Noncommutative Geometry

Transcription

1 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

2 TABLE OF CONTENTS PREFACE xiii INTRODUCTION 1 NONCOMMUTATIVE SPACES AND MEASURE THEORY Heisenberg and the Noncommutative Algebra of Physical Quantities Associated to a Microscopic System Statistical State of a Macroscopic System and Quantum Statistical Mechanics Modular Theory and the Classification of Factors 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 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 v

3 vi II. TOPOLOGY AND ^-THEORY C*-algebras and their ic-theory Elementary Examples of Quotient Spaces 85 «. Open Covers of manifolds 86 ß. The dual of the infinite dihedral group T = Zx 2/ The Space X of Penrose Tilings Duals of Discrete Croups and the Novikov Conjecture The Tangent Groupoid of a Manifold 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 Wrong-way functoriality for if-oriented maps The Orbit Space of a Group Action 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)) The Longitudinal Index Theorem for Foliations 129 «. Construction of Ind(D) e K 0 (J) 130 ß. Significance of the C*-algebra index 132 y. The longitudinal index theorem 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

4 vü III. CYCUC COHOMOLOGY AND DIFFERENTIAL GEOMETRY 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 Examples 207 a. JA = C 00 (V), V a compact smooth manifold 207 ß. The cyclic cohomology of the noncommutative torus JA = JA g, 0 G R/1 212 y. The cyclic cohomology of the group ring Cr for T a discrete group Cyclic cohomology of Q 00 (V x T) Pairing of Cyclic Cohomology with X>Theory 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 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 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 = i 5 [V/F] 257 y. Invariant measures on the flow of weights 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 S 1 Spaces 282 B. Appendix: Locally Convex Algebras 283 C. Appendix: Stability under Holomorphic Functional Calculus 285

5 viii IV. QUANTIZED CALCULUS 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 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 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 The harmonic measure and non-normality of the Dixmier trace 326 s. Cantor sets, Dixmier trace and Minkowski measure 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 Fredholm Modules and Rank-One Discrete Groups EUiptic Theory on the Noncommutative Torus l 2 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 a H 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 K 0 (A) 378 e. Entire cyclic cohomology of S The Chern Character of 0-summable Fredholm Modules 390 a. Fredholm modules and K-cycles 391 ß. The supergroup E 1,1 and the convolution algebra of operator-valued distributions on [0, +oo[ 395 y. The Chern character of.k-cycles The index formula 403 f. The JLO cocycle 405

6 ix 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 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 Algebraic isomorphism and spatial isomorphism 455 E. The first two examples of type II j factors, the hyperfinite factor and the property T Representations of C*-algebras The Algebraic Framework for Noncommutative Integration and the Theory of Weights The Factors of Powers, Araki and Woods, and of Krieger The Radon-Nikodym Theorem and Factors of Type III A 469 a. The Radon-Nikodym theorem 469 ß. The factors of type III A 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 R 01 of type n M ' 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 The Flow of Weights: mod(m) 493 a. The discrete decomposition of factors of type III ß. Continuous decomposition of type III factors 494 y. Functorial definition of the flow of weights 495

7 X 8. Virtual groups and the flow of weights as modular spectrum The Classification of Amenable Factors 499 a. Factors of type IIj 499 ß. Factors of type II«, 501 y. Factors of type III A, A e ]0,1[ 502 ö. Factors of type III E. Factors of type 11^ Subfactors of Type II X Factors 505 a. Index of subfactors 505 ß. Positive Markov traces on Hecke algebras 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 Riemannian Manifolds and the Dirac Operator Positivity in Hochschild Cohomology and Inequalities for the Yang-Mills Action Product of the Continuum by the Discrete and the Symmetry Breaking Mechanism 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 Bivector potentials on an (J4,2)-bimodule (Sj, D, y) 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

8 xi BIBUOGRAPHY 613 NOTATION AND CONVENTIONS 645 INDEX 649