Noncommutative

Geometry

Alain Connes College de France Institut des Hautes Etudes Scientifiques Paris, France

ACADEMIC PRESS, INC. Harcourt Brace & Company, Publishers San Diego New York Boston London Sydney Tokyo Toronto

TABLE

OF

CONTENTS

PREFACE xiii

INTRODUCTION 1

NONCOMMUTATIVE SPACES AND MEASURE THEORY 33

1. Heisenberg and the Noncommutative Algebra of Physical Quantities Associated to a Microscopic System 33

2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics 39

3. Modular Theory and the Classification of Factors 43 4. Geometrie Examples of von Neumann Algebras: Measure

Theory of Noncommutative Spaces 45 a. Classical Lebesgue measure theory 46 ß. Foliations 49 y. The von Neumann algebra of a foliation 50

5. The Index Theorem for Measured Foliations 59 a. Transverse measures for foliations 60 ß. The Ruelle-Sullivan cycle and the Euler number of a

measured foliation 64 y. The index theorem for measured foliations 69

A. Appendix: Transverse Measures and Averaging Sequences 72 B. Appendix: Abstract Transverse Measure Theory 72 C. Appendix: Noncommutative Spaces and Set Theory 74

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II. TOPOLOGY AND ^-THEORY 79 1. C*-algebras and their iC-theory 80 2. Elementary Examples of Quotient Spaces 85

«. Open Covers of manifolds 86 ß. The dual of the infinite dihedral group T = Zx 2/2 87

3. The Space X of Penrose Tilings 88 4. Duals of Discrete Croups and the Novikov Conjecture 94 5. The Tangent Groupoid of a Manifold 99 6. Wrong-way Functoriality in .K-theory as a Deformation 107

a. The index groupoid of a linear map 107

ß. Construction of f\ e E(T*M e f*TN,N) 108 y. K-orientations of vector bundles and maps 109 8. Wrong-way functoriality for if-oriented maps 110

7. The Orbit Space of a Group Action 111 8. The Leaf Space of a Foliation 118

a. Construction of C*(V,F) 118 ß. Closed transversals and idempotents of C* (V, F) 120 y. The analytic assembly map \x : K^T(BG) - K(C*(V,F)) 125

9. The Longitudinal Index Theorem for Foliations 129 «. Construction of Ind(D) e K0(J) 130 ß. Significance of the C*-algebra index 132 y. The longitudinal index theorem 133

10. The Analytic Assembly Map and Lie Groups 136 a. Geometrie cycles for smooth groupoids 136 ß. Lie groups and deformations 140 y. The G-equivariant index of elliptic Operators on

homogeneous Spaces of Lie groups 142 ö. The iC-theory K(C*(G)) for Lie groups 148 s. The general conjecture for smooth groupoids 151

A. Appendix: C*-modules and Strong Morita Equivalence 152 B. Appendix: £-theory and Deformations of Algebras 158

a. Deformations of C*-algebras and asymptotic morphisms 159

ß. Composition of asymptotic morphisms 161 y. Asymptotic morphisms and exaet sequences of

C*-algebras 163 S. The cone of a map and half-exactness 165 E. £-theory 168

C. Appendix: Crossed Products of C* -algebras and the Thom Isomorphism 171

D. Appendix: Penrose Tilings 175

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III. CYCUC COHOMOLOGY AND DIFFERENTIAL GEOMETRY 179 1. Cyclic Cohomology 182

a. Characters of cycles and the cup product in HC* 183 ß. Cobordisms of cycles and the Operator B ..194 y. The exact couple relating HC* {JA) to Hochschild

cohomology 199 2. Examples 207

a. JA = C00 (V), V a compact smooth manifold 207 ß. The cyclic cohomology of the noncommutative torus

JA = JAg, 0 G R/1 212 y. The cyclic cohomology of the group ring Cr for T a discrete

group 213 5. Cyclic cohomology of Q00 (V x T) 216

3. Pairing of Cyclic Cohomology with X>Theory 223 4. The Higher Index Theorem for Covering Spaces 233

a. The smooth groupoid of a covering space 233 ß. The group ring 3ir 234 y. The index theorem 237

5. The Novikov Conjecture for Hyperbolic Groups 238 a. Word hyperbolic groups 238 ß. The Haagerup inequality 241 y. Extension to C* {T) of K-theory invariants 242

6. Factors of Type III, Cyclic Cohomology and the Godbillon-Vey Invariant 244

a. Extension of densely defined cyclic cocycles on Banach algebras 246

ß. The Bott-Thurston cocycle and the equality GV = i5£[V/F] 257

y. Invariant measures on the flow of weights 259 7. The Transverse Fundamental Class for Foliations and

Geometrie Corollaries 263 a. The transverse fundamental class 263 ß. Geometrie corollaries 268 y. Index formula for longitudinal elliptic Operators 271

A. Appendix: The Cyclic Category A 274 a. The simplicial category A 275 ß. The cyclic category A 276 y. The A-module JA^ associated to an algebra JA 280 ö. Cyclic Spaces and S1 Spaces 282

B. Appendix: Locally Convex Algebras 283 C. Appendix: Stability under Holomorphic Functional

Calculus 285

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IV. QUANTIZED CALCULUS 287 1. Quantized Differential Calculus and Cyclic Cohomology 292

a. The cycle associated to a Fredholm module 292 ß. The periodicity Operator S and the Chern character 294 y. Pairing with K-theory and index formula 296

2. The Dixmier Trace and the Hochschild Class of the Character 299

a. General properties of interpolation ideals £(P^ 299 ß. The Dixmier trace 303 y. The residue formula for the Hochschild class of the

character of Fredholm modules 308 <5. Growth of algebras and degree of summability of

iC-cycles 310 3. Quantized Calculus in One Variable and Fractal Sets 313

<x. Quantized calculus in one variable 314 ß. The class of df in £P-x/£%'°° 317 y. The Dixmier trace of f(Z)\dZ\p 321 6. The harmonic measure and non-normality of the Dixmier

trace 326 s. Cantor sets, Dixmier trace and Minkowski measure 327

4. Conformal Manifolds 331 a. Quantized calculus on conformal manifolds 331 ß. Perturbation of Fredholm modules by the commutant von

Neumann algebra 335 y. The 4-dimensional analogue of the Polyakov action 338

5. Fredholm Modules and Rank-One Discrete Groups 340 6. EUiptic Theory on the Noncommutative Torus l2

e and the Quantum Hall Effect 347

a. EUiptic theory on T | 348 ß. The quantum Hall effect 355 y. The work of J. Bellissard on the integrality of aH 357

7. Entire Cyclic Cohomology 366 a. Entire cyclic cohomology of Banach algebras 367 ß. Infinite-dimensional cycles 371 y. Traces on QA and TA 3 74 <5. Pairing with K0(A) 378 e. Entire cyclic cohomology of S1 381

8. The Chern Character of 0-summable Fredholm Modules 390 a. Fredholm modules and K-cycles 391 ß. The supergroup E1,1 and the convolution algebra £ of

operator-valued distributions on [0, +oo[ 395 y. The Chern character of .K-cycles 399 5. The index formula 403 f. The JLO cocycle 405

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9. 0-summable K-cycles, Discrete Croups and Quantum Field Theory 407

«. Discrete subgroups of Lie groups 407 ß. Supersymmetric quantum field theory 415

A. Appendix: Kasparov's Bivariant Theory 428 B. Appendix: Real and Complex Interpolation of Banach

Spaces 436 C. Appendix: Normed Ideals of Compact Operators 439 D. Appendix: The Chern Character of Deformations of

Algebras 443

V. OPERATOR ALGEBRAS 447 1. The Papers of Murray and von Neumann 448

ex. Examples of von Neumann algebras 449 ß. Reduction theory 452 y. Comparison of subrepresentations, comparison of

projections and the relative dimension function 453 5. Algebraic isomorphism and spatial isomorphism 455 E. The first two examples of type II j factors, the hyperfinite

factor and the property T 456 2. Representations of C*-algebras 457 3. The Algebraic Framework for Noncommutative Integration

and the Theory of Weights 460 4. The Factors of Powers, Araki and Woods, and of Krieger 463 5. The Radon-Nikodym Theorem and Factors of Type IIIA 469

a. The Radon-Nikodym theorem 469 ß. The factors of type IIIA 472

6. Noncommutative Ergodic Theory 475 a. Rokhlin's theorem 476 ß. Entropy 477 y. Approximately inner automorphisms 482 ö. Centrally trivial automorphisms 482 e. The obstruction y(0) 484 £,. The list of automorphisms of R up to outer conjugacy 485 q. Automorphisms of the Araki-Woods factor R01 of type

nM ' 487 7. Amenable von Neumann Algebras 488

a. Approximation by finite-dimensional algebras 488 ß. The properties P of Schwartz, E of Hakeda and Tomiyama,

and injectivity 489 y. Semidiscrete von Neumann algebras 490

8. The Flow of Weights: mod(M) 493 a. The discrete decomposition of factors of type III0 493 ß. Continuous decomposition of type III factors 494 y. Functorial definition of the flow of weights 495

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8. Virtual groups and the flow of weights as modular spectrum 497

9. The Classification of Amenable Factors 499 a. Factors of type IIj 499 ß. Factors of type II«, 501 y. Factors of type IIIA, A e ]0,1[ 502 ö. Factors of type III0 503 E. Factors of type 11^ 504

10. Subfactors of Type IIX Factors 505 a. Index of subfactors 505 ß. Positive Markov traces on Hecke algebras 508

11. Hecke Algebras, Type III Factors and Statistical Theory of Prime Numbers 510

a. Description of the system and its phase transition 510 ß. Bosonic second quantization and prime numbers as a

subset of R 515 y. Products of trees and the noncommutative Hecke

algebra 518 A. Appendix: Crossed Products of von Neumann Algebras 524 B. Appendix: Correspondences 526

a. Half densities and the identity correspondence 527 ß. Correspondences and *-homomorphisms 529 y. Coefficients of correspondences and completely positive

maps 531 ö. Composition of correspondences 533 e. Correspondences, hyperfiniteness and property T 536

VI. THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY . . . 539 1. Riemannian Manifolds and the Dirac Operator 543 2. Positivity in Hochschild Cohomology and Inequalities for the

Yang-Mills Action 556 3. Product of the Continuum by the Discrete and the Symmetry

Breaking Mechanism 561 4. The Notion of Manifold in Noncommutative Geometry 585

a. The classical notion of manifold 585 ß. Bivariant K-theory and Poincare duality 587 y. Poincare duality and cyclic cohomology 591 6. Bivector potentials on an (J4,2)-bimodule (Sj, D, y) 594

5. The Standard 1/(1) x 51/(2) x 51/(3) Model 595 a. The dictionary 596 ß. The Standard model 598 y. Geometrie strueture of the finite space F 601 ö. Geometrie strueture of the Standard model 604 e. Unimodularity condition and hypercharges 609

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BIBUOGRAPHY 613

NOTATION AND CONVENTIONS 645

INDEX 649