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Analytic K-Homology

By: John Roe, Nigel Higson

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Hardcover | 1 February 2001

At a Glance

Hardcover
424 Pages

Dimensions(cm)
24.7 x 16.1 x 2.8

Hardcover

RRP $575.00

$408.75

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Analytic K-homology draws together ideas from algebraic topology, functional analysis and geometry. It is a tool - a means of conveying information among these three subjects - and it has been used with specacular success to discover remarkable theorems across a wide span of mathematics. The purpose of this book is to acquaint the reader with the essential ideas of analytic K-homology and develop some of its applications. It includes a detailed introduction to the necessary functional analysis, followed by an exploration of the connections between K-homology and operator theory, coarse geometry, index theory, and assembly maps, including a detailed treatment of the Atiyah-Singer Index Theorem. Beginning with the rudiments of C - algebra theory, the book will lead the reader to some central notions of contemporary research in geometric functional analysis. Much of the material included here has never previously appeared in book form.

Industry Reviews

The book gives a well-written introduction to the essential ideas of analytic K-homology and develops some of its applications.--Mathematical Reviews

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You Can Find This Book In

Non-Fiction Mathematics Topology Algebraic Topology Calculus & Mathematical Analysis Functional Analysis & Transforms Applied Mathematics Geometry Algebraic Geometry Science Physics

C*-Algebras and Operator Theory	p. 1
Bounded Operators and Functional Calculus	p. 1
Positive Operators and the Strong Operator Topology	p. 6
C*-Algebras	p. 8
The GNS Construction	p. 12
Representations of Commutative C*-Algebras	p. 14
Abstract C*-Algebras	p. 16
Ideals and Quotients	p. 17
Unbounded Operators	p. 19
Exercises	p. 22
Notes	p. 27
Index Theory and Extensions	p. 29
Fredholm Operators and the Calkin Algebra	p. 29
The Essential Spectrum	p. 32
The Toeplitz Extension	p. 34
Essentially Normal Operators	p. 36
C*-Algebra Extensions	p. 38
Extensions and the Calkin Algebra	p. 40
The Extension Semigroup	p. 41
Geometric Examples of Extensions	p. 44
Exercises	p. 49
Notes	p. 54
Completely Positive Maps	p. 55
Completely Positive Maps	p. 55
Quasicentral Approximate Units	p. 57
Nuclearity	p. 60
Voiculescu's Theorem	p. 63
Block-Diagonal Maps	p. 65
Proof of Voiculescu's Theorem	p. 68
Property T and Ext	p. 71
Kasparov's Technical Theorem	p. 76
Exercises	p. 79
Notes	p. 83
K-Theory	p. 85
The Group K[subscript 0](A)	p. 85
K[subscript 0] for Non-Unital Algebras	p. 90
Relative K-Theory and Excision	p. 92
Homotopy	p. 97
Higher K-Theory	p. 98
Inner Automorphisms	p. 101
Products	p. 103
Another Description of K[subscript 1]	p. 106
Bott Periodicity	p. 110
Exercises	p. 113
Notes	p. 120
Duality Theory	p. 123
Extension Groups and Dual C*-Algebras	p. 123
K-Homology	p. 125
Relative K-Homology	p. 130
Excision in K-Homology	p. 133
Example: the Theta Curve	p. 137
Exercises	p. 138
Notes	p. 139
Coarse Geometry and K-Homology	p. 141
Coarse Structures	p. 141
Coarse Geometry of Cones	p. 145
The C*-Algebra of a Coarse Space	p. 147
K-Theory for Metric Coarse Structures	p. 152
K-Theory for Topological Coarse Structures	p. 157
The Homotopy Invariance of K-Homology	p. 160
Exercises	p. 162
Notes	p. 165
The Brown-Douglas-Fillmore Theorem	p. 167
Generalized Homology Theories	p. 168
The Index Pairing	p. 170
Steenrod Homology Theory	p. 180
The Cluster Axiom for K-Homology	p. 183
The Brown-Douglas-Fillmore Theorem	p. 186
The Universal Coefficient Theorem	p. 188
Exercises	p. 196
Notes	p. 198
Kasparov's K-Homology	p. 199
Fredholm Modules	p. 199
The Kasparov Groups	p. 204
Normalization of Fredholm Modules	p. 208
Kasparov Theory and Duality	p. 212
Relative K-Homology	p. 215
Schrodinger Pairs	p. 219
The Index Pairing	p. 223
Exercises	p. 233
Notes	p. 237
The Kasparov Product	p. 239
The Product of Fredholm Operators	p. 239
The Definition of the Kasparov Product	p. 243
Index One Operators and Homotopy Invariance	p. 248
Stability	p. 252
Bott Periodicity	p. 253
Boundary Maps and the Kasparov Product	p. 259
The Kasparov Product and the Index Pairing	p. 264
Exercises	p. 265
Notes	p. 267
Elliptic Differential Operators	p. 269
First-Order Differential Operators	p. 269
Symmetric and Selfadjoint Differential Operators	p. 271
Wave Operators	p. 274
Ellipticity	p. 279
Elliptic Operators on Open Manifolds	p. 284
The Homology Class of a Selfadjoint Operator	p. 286
Elliptic Operators and the Kasparov Product	p. 290
The Homology Class of a Symmetric Operator	p. 293
Exercises	p. 297
Notes	p. 303
Index Theory	p. 305
Dirac Operators	p. 306
Spin[superscript c]-Manifolds	p. 311
Even-Dimensional Spin[superscript c]-Manifolds	p. 318
Index Theory for Hypersurfaces	p. 320
The Index Theorem for Spin[superscript c]-Manifolds	p. 326
Toeplitz Index Theorems	p. 329
Index Theory on Strongly Pseudoconvex Domains	p. 333
Exercises	p. 341
Notes	p. 345
Higher Index Theory	p. 347
Metrics of Positive Scalar Curvature	p. 347
Non-Positive Sectional Curvature	p. 350
Coarse Geometry and Assembly Maps	p. 352
Scaleable Spaces and the Baum-Connes Conjecture	p. 357
Equivariant Assembly	p. 363
The Descent Principle	p. 369
Exercises	p. 373
Notes	p. 375
Gradings	p. 377
Graded Vector Spaces and Algebras	p. 377
Graded Tensor Products	p. 378
Multigradings	p. 379
Hermitian Modules and K-Theory	p. 380
Graded Hermitian Modules	p. 384
Notes	p. 385
Real K-Homology	p. 387
Real C*-Algebras	p. 387
K-Theory for Real C*-Algebras	p. 388
K-Homology for Real C*-Algebras	p. 389
Notes	p. 390
References	p. 391
Index	p. 401
Table of Contents provided by Syndetics. All Rights Reserved.

Analytic K-Homology

At a Glance

Hardcover

Industry Reviews

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