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| TABIF OF CONTENTS PREFACE INTRODUCTION I NONCOMMUTATIVE SPACES AND MEASURE THEORY 1. Heisenberg and the Noncommutative Algebra of Physical Quantities Associated to a Microscopic System 2. Statistical State of a Macroscopic System and Quantum Statistical Mechanics 3. Modular Theory and the Classification of Factors 4. Geometric Examples of yon Neumann Algebras: Measure Theory of Noncommutative Spaces α Classical Lebesgue measure theory β. Foliations γ. The von Neumann algebra of a foliation 5. The Index Theorem for Measured Foliations α. Transverse measures for follations β. The Ruelle-Sullivan cycle and the ELder number of a measured foliation γ. The Index theorem for measured foliations A. Appendix: Transverse Measures and Averaging Sequences B. AppendL, c Abstract Transverse Measure Theory C. Appendix: Noncommutative Spaces and Set Theory II. TOPOLOGY AND K-THEORY.. 1. C*-algebras and their K-theory 2. Elementary Examples of Quotient Spaces α. Open covers of manifolds β. The dual of the infinite dihedral group F = Z >~ Z/2 3. The Space X of Penrose Tilings 4. Duals of Discrete Groups and the Novikov Conjecture 5. The Tangent Groupoid of a Manifold 6. Wrong-way Functoriality in K-theory as a Deformation α. The index groupoid of a linear map β. Construction off！ ~ E（T*M ~ f*TN, N） γ. K-orientations of vector bundles and maps δ. Wrong-way functoriality for K-oriented maps 7. The Orbit Space of a Group Action 8. The Leaf Space of a Foliation α. 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 Foliations α. Construction of lnd（D） ~ Ko（J） β. Significance of the C*-algebra Index γ. The longitudinal index theorem 10. The Analytic Assembly Map and Lie Groups α. Geometric cycles for smooth groupoids β. Lie groups and deformations γ. The G-equivariant index of elliptic operators on homogeneous spaces of Lie groups δ. The K-theory K （C* （G）） for Lie groups ε. The general conjecture for smooth groupoids A. Appendix: C*-modules and Strong Morita Equivalence B. Appendix: E-theory and Deformations of Algebras α. Deformations of C*-algebras and asymptotic morphisms β. Composition of asymptotic morphisms γ. Asymptotic morphisms and exact sequences of C*-algebras δ. The cone of a map and half-exactness ε. E-theory C. Appendix: Crossed Products of C*-algebras and the Thorn Isomorphism D. Appendix: Penrose Tilings III. CYCLIC COHOMOLOGY AND DIFFERENTIAL GEOMETRY IV. QUANTIZED CALCULUS V.OPERATOR ALGEBRAS VI THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY BIBLIOGRAPHY NOTATION AND CONVENTIONS INDEX |
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