The theory and applications of C� -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� -algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C� -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� -algebras, a class of C� -algebras that arises most naturally. For example, a large class of simple amenable C� -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� -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. 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� -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� -algebras. Besides being as an introduction to the theory of the classification of amenable C� -algebras, it is a comprehensive reference for those more familiar with the subject.
The basics of C*-algebras; amenable C*-algebras and K-theory; AF-algebras and ranks of C*-algebras; classification of simple AT-algebras; C*-algebra extensions; classification of simple amenable C*-algebras.