Theory of operator algebras. III
type: Item
instance of: scholarly article
title: Theory of operator algebras. III
zbMATH Open document ID: 1059.46032
author: Masamichi Takesaki
published in: Encyclopaedia of Mathematical Sciences
publication date: 2002-11-24T00:00:00Z
review text: As author stated ''the theory of operator algebras should be viewed as a number theory in analysis'' from the point of a historical or cultural view. This is the third volume of an a modern book written by one of main contributors to the theory of operator algebras and is useful for researchers. The first volume deals with the similarity between the theory of measures on a locally compact space and the theory of operator algebras and may be regarded as a non-commutative integration, cf. \textit{M. Takesaki} [Theory of operator algebras I. 2nd printing of the 1979 ed. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 124(5). Berlin: Springer)(2002; Zbl 0990.46034)]. In the second volume one can find an investigation of structure of von Neumann algebras of type III and their automorphism groups, cf. \textit{M. Takesaki} [Theory of operator algebras II. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 125(VI). Berlin: Springer)(2003; Zbl 1059.46031)]. The third volume is a comprehensive account of structure analysis of approximately finite dimensional factors and their automorphism groups and consists of seven chapters (13-19), notation index, and subject index, and also jointly with vol. II consists of an author preface, an appendix containing twenty five sections and a bibliography containing 334 references. Each chapter begins with a clear introduction describing the content of that chapter, contains several interesting exercises and is concluded with a section of rich historical notes. Some materials in the theory of operator algebras such as the classification of nuclear \(C^{*}\)-algebras, free probability theory, entropy theory, and \(K\)-theory for \(C^{*}\)-algebras are not included in the book and Encyclopaedia subseries editors will plan to include them and other topics on operator algebras and non-commutative geometry in the other volumes. In the following paragraphs we briefly d…
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1834432
zbMATH Keywords: ergodic transformation group
zbMATH Keywords: Krieger's construction
zbMATH Keywords: orbit structure
zbMATH Keywords: approximately finite measured groupoids
zbMATH Keywords: amenable group
zbMATH Keywords: inductive limit
zbMATH Keywords: infinite tensor product
zbMATH Keywords: group von Neumann algebras
zbMATH Keywords: strongly stable factors
zbMATH Keywords: maximal abelian subalgebra
zbMATH Keywords: completely positive approximation
zbMATH Keywords: nuclear \(C^{*}\)-algebra
zbMATH Keywords: injective von Neumann algebra
zbMATH Keywords: Grothendieck-Haagerup-Pisier inequality
zbMATH Keywords: approximately finite dimensionality
zbMATH Keywords: non-commutative ergodic theory
zbMATH Keywords: non-commutative Rokhlin type theorem
zbMATH Keywords: stability of outer conjugacy classes
zbMATH Keywords: the flow of weights
zbMATH Keywords: asymptotic centralizer
zbMATH Keywords: AF-algebra
zbMATH Keywords: index of subfactor
zbMATH Keywords: classification of subfactors
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1834432