C*-algebras by Example

Front Cover
American Mathematical Soc., 1996 - Mathematics - 309 pages
The subject of C*-algebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of Brown, Douglas, and Fillmore on extensions of C*-algebras and Elliott's use of K-theory to provide a useful classification of AF algebras. These results were the beginning of a marvelous new set of tools for analyzing concrete C*-algebras.
This book is an introductory graduate level text which presents the basics of the subject through a detailed analysis of several important classes of C*-algebras. The development of operator algebras in the last twenty years has been based on a careful study of these special classes. While there are many books on C*-algebras and operator algebras available, this is the first one to attempt to explain the real examples that researchers use to test their hypotheses. Topic include AF algebras, Bunce-Deddens and Cuntz algebras, the Toeplitz algebra, irrational rotation algebras, group C*-algebras, discrete crossed products, abelian C*-algebras (spectral theory and approximate unitary equivalence) and extensions. It also introduces many modern concepts and results in the subject such as real rank zero algebras, topological stable rank, quasidiagonality, and various new constructions.
These notes were compiled during the author's participation in the special year on C*-algebras at the Fields Institute of Mathematics during the 1994-1995 academic year. The field of C*-algebras touches upon many other areas of mathematics such as group representations, dynamical systems, physics, K-theory, and topology. The variety of examples offered in this text expose the student to many of these connections. A graduate student with a solid course in functional analysis should be able to read this book. This should prepare them to read much of the current literature. This book is reasonably self-contained, and the author has provided results from other areas when necessary.
 

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Contents

The Basics of Calgebras
1
12 Banach Algebras Basics
3
13 Commutative Calgebras
7
14 Positive Elements
9
15 Ideals Quotients and Homomorphisms
12
16 Weak Topologies
15
17 The Density Theorems
19
18 Some Operator Theory
23
Irrational Rotation Algebras
166
VI2 Projections in Aθ
170
VI3 An AF Algebra
172
VI4 Bergs Technique
174
VI5 Imbedding Aθ into Uθ
177
Group Calgebras
182
VII2 Amenability
185
VII3 Primitive Ideals
190

19 Representations of Calgebras
26
110 Calgebras of Compact Operators
36
Normal Operators and Abelian Calgebras
46
II2 The L Functional Calculus
48
II3 Multiplicity Theory
53
II4 The Weylvon NeumannBerg Theorem
57
II5 Voiculescus Theorem
64
AF Calgebras
74
III2 AF algebras
75
III3 Perturbations
79
1114 Ideals and Quotients
84
1115 Examples
86
III6 Extensions
91
Ktheory for AF Calgebras
97
IV2 K0
100
IV3 Dimension Groups
102
IV4 Elliotts Theorem
109
IV5 Applications
112
IY6 Riesz Groups
118
FV7 The EffrosHandelmanShen Theorem
120
IV8 Blackadars Simple Unital Projectionless Calgebra
124
Calgebras of Isometries
132
V2 Isometrics
136
V3 BunceDeddens Algebras
137
V4 Cuntz Algebras
144
V5 Simple Infinite CAIgebras
147
V6 Classification of Cuntz Algebras
150
V7 Real Rank Zero
156
VII4 A Crystallographic Group
193
VII5 The Discrete Heisenberg Group
200
VII6 The Free Group
203
VII7 The reduced Calgebra of the free group
206
VII8 CrF2 is projectionless
210
Discrete Crossed Products
216
VIII2 Crossed Products by Z
222
VIII3 Minimal Dynamical Systems
223
VIII4 Odometers
230
VIII5 Ktheory for Crossed Products
232
VIII6 AF subalgebras of Crossed Products
235
VIII7 Crossed Product subalgebras of AF algebras
238
VIII8 Topological stable rank
244
VIII9 An order 2 automorphism
247
BrownDouglasFillmore Theory
252
IX2 An Addition and Zero Element for ExtX
254
IX3 Some Special Cases
258
IX4 Positive Maps
259
IX5 ExtX is a Group
266
IX6 First Topological Properties
268
IX7 Ext for Planar Sets
273
IX8 Quasidiagonality
276
IX9 Homotopy Invariance
280
IX10 The MayerVietoris Sequence
283
IX11 Examples
288
References
297
Index
301
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