊗ (a1, . . . ,ap+q )     = τ μ(δf ⊗ h) (ak+1, . . . ,an a0, . . . ,aj ) ⊗ (aj +1, . . . ,ak ) 0≤j ≤k≤n

 + (−1)

p+1

=τ



τ μ∂f δh



 (ak+1, . . . ,an,a0, . . . ,aj )⊗(aj +1, . . . ,ak )

0≤j ≤k≤n 

f (b (aq+j +1, . . . ,aj )h(aj +1, . . . ,aj +q ))

j

+ (−1)p+1 τ



f (aj +q+1, . . . ,ap+q ,a0, . . . ,aj −1 )h(b (aj , . . . ,aj +q ))

j

(10.5.7) and the left-hand side is (−1)deg(ϕ)+1 ϕ ◦ b(a0, . . . ,ap+q ) = (−1)p+q τ μ(f ⊗ h)I (b(a0, . . . ,ap+q )) = (−1)p+q τ μ(f ⊗ h)(b ⊗ id + id ⊗ b )I (a0, . . . ,ap+q )  f (b (aj +q+1, . . . ,ap+q ,a0, . . . ,aj )h(aj +1, . . . ,aj +q ) = (−1)p+q τ j

+ (−1)

p+q

τ



f (aj +q+1, . . . ,ap+q ,a0, . . . ,aj +1 )h(b (aj , . . . ,aj +q )).

j

(10.5.8) (ii) Basically, this follows from J I = 0: A⊗C

I −→ #⊗id

C⊗C ↓ C⊗C⊗C

f .g⊗h −→

−→

L⊗M ↑ L⊗L⊗M

τμ −→ μ⊗id

k (10.5.9)

The cochains in (ii) are τ (∂(f g)h) = τ μ(μ ⊗ id)(f ⊗ g ⊗ h)(# ⊗ id)I, τ (∂(f )gh) = τ μ(id ⊗ μ)(f ⊗ g ⊗ h)(# ⊗ id)I, τ ((∂g)hf ) = τ μ(μ ⊗ id)σ −1 (f ⊗ g ⊗ h)σ (# ⊗ id)I ;

(10.5.10)

the hypotheses that τ |[L,M] = 0 implies that the maps τ μ(μ ⊗ id),

τ μ(id ⊗ μ),

τ μ(1 ⊗ μ)σ −1

(10.5.11)

264

Cyclic Cochains

are the same. For instance, τ μ(id ⊗ μ)σ −1 (x,y,z) = τ μ(y,mx) = τ (ymx) = τ (xym) = τ μ(id ⊗ μ)(x,y,m).

(10.5.12)

So we take the alternating sum and use the identity J I = 0.

10.6 Analogue of 1 R for the Bar Construction For a unital algebra A, we work with A⊗ A¯ ⊗n ; whereas for a non-unital algebra A, we consider A[1] ⊗ A⊗n . The bar construction of the augmented algebra A+ = k ⊕ A is the ⊗n with differential d = b and differential graded coalgebra B = ⊕∞ n=0 A comultiplication #(a1, . . . ,an ) =

n 

(a1, . . . ,ai ) ⊗ (ai+1, . . . ,an ).

(10.6.1)

i=1

With C n (A,L) = Hom(Bn,L), we have C(A,L) = ⊕C n (A,L) and inside C(A,L), we can do curvature calculations. With ρ ∈ C 1 (A,L) we can build ω = δρ + ρ 2 so that δω = −ad(ρ)(ωn ). If τ is a trace on L, then trτ is a trace on C(A,L), and trτ (γ ) = N τ γ ∈ Cλ (A) is a cyclic cochain such that δtrτ (ωn ) = 0. In Corollary 11.2.2 we achieve our goal by defining Connes’s S operator such that



S trτ

ωn n!



= trτ



ωn+1 (n + 1)!

 ,

(10.6.2)

where the S operator has degree 2 and is a periodicity map, as in Section 7.7. Proposition 10.6.1 Let R = C(A,L). Then there exists an R-bimodule 1 R such that f ∂(gh)k = f (∂g)hk + f g(∂h)k.

(10.6.3)

Proof The first step is to take R = C(A,L) and produce 1 R. The differentials relate to bar cochains R ⊗4

b ⊗3 b ⊗2 b R R −→ −→ −→

(10.6.4)

10.6 Analogue of 1 R for the Bar Construction

265

With B the tensor algebra as above, B is co-unital, so we have an exact sequence B

# −→

B ⊗2 P ↓ B⊗A[1] ⊗ B

#⊗id−id⊗ −→ I

B

#⊗id⊗id−id⊗#⊗id+id⊗id⊗# −→

B ⊗4

(10.6.5) In this diagram, we take I to distribute two tensor factors into a sum of three tensor factors  I ((a1, . . . ,ap ) ⊗ (ap+1, . . . ,an )) = (a1, . . . ,aj ) ⊗ (aj +1, . . . ,ak ) 0≤j ≤p