Introduction To The Classification Of Amenable C*-algebras, An (PDF)

The theory and applications of Câ-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 Câ-algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of Câ-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 Câ-algebras, a class of Câ-algebras that arises most naturally. For example, a large class of simple amenable Câ-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 Câ-algebras â the first such attempt. The first three chapters present the basics of the theory of Câ-algebras which are particularly important to the theory of the classification of amenable Câ-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 Câ-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 Câ-algebras. Besides being as an introduction to the theory of the classification of amenable Câ-algebras, it is a comprehensive reference for those more familiar with the subject.