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Noncommutative Geometry Alain Connes

Noncommutative GeometryAlain ConnesContentsPreface6 Introduction71. measure theory (Chapters I and V)82. Topology andK- theory (Chapter II)143. Cyclic cohomology (Chapter III)194. The quantized calculus (Chapter IV)255. The metric aspect of Noncommutative geometry34 Chapter 1. Noncommutative Spaces and measure Theory391. Heisenberg and the Noncommutative Algebra of Physical Quantities402. Statistical State of a Macroscopic System and Quantum StatisticalMechanics453. Modular theory and the Classification of Factors484.

1. MEASURE THEORY (CHAPTERS I AND V) 8 Let us now discuss in more detail the extension of the classical tools of analysis to the noncommutative case.

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Transcription of Noncommutative Geometry Alain Connes

1 Noncommutative GeometryAlain ConnesContentsPreface6 Introduction71. measure theory (Chapters I and V)82. Topology andK- theory (Chapter II)143. Cyclic cohomology (Chapter III)194. The quantized calculus (Chapter IV)255. The metric aspect of Noncommutative geometry34 Chapter 1. Noncommutative Spaces and measure Theory391. Heisenberg and the Noncommutative Algebra of Physical Quantities402. Statistical State of a Macroscopic System and Quantum StatisticalMechanics453. Modular theory and the Classification of Factors484.

2 Geometric Examples of von Neumann Algebras : measure theory ofNoncommutative Spaces515. The Index Theorem for Measured Foliations64 Appendix A : Transverse Measures and Averaging Sequences77 Appendix B : Abstract Transverse measure Theory78 Appendix C : Noncommutative Spaces and Set Theory79 Chapter 2. Topology -algebras and theirK-theory862. Elementary Examples of Quotient Spaces903. The SpaceXof Penrose Tilings944. Duals of Discrete Groups and the Novikov Conjecture995. The Tangent Groupoid of a Manifold1046. Wrong-way Functoriality inK- theory as a Deformation1117.

3 The Orbit Space of a Group Action1168. The Leaf Space of a Foliation1239. The Longitudinal Index Theorem for Foliations13410. The Analytic Assembly Map and Lie Groups141 Appendix A :C -modules and Strong Morita Equivalence156 Appendix B :E- theory and Deformations of Algebras163 Appendix C : Crossed Products ofC -algebras and the Thom Isomorphism1753 CONTENTS4 Appendix D : Penrose Tilings179 Chapter 3. Cyclic Cohomology and Differential Geometry1831. Cyclic Cohomology1872. Examples2123. Pairing of Cyclic Cohomology withK-Theory2294.

4 The Higher Index Theorem for Covering Spaces2385. The Novikov Conjecture for Hyperbolic Groups2446. Factors of Type III, Cyclic Cohomology and the Godbillon-Vey Invariant2507. The Transverse Fundamental Class for Foliations and Geometric Corollaries269 Appendix A. The Cyclic Category 280 Appendix B. Locally Convex Algebras290 Appendix C. Stability under Holomorphic Functional Calculus291 Chapter 4. Quantized Calculus2931. Quantized Differential Calculus and Cyclic Cohomology2982. The Dixmier Trace and the Hochschild Class of the Character3053.

5 Quantized Calculus in One Variable and Fractal Sets3204. Conformal Manifolds3395. Fredholm Modules and Rank-One Discrete Groups3486. Elliptic theory on the Noncommutative TorusT2 and the Quantum HallEffect3567. Entire Cyclic Cohomology3758. The Chern Character of -summable Fredholm Modules3999. -summableK-cycles, Discrete Groups and Quantum Field Theory417 Appendix A. Kasparov s Bivariant Theory439 Appendix B. Real and Complex Interpolation of Banach Spaces447 Appendix C. Normed Ideals of Compact Operators450 Appendix D.

6 The Chern Character of Deformations of Algebras454 Chapter 5. Operator algebras4581. The Papers of Murray and von Neumann4592. Representations ofC -algebras4693. The Algebraic Framework for Noncommutative Integration and the Theoryof Weights4724. The Factors of Powers, Araki and Woods, and of Krieger4755. The Radon-Nikod ym Theorem and Factors of Type III 4806. Noncommutative Ergodic Theory4877. Amenable von Neumann Algebras5008. The Flow of Weights: Mod(M)5059. The Classification of Amenable Factors51210. Subfactors of Type II1 Factors51711.

7 Hecke Algebras, Type III Factors and Statistical theory of Prime Numbers523 CONTENTS5 Appendix A. Crossed Products of von Neumann Algebras537 Appendix B. Correspondences539 Chapter 6. The metric aspect of Noncommutative geometry5521. Riemannian Manifolds and the Dirac Operator5552. Positivity in Hochschild Cohomology and Inequalities for the Yang MillsAction5693. Product of the Continuum by the Discrete and the Symmetry BreakingMechanism5744. The Notion of Manifold in Noncommutative Geometry5985. The StandardU(1) SU(2) SU(3) Model609 Bibliography627 PrefaceThis book is the English version of the French G eom etrie non commutative pub-lished by InterEditions Paris (1990).

8 After the initial translation by Berberian,a considerable amount of rewriting was done and many additions made, multiplyingby the size of the original manuscript. In particular the present text containsseveral unpublished results. My thanks go first of all to C ecile whose patience andcare for the manuscript have been essential to its completion. This second version ofthe book greatly benefited from the important modifications suggested by M. Rieffel,D. Sullivan, Loday, J. Lott, J. Bellissard, P. B. Cohen, R. Coquereaux, J.

9 Dixmier,M. Karoubi, P. Kr ee, H. Bacry, P. de la Harpe, A. Hof, G. Kasparov, J. Cuntz, D. Tes-tard, D. Kastler, T. Loring, J. Pradines, V. Nistor, R. Plymen, R. Brown, C. Kassel,and M. Gerstenhaber. Patrick Ion and Arthur Greenspoon played a decisive r ole inthe finalisation of the book, clearing up many mathematical imprecisions and consid-erably smoothing the initial manuscript. I wish to express my deep gratitude for theirgenerous help and their insight. Finally, my thanks go to Marie Claude for her help increating the picture on the cover of the book, to Gilles who took the photograph, andto Bonnie Ion and Fran coise for their help with the bibliography.

10 Many thanks go alsoto Peter Renz who orchestrated the whole Connes30 June 1994 Paris6 IntroductionThe correspondence between geometric spaces and commutative algebras is a familiarand basic idea of algebraic Geometry . The purpose of this book is to extend thiscorrespondence to the Noncommutative case in the framework of real analysis. Thetheory, called Noncommutative Geometry , rests on two essential points:1. The existence of many natural spaces for which the classical set-theoretic toolsof analysis, such as measure theory , topology, calculus, and metric ideas lose theirpertinence, but which correspond very naturally to a Noncommutative algebra.


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