An Introduction To The Classification Of Amenable C Algebras

The 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."

This book is directed towards graduate students that wish to start from the basic theory of C*-algebras and advance to an overview of some of the most spectacular results concerning the structure of nuclear C*-algebras. The text is divided into three parts. First, elementary notions, classical theorems and constructions are developed. Then, essential examples in the theory, such as crossed products and the class of quasidiagonal C*-algebras, are examined, and finally, the Elliott invariant, the Cuntz semigroup, and the Jiang-Su algebra are defined. It is shown how these objects have played a fundamental role in understanding the fine structure of nuclear C*-algebras. To help understanding the theory, plenty of examples, treated in detail, are included. This volume will also be valuable to researchers in the area as a reference guide. It contains an extensive reference list to guide readers that wish to travel further.

This book examines some recent developments in the theory of -algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable -algebras, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, -theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.

This book collects the notes of the lectures given at an Advanced Course on Dynamical Systems at the Centre de Recerca Matemàtica (CRM) in Barcelona. The notes consist of four series of lectures. The first one, given by Andrew Toms, presents the basic properties of the Cuntz semigroup and its role in the classification program of simple, nuclear, separable C*-algebras. The second series of lectures, delivered by N. Christopher Phillips, serves as an introduction to group actions on C*-algebras and their crossed products, with emphasis on the simple case and when the crossed products are classifiable. The third one, given by David Kerr, treats various developments related to measure-theoretic and topological aspects of crossed products, focusing on internal and external approximation concepts, both for groups and C*-algebras. Finally, the last series of lectures, delivered by Thierry Giordano, is devoted to the theory of topological orbit equivalence, with particular attention to the classification of minimal actions by finitely generated abelian groups on the Cantor set.

In this paper the author proves Generalized Homotopy Lemmas. These type of results play an important role in the classification theory of $*$-homomorphisms up to asymptotic unitary equivalence. Table of Contents: Prelude; The basic homotopy lemma for higher dimensional spaces; Purely infinite simple $C^*$-algebras; Approximate homotopy; Super homotopy; Postlude; Bibliography. (MEMO/205/963)

The 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.

This volume contains translations of papers that originally appeared in the Japanese journal 'Sugaku'. The papers range over a variety of topics, including operator algebras, analysis, and statistics.

A unital separable -algebra, is said to be locally AH with no dimension growth if there is an integer satisfying the following: for any and any compact subset there is a unital -subalgebra, of with the form , where is a compact metric space with covering dimension no more than and is a projection, such that The authors prove that the class of unital separable simple -algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple -algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.