C*-Algebras by Example
This is a graduate text published in the
Fields Institute Monograph Series volume 6
by the American Mathematical Society.
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Reviews
Robert Doran, Math. Reviews MR97i:46095
William Arveson, Bull. Amer. Math. Soc. 34 (1997), 435-439.
Simon Wassermann, Bull. London Math. Soc. 30 (1998), 210-213.
Andrej Bulinski, Zentralblatt 0956.46034
Index
The Basics of C*-algebras 1
Definitions 1
Banach Algebra Basics 3
Commutative C*-algebras 7
Positive Elements 9
Ideals, Quotients and Homomorphisms 12
Weak Topologies 15
The Density Theorems 19
Some Operator Theory 23
Representations of C*-algebras 26
C*-algebras of Compact Operators 36
Exercises 40
Normal Operators and Abelian C*-algebras 46
Spectral Theory 46
The L-infinity Functional Calculus 48
Multiplicity Theory 53
The Weyl--von Neumann--Berg Theorem 57
Voiculescu's Theorem 64
Exercises 72
AF C*-algebras 74
Finite Dimensional C*-algebras 74
AF Algebras 75
Perturbations 79
Ideals and Quotients 84
Examples 86
Extensions 91
Exercises 95
K-theory for AF C*-algebras 97
Idempotents 97
K_0 100
Dimension Groups 102
Elliott's Theorem 109
Applications 112
Riesz groups 118
The Effros--Handelman--Shen Theorem 120
Blackadar's Simple Unital Projectionless C*-algebra 124
Exercises 129
C*-Algebras of Isometries 132
Toeplitz Operators 132
Isometries 136
Bunce--Deddens Algebras 137
Cuntz Algebras 144
Simple Infinite C*-algebras 147
Classification of Cuntz Algebras 150
Real Rank Zero 156
Exercises 162
Irrational Rotation Algebras 166
The irrational rotation algebras 166
Projections in irrational rotation algebras 170
An AF algebra 172
Berg's technique 174
Imbedding into AF algebras 177
Exercises 180
Group C*-Algebras 182
Group Representations 182
Amenability 185
Primitive Ideals 190
A Crystallographic Group 193
The Discrete Heisenberg Group 200
The Free Group 203
The Reduced C*-algebra of the Free Group 206
C*_r(F_2) is Projectionless 210
Exercises 214
Discrete Crossed Products 216
Crossed Products 216
Crossed Products by Z 222
Minimal Dynamical Systems 223
Odometers 230
K-theory of Crossed Products 232
AF Subalgebras of Crossed Products 235
Crossed Product subalgebras of AF Algebras 238
Topological Stable Rank 244
An Order 2 Automorphism 247
Exercises 250
Brown--Douglas--Fillmore Theory 252
Extensions 252
An Addition and Zero Element for Ext(X) 254
Some Special Cases 258
Positive maps 259
Ext(X) is a group 266
First Topological Properties 268
Ext for planar sets 273
Quasidiagonality 281
Homotopy Invariance 286
The Mayer--Vietoris Sequence 289
Examples 294
Exercises 299
References 303
Index 307
Preface
These notes were developed in the fall of 1993 for a graduate course
on C*-algebras. The subject of C*-algebras received a dramatic
revitalization in the 1970s by the introduction of topological methods
due the deep 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 and Kasparov's melding of the two into
KK-theory. These results were the beginning of a marvelous new set of
tools for analyzing concrete C*-algebras. The subject flourished by
virtue of a rich and varied group of examples which were the perfect
fodder for these new methods. Moreover, these examples served as a
beacon for the kinds of general tools needed to study C*-algebras.
Today, a student cannot get very far in the C*-algebra literature
without being somewhat familiar with the lexicon of examples that now
dot the landscape.
These notes are not intended as a systematic study of the general
theory of C*-algebras, nor of K-theory. There are several excellent
books on both of these aspects. Rather, I develop a modicum of the
general theory for the sake of self-containment and then launch into
a study of various important classes of examples.
A number of choices had to be made. I have developed the theory of
AF algebras to quite an extent. However, in some of the other topics,
such as irrational rotation algebras and Cuntz algebras, I have
limited myself to the more modest goal of obtaining enough information
to classify the algebras within their narrow context. Because this is
not a course in K-theory, certain stronger results that require a more
systematic study have been omitted. I have also given a detailed
treatment of the Brown--Douglas--Fillmore theory. The discussion is
limited to the commutative case as in their original work because of
the technical simplifications. However the informed reader will
notice that a number of the proofs have been influenced by the more
general theory.
The hope is that these notes will open a student's eyes to some of the
power, beauty and variety of these amazing algebras. This is just
a glimpse into an exciting area of current research interest.
These notes were compiled during the author's participation in the
special year on C*-algebras at the Fields Institute of Mathematics
held in Waterloo, Canada during the 1994--95 academic year. The
author wishes to thank the Fields Institute and the University of
Waterloo for providing the release time that made this project
possible. I also wish to thank everyone who provided feedback on
the early drafts of these notes. I am especially indebted to Keith
Taylor, who provided me with some wonderful material on the
C*-algebras of crystallographic groups and Alan Paterson for
advice on amenability. I am also grateful to Florin Boca, who
provided me with detailed notes on showing that the irrational
rotation algebras are limit circle algebras even though, in the
end, I decided not to include them. I must also mention Ileana
Ionascu and Raul Curto, who read the whole manuscript and made many
detailed comments. In addition, I thank Ola Bratteli, Larry Brown,
John Conway, Don Hadwin, Dick Kadison, Ian Putnam, Norberto Salinas
and Hong Sheng Yin who put me right at various points of the book.
In spite of all their help, I know that I have been unable to get all
the minor blemishes out. For this, I must bear the burden myself.
The text was prepared using \AmS-\LaTeX, and typeset in 11pt Times
roman. The commutative diagrams and figures were produced using the
macro package \XY-pic version 3.2 written by Kris Rose and Ross Moore.
Kenneth R. Davidson
Waterloo, February, 1996
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