內容簡介
《非交換幾何》主要內容：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 In particular the present text contains several unpublished results.
目錄
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
1.Cyclic Cohomology
α.Characters of cycles and the cup product in HC*
β.Cobordisms of cycles and the operator B
γ.The exact couple relating HC* （A） to Hochschild cohomology
2.Examples
α.A = C（V）, V a compact smooth manifold
β.The cyclic cohomology of the noncommutative torus A = Aθ1θεR/Z
γ.The cyclic cohomology of the group ring CF for F a discrete group
δ.Cyclic cohomology of C（V>F）
3.Pairing of Cyclic Cohomology with K-Theory
4.The Higher Index Theorem for Covering Spaces
α.The smooth groupoid of a covering space
β.The group ring RF
γ.The index theorem
5.The Novikov Conjecture for Hyperbolic Groups
α.Word hyperbolic groups
β.The Haagerup inequality
γ.Extension to C* （F） of K-theory invariants
6.Factors of Type III, Cyclic Cohomology and the Godbillon-Vey lnvariant
α.Extension of densely defined cyclic cocycles on Banach algebras
β.The Bott-Thurston cocycle and the equaliW GV = i[V/F]
γ.Invariant measures on the flow of weights
7.The Transverse Fundamental Class for Foliations and Geometric Corollaries
α.The transverse fundamental class
β.Geometric corollaries
γ.Index formula for longitudinal elliptic operators
A.Appendix: The Cyclic Category A
α.The simplicial category
β.The cyclic category
γ.The A-module A associated to an algebra A
δ.Cyclic spaces and $1 spaces
B.Appendix: Locally Convex Algebras
C.Appendix: StabiliW under Holomorphic Functional Calculus
Ⅳ QUANTIZED CALCULUS
1.Quantized Differential Calculus and Cyclic Cohomology
a.The cycle associated to a Fredholm module
β.The periodicity operator S and the Chern character
r.Pairing with K-theory and index formula
2.The Dixmler Trace and the Hochschiid Class of the Character
a.General properties of interpolation ideals
β.The Dlxmier trace
γ.The residue formula for the Hochschild class of the character of Fredholm modules
δ.Growth of algebras and degree of summability of K-cycles
ε.Quantized Calculus m One Variable and Fractal Sets
a.Quantized calculus in one variable
β.The class of d f in
γ.The Dixmier trace of f（Z）|dZ|p
δ.The harmonic measure and non-normality of the Dlxmler trace
ε.Cantor sets, Dixrnier trace and Minkowsld measure
4.Conformal Manifolds
a.Quantized calculus on conformal manifolds
β.Perturbation of Fredholm modules by the commutant von Neumann algebra
γ.The 4-dimensional analogue of the Polyakov action
5.Fredholm Modules and Rank-One Discrete Groups
6.Elliptic Theory on the Noncommutative Torus and the Quantum Hall Effect
a.Elliptic theory on
β.The quantum Hall effect
γ.The work of J.Bellissard on the integrality of （rH
7.Entire Cyclic Cohomology
α.Entire cyclic cohomology of Banach algebras
β.Infinite-dimensional cycles
γ.Traces on QA and ΕA
δ.Pairing with Ko （ A ）
ε.Entire cyclic cohomology of S1
8.The Chern Character of θ-summable Fredholm Modules
α.Fredholm modules and K-cycles
β.The supergroup R1,1 and the convolution algebra of operator-valued distributions on
γ.The Chern character of K-cycles
δ.The index formula
ε.The JLO cocycle
9.θ-summable K-cycles, Discrete Groups and Quantum Field Theory
α Discrete subgroups of Lie groups
β.Supersymmetr/c quantum field theory
A.Appendix: Kasparov's Bivariant Theory
B.Appendix: Real and Complex Interpolation of Banach Spaces
C.Appendix: Nonned Ideals of Compact Operators
D.Appendix: The Chern Character of Deformations of Algebras
Ⅴ OPERATOR ALGEBRAS
1.The Papers of Murray and yon Neumann
α.Examples of yon Neumann algebras
β.Reduction theory
γ.Comparison of subrepresentations, comparison of projections and the relative dimension function
δ.Algebraic isomorphism and spatial isomorphism
ε.The Krst two examples of type II1 factors, the hyperfinite factor and the property F
2.Representations of C*-algebras
3.The Algebraic Framework for Noncommutative Integration and the Theory of Weights
4.The Factors of Powers, Arald and Woods, and of Krieger
5.The Radon-Ntkodyn Theorem and Factors of Type HI~~
α The Radon-Nikodym theorem
β.The factors of type IIIλ
6.Noncommutative Erg0dic Theory
α.Rokhlin's theorem
β.Entropy
γ.Approximately inner automorphisms
δ.Centrally trivial automorphisms
ε.The obstruction γ（θ）
ζ.The list of automorphisms of R up to outer conjugacy
η.Automorphisms of the Araki-Woods factor R0,1 of type
7.Amenable yon Neumann Algebras
α.Approximation by finite-dimensional algebras
β.The properties P of Schwartz, E of Hakeda and Tomiyama, and injectivity
γ.Semidiscrete von Neumann algebras
8.The Flow of Weights: mod（M）
a.The discrete decomposition of factors of type IIIO
β.Continuous decomposition of type IIl factors
γ.Functorial definition of the flow of weights
Ⅵ THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY~
