Analytic K-Homology

by Higson, Nigel; Roe, John

• ISBN13: 9780198511762
• ISBN10: 0198511760
• Format: Hardcover
• Copyright: 2001-02-15
• Publisher: Oxford University Press

Summary

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. Thepurpose 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, coarsegeometry, 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 materialincluded here has never previously appeared in book form.

Table of Contents

C*-Algebras and Operator Theory	1	(29)
Bounded Operators and Functional Calculus	1	(5)
Positive Operators and the Strong Operator Topology	6	(2)
C*-Algebras	8	(4)
The GNS Construction	12	(2)
Representations of Commutative C*-Algebras	14	(2)
Abstract C*-Algebras	16	(1)
Ideals and Quotients	17	(2)
Unbounded Operators	19	(3)
Exercises	22	(5)
Notes	27	(2)
Index Theory and Extensions	29	(26)
Fredholm Operators and the Calkin Algebra	29	(3)
The Essential Spectrum	32	(2)
The Toeplitz Extension	34	(2)
Essentially Normal Operators	36	(2)
C*-Algebra Extensions	38	(2)
Extensions and the Calkin Algebra	40	(1)
The Extension Semigroup	41	(3)
Geometric Examples of Extensions	44	(5)
Exercises	49	(5)
Notes	54	(1)
Completely Positive Maps	55	(30)
Completely Positive Maps	55	(2)
Quasicentral Approximate Units	57	(3)
Nuclearity	60	(3)
Voiculescu's Theorem	63	(2)
Block-Diagonal Maps	65	(3)
Proof of Voiculescu's Theorem	68	(3)
Property T and Ext	71	(5)
Kasparov's Technical Theorem	76	(3)
Exercises	79	(4)
Notes	83	(2)
K-Theory	85	(38)
The Group K0(A)	85	(5)
K0 for Non-Unital Algebras	90	(2)
Relative K-Theory and Excision	92	(5)
Homotopy	97	(1)
Higher K-Theory	98	(3)
Inner Automorphisms	101	(2)
Products	103	(3)
Another Description of K1	106	(4)
Bott Periodicity	110	(3)
Exercises	113	(7)
Notes	120	(3)
Duality Theory	123	(18)
Extension Groups and Dual C*-Algebras	123	(2)
K-Homology	125	(5)
Relative K-Homology	130	(3)
Excision in K-Homology	133	(4)
Example: the Theta Curve	137	(1)
Exercises	138	(1)
Notes	139	(2)
Coarse Geometry and K-Homology	141	(26)
Coarse Structures	141	(4)
Coarse Geometry of Cones	145	(2)
The C*-Algebra of a Coarse Space	147	(5)
K-Theory for Metric Coarse Structures	152	(5)
K-Theory for Topological Coarse Structures	157	(3)
The Homotopy Invariance of K-Homology	160	(2)
Exercises	162	(3)
Notes	165	(2)
The Brown-Douglas-Fillmore Theorem	167	(32)
Generalized Homology Theories	168	(2)
The Index Pairing	170	(10)
Steenrod Homology Theory	180	(3)
The Cluster Axiom for K-Homology	183	(3)
The Brown-Douglas-Fillmore Theorem	186	(2)
The Universal Coefficient Theorem	188	(8)
Exercises	196	(2)
Notes	198	(1)
Kasparov's K-Homology	199	(40)
Fredholm Modules	199	(5)
The Kasparov Groups	204	(4)
Normalization of Fredholm Modules	208	(4)
Kasparov Theory and Duality	212	(3)
Relative K-Homology	215	(4)
Schrodinger Pairs	219	(4)
The Index Pairing	223	(10)
Exercises	233	(4)
Notes	237	(2)
The Kasparov Product	239	(30)
The Product of Fredholm Operators	239	(4)
The Definition of the Kasparov Product	243	(5)
Index One Operators and Homotopy Invariance	248	(4)
Stability	252	(1)
Bott Periodicity	253	(6)
Boundary Maps and the Kasparov Product	259	(5)
The Kasparov Product and the Index Pairing	264	(1)
Exercises	265	(2)
Notes	267	(2)
Elliptic Differential Operators	269	(36)
First-Order Differential Operators	269	(2)
Symmetric and Selfadjoint Differential Operators	271	(3)
Wave Operators	274	(5)
Ellipticity	279	(5)
Elliptic Operators on Open Manifolds	284	(2)
The Homology Class of a Selfadjoint Operator	286	(4)
Elliptic Operators and the Kasparov Product	290	(3)
The Homology Class of a Symmetric Operator	293	(4)
Exercises	297	(6)
Notes	303	(2)
Index Theory	305	(42)
Dirac Operators	306	(5)
Spinc-Manifolds	311	(7)
Even-Dimensional Spinc-Manifolds	318	(2)
Index Theory for Hypersurfaces	320	(6)
The Index Theorem for Spinc-Manifolds	326	(3)
Toeplitz Index Theorems	329	(4)
Index Theory on Strongly Pseudoconvex Domains	333	(8)
Exercises	341	(4)
Notes	345	(2)
Higher Index Theory	347	(30)
Metrics of Positive Scalar Curvature	347	(3)
Non-Positive Sectional Curvature	350	(2)
Coarse Geometry and Assembly Maps	352	(5)
Scaleable Spaces and the Baum-Connes Conjecture	357	(6)
Equivariant Assembly	363	(6)
The Descent Principle	369	(4)
Exercises	373	(2)
Notes	375	(2)
Appendix A Gradings	377	(10)
A.1 Graded Vector Spaces and Algebras	377	(1)
A.2 Graded Tensor Products	378	(1)
A.3 Multigradings	379	(1)
A.4 Hermitian Modules and K-Theory	380	(4)
A.5 Graded Hermitian Modules	384	(1)
A.6 Notes	385	(2)
Appendix B Real K-Homology	387	(4)
B.1 Real C*-Algebras	387	(1)
B.2 K-Theory for Real C*-Algebras	388	(1)
B.3 K-Homology for Real C*-Algebras	389	(1)
B.4 Notes	390	(1)
References	391	(10)
Index	401