An Introduction to the Classification of Amenable C*-Algebras

by Lin, Huaxin

• ISBN13: 9789810246808
• ISBN10: 9810246803
• Format: Hardcover
• Copyright: 2001-09-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

Introduces the recent development of the theory of the classification of amenable C*-algebras-- the first such attempt. The first three chapters present the basics of the theory of C*-algebras which are particularly important to the theory of the classification of amenable C*-algebras.

Table of Contents

Preface	vii
The Basics of C*-algebras	1	(66)
Banach algebras	1	(8)
C*-algebras	9	(3)
Commutative C*-algebras	12	(4)
Positive cones	16	(4)
Approximate identities, hereditary C*-subalgebras and quotients	20	(5)
Positive linear functionals and a Gelfand-Naimark theorem	25	(7)
Von Neumann algebras	32	(6)
Enveloping von Neumann algebras and the spectral theorem	38	(4)
Examples of C*-algebras	42	(9)
Inductive limits of C*-algebras	51	(9)
Exercises	60	(5)
Addenda	65	(2)
Amenable C*-algebras and K-theory	67	(46)
Completely positive linear maps and the Stinespring representation	67	(5)
Examples of completely positive linear maps	72	(4)
Amenable C*-algebras	76	(6)
K-theory	82	(7)
Perturbations	89	(8)
Examples of K-groups	97	(6)
K-theory of inductive limits of C*-algebras	103	(5)
Exercises	108	(3)
Addenda	111	(2)
AF-algebras and Ranks of C*-algebras	113	(52)
C*-algebras of stable rank one and their K-theory	113	(7)
C*-algebras of lower rank	120	(7)
Order structure of K-theory	127	(6)
AF-algebras	133	(7)
Simple C*-algebras	140	(6)
Tracial topological rank	146	(8)
Simple C*-algebras with TR(A) ≤ 1	154	(6)
Exercises	160	(2)
Addenda	162	(3)
Classification of Simple AT-algebras	165	(46)
Some basics about AT-algebras	165	(5)
Unitary groups of C*-algebras with real rank zero	170	(7)
Simple AT-algebras with real rank zero	177	(5)
Unitaries in simple C*-algebra with RR(A) = 0	182	(4)
A uniqueness theorem	186	(6)
Classification of simple AT-algebras	192	(4)
Invariants of simple AT-algebras	196	(8)
Exercises	204	(4)
Addenda	208	(3)
C*-algebra Extensions	211	(58)
Multiplier algebras	211	(6)
Extensions of C*-algebras	217	(4)
Completely positive maps to Mn(C)	221	(6)
Amenable completely positive maps	227	(6)
Absorbing extensions	233	(10)
A stable uniqueness theorem	243	(7)
K-theory and the universal coefficient theorem	250	(5)
Characterization of KK-theory and a universal multi-coefficient theorem	255	(4)
Approximately trivial extensions	259	(6)
Exercises	265	(4)
Classification of Simple Amenable C*-algebras	269	(38)
An existence theorem	269	(10)
Simple AH-algebras	279	(9)
The classification theorems	288	(7)
Invariants and some isomorphism theorems	295	(12)
Bibliography	307	(10)
Index	317