## C*-algebras by ExampleThe 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

1 | |

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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### Common terms and phrases

abelian algebra apply approximate automorphism basis belongs C*-algebra calculation called choose circle Clearly closed commutes compact compact operators complete Consider construction contains continuous contractive converges Corollary corresponding crossed product define definition denote dense determined diagonal diagram easy element equal evident exact example Exercise Ext(X extension fact finite dimensional follows function given Hence Hilbert space homomorphism ideal identity imbedding implies infinite injective integer invariant invertible irreducible representations isometry isomorphism Lemma lies limit linear matrix units measure minimal Moreover multiplicity non-zero norm normal operator Note Notice obtain operator partial particular positive projection Proof Proposition quasidiagonal quotient range representation result satisfies separable sequence shift simple spectrum subalgebra subset sufficiently Suppose taking Theorem topology trace trivial unique unit unitarily equivalent unitary unitary operator vector verify weak whence yields zero