## An Introduction to the Classification of Amenable C*-algebrasThe theory and applications of C Oeu -algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C Oeu -algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C Oeu -algebras (up to isomorphism) by their K -theoretical data. It started with the classification of AT -algebras with real rank zero. Since then great efforts have been made to classify amenable C Oeu -algebras, a class of C Oeu -algebras that arises most naturally. For example, a large class of simple amenable C Oeu -algebras is discovered to be classifiable. The application of these results to dynamical systems has been established. This book introduces the recent development of the theory of the classification of amenable C Oeu -algebras OCo the first such attempt. The first three chapters present the basics of the theory of C Oeu -algebras which are particularly important to the theory of the classification of amenable C Oeu -algebras. Chapter 4 otters the classification of the so-called AT -algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C Oeu -algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH -algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C Oeu -algebras. Besides being as an introduction to the theory of the classification of amenable C Oeu -algebras, it is a comprehensive reference for those more familiar with the subject. Sample Chapter(s). Chapter 1.1: Banach algebras (260 KB). Chapter 1.2: C*-algebras (210 KB). Chapter 1.3: Commutative C*-algebras (212 KB). Chapter 1.4: Positive cones (207 KB). Chapter 1.5: Approximate identities, hereditary C*-subalgebras and quotients (230 KB). Chapter 1.6: Positive linear functionals and a Gelfand-Naimark theorem (235 KB). Chapter 1.7: Von Neumann algebras (234 KB). Chapter 1.8: Enveloping von Neumann algebras and the spectral theorem (217 KB). Chapter 1.9: Examples of C*-algebras (270 KB). Chapter 1.10: Inductive limits of C*-algebras (252 KB). Chapter 1.11: Exercises (220 KB). Chapter 1.12: Addenda (168 KB). Contents: The Basics of C Oeu -Algebras; Amenable C Oeu -Algebras and K -Theory; AF- Algebras and Ranks of C Oeu -Algebras; Classification of Simple AT -Algebras; C Oeu -Algebra Extensions; Classification of Simple Amenable C Oeu -Algebras. Readership: Researchers and graduate students in operator algebras." |

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### Contents

The Basics of Calgebras | 1 |

Amenable Calgebras and ftTtheory | 67 |

j4Falgebras and Ranks of Calgebras | 113 |

Classification of Simple j4Talgebras | 165 |

Calgebra Extensions | 211 |

Classification of Simple Amenable Calgebras | 269 |

307 | |

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### Common terms and phrases

a-unital abelian group amenable C*-algebras approximate identity assume AT-algebra Banach algebra Borel C*-algebra with TR(A compact Hausdorff space completely positive linear contractive completely positive converges Corollary CW complex Define Definition Denote dense diagram direct sum embedding equivalent exists finite dimensional range finite spectrum finite subset T C A Hausdorff space Hence Her(a hereditary C*-subalgebra Hilbert space homomorphism h ideal implies inductive limit infinite injective integer isomorphism Ki(A Ko(A Ko(B Lemma Mk(A Mn(A Moreover morphism mutually orthogonal projections non-unital nonzero projection norm Note numbers obtain ordered group partial isometry positive elements positive linear functional positive linear map projection q Proof Proposition quotient map real rank zero representation RR(A self-adjoint sequence of contractive short exact sequence sp(a stable rank stably finite subalgebra subgroup Suppose surjective Theorem tracial tsr(A unit ball unital C*-algebra unitary von Neumann algebra write