非交換幾何

前言

　　This book is the English version of the French “Geometrie non commutative” published by InterEditions Paris （1990）. After the excellent initial translation by S.K. Berberian， a considerable amount of rewriting was done and many additions made， multiplying by 3.8 the size of the original manuscript. In particular the present text contains several unpublished results.　　My thanks go first of all to Cecile whose patience and care for the manuscript have been essential to its completion. This second version of the book greatly benefited from the modifications suggested by many people： foremost was Marc Rieffel， but important contributions were made by D. Sullivan， J.-L. Loday，J. Lott， J. Bellissard， P. B. Cohen， R. Coquereaux， J. DLxmier， M. Karoubi， P. Kree，H. Bacry， P. de la Harpe， A. Hof， G. Kasparov， J. Cuntz， D. Testard， D. Kastler，T. Loring， J. Pradines， V. Nistor， R. PIymen， R. Brown， C. Kassel， and M. Gerstenhaber， with several of whom I have shared the pleasure of collaboration.　　Patrick Ion and Arthur Greenspoon played a decisive r61e in the finalisation of the book， clearing up many mathematical imprecisions and considerably smoothing the initial manuscript. I wish to express my deep gratitude for their generous help and their insight.　　Finally， my thanks go to Marie Claude for her help in creating the picture on the cover of the book， to Gilles who took the photograph， and to Bonnie Ion and Francoise for their help with the bibliography. Many thanks go also to Peter Renz who orchestrated the whole thing.

內(nèi)容概要

his book is the English version of the French “Geometrie non commutative” published by InterEditions Paris （1990）. After the excellent initial translation by S.K. Berberian, a considerable amount of rewriting was done and many additions made, multiplying by 3.8 the size of the original manuscript. In particular the present text contains several unpublished results.

書籍目錄

TABIF OF CONTENTSPREFACEINTRODUCTIONI NONCOMMUTATIVE SPACES AND MEASURE THEORY1. Heisenberg and the Noncommutative Algebra of Physical Quantities Associated to a Microscopic System2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics3. Modular Theory and the Classification of Factors4. Geometric Examples of yon Neumann Algebras: Measure Theory of Noncommutative Spacesα Classical Lebesgue measure theoryβ. Foliationsγ. The von Neumann algebra of a foliation5. The Index Theorem for Measured Foliationsα. Transverse measures for follationsβ. The Ruelle-Sullivan cycle and the ELder number of a measured foliationγ. The Index theorem for measured foliationsA. Appendix: Transverse Measures and Averaging SequencesB. AppendL, c Abstract Transverse Measure TheoryC. Appendix: Noncommutative Spaces and Set TheoryII. TOPOLOGY AND K-THEORY..1. C*-algebras and their K-theory2. Elementary Examples of Quotient Spacesα. Open covers of manifoldsβ. The dual of the infinite dihedral group F = Z >~ Z/23. The Space X of Penrose Tilings4. Duals of Discrete Groups and the Novikov Conjecture5. The Tangent Groupoid of a Manifold6. Wrong-way Functoriality in K-theory as a Deformationα. The index groupoid of a linear mapβ. Construction off！ ~ E（T*M ~ f*TN, N）γ. K-orientations of vector bundles and mapsδ. Wrong-way functoriality for K-oriented maps7. The Orbit Space of a Group Action8. The Leaf Space of a Foliationα. Construction of C* （V, F）β. Closed transversals and idempotents of C* （V, F）γ. The analytic assembly map/~ : K.,r（BG） -. K（C*（V,F））...9. The Longitudinal Index Theorem for Foliationsα. Construction of lnd（D） ~ Ko（J）β. Significance of the C*-algebra Indexγ. The longitudinal index theorem10. The Analytic Assembly Map and Lie Groupsα. Geometric cycles for smooth groupoidsβ. Lie groups and deformationsγ. The G-equivariant index of elliptic operators on homogeneous spaces of Lie groupsδ. The K-theory K （C* （G）） for Lie groupsε. The general conjecture for smooth groupoidsA. Appendix: C*-modules and Strong Morita EquivalenceB. Appendix: E-theory and Deformations of Algebrasα. Deformations of C*-algebras and asymptotic morphismsβ. Composition of asymptotic morphismsγ. Asymptotic morphisms and exact sequences of C*-algebrasδ. The cone of a map and half-exactnessε. E-theoryC. Appendix: Crossed Products of C*-algebras and the Thorn IsomorphismD. Appendix: Penrose TilingsIII. CYCLIC COHOMOLOGY AND DIFFERENTIAL GEOMETRYIV. QUANTIZED CALCULUSV.OPERATOR ALGEBRASVI THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRYBIBLIOGRAPHYNOTATION AND CONVENTIONSINDEX

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