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