Classification of Nuclear C*-Algebras. Entropy in Operator Algebras

ISBN-13: 9783642076053 / Angielski / Miękka / 2010 / 198 str.

M. Rordam; E. Stormer

Opis

This EMS volume consists of two parts, written by leading scientists in the field of operator algebras and non-commutative geometry. The first part, written by M.Rordam, is on Elliott's classification program for nuclear C*-algebras. The emphasis is on the work of Kirchberg and the spectacular results by Kirchberg and Phillips giving a nearly complete classification, in terms of K-theoretic invariants, in the purely infinite case. This part of the program is described with almost full proofs beginning with Kirchberg's tensor product theorems and Kirchberg's embedding theorem for exact C*-algebras. The classification of finite simple C*-algebras starting with AF-algebras, and continuing with AT- and AH-algebras is covered, but mostly without proofs. The second part, written by E.Stormer, is a survey of the theory of of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation by Connes and Stormer in 1975 till 2001.

Recenzje

From the reviews:

"... These notes [by E.Stormer] describe the main approaches to noncommutative entropy, together with several ramifications and variants. The notion of generator and variational principle are used to give applications to subfactors and C*-algebra formalism of quantum statistical mechanics. The author considers the most frequently studied examples, including Bernoulli shifts, Bogolyubov automorphisms, dual automorphisms on crossed products, shifts on infinite free products, and binary shifts on the CAR-algebra. The mathematical techniques and ideas are beautifully exposed, and the whole paper is a rich resource on the subject, either for the expert or the beginner. ..."

V.Deaconu, Mathematical Reviews 2004

"... the author gives a clear presentation of the dramatic developments in the classification theory for simple C*-algebras that have taken place over the past 25 years or so. ... As there is such a large amount of literature on the subject, this monograph article is particularly useful to the relative novice who wants to know the fundamental results in the theory without wading through a massive amount of detail. ...This monograph-length article is extremely well-written, filled with concrete examples, and has an exhaustive bibliography. I recommend it as an excellent introduction to graduate students and other mathematicians who want to bring themselves up-to-date on the subject. .."

J.A.Packer, Mathematical Reviews 2004

"Both contributions to this volume are high-end, excellently written research reviews, reflecting very thoroughly the current status in the respectively treated subbranches of the quickly evolving complex field of C* algebra theory. They both give a beautiful lay-out of the vast research program in the field which has been going on for decades ... as well as to the standard works. ... an excellent, very thorough, concise and needed overview for the researcher who is active in this field." (Mark Sioen, Bulletin of the Belgian Mathematical Society, 2007)

Spis treści

I. Classification of Nuclear, Simple C*-algebras.- II. A Survey of Noncommutative Dynamical Entropy.

Blurb

This EMS volume consists of two parts, written by leading scientists in the field of operator algebras and non-commutative geometry. The first part, written by M.Rordam entitled "Classification of Nuclear, Simple C*-Algebras" is on Elliotts classification program. The emphasis is on the classification by Kirchberg and Phillips of Kirchberg algebras: purely infinite, simple, nuclear separable C*-algebras. This classification result is described almost with full proofs starting from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for exact C*-algebras. The classificatin of finite simple C*-algebras starting with AF-algebras, and continuing with AF- and AH-algberas) is covered, but mostly without proofs. The second part, written by E.Stormer entitled "A Survey of Noncommutative Dynamical Entropy" is a survey of the theory of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation by Connes and Stormer in 1975 till 2001. The main definitions and resuls are discussed and illustrated with the key examples in the theory. This book will be useful to graduate students and researchers in the field of operator algebras and related areas.