Theory of operator algebras. II
type: Item
instance of: scholarly article
title: Theory of operator algebras. II
zbMATH Open document ID: 1059.46031
author: Masamichi Takesaki
published in: Encyclopaedia of Mathematical Sciences
publication date: 2002-11-24T00:00:00Z
review text: The fundamental work on the theory of operator algebras was first started by John von Neumann and F. J. Murray in the 1930's. By an operator algebra we mean a closed, not necessarily self-adjoint, subalgebra of \(B(H)\) for some Hilbert space \(H\). Operator algebras are, in a certain sense, a generalization of the algebra of complex matrices. This is the second volume of an advanced textbook written by one of the most active researchers in the theory of operator algebras and is useful for graduate students and specialists. The first volume is an account of 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)]. The third volume is focused on structure analysis of approximately finite dimensional factors and their automorphism groups, cf. \textit{M. Takesaki} [Theory of operator algebras III. (Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-Commutative Geometry. 127(VIII). Berlin: Springer)(2003; Zbl 1059.46032)]. In the second volume which is well organized, one can find an investigation of structure of von Neumann algebras of type III and their automorphism groups and consists of seven chapters (6-12), notation index, and subject index, and also jointly with vol. III 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. There are referred to some publications listing merely in the bibliography of the first volume. In the following paragraphs we briefly describe the chapters of th…
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1834431
zbMATH Keywords: Hilbert algebra
zbMATH Keywords: Tomita algebra
zbMATH Keywords: direct integral
zbMATH Keywords: weight
zbMATH Keywords: semi-cyclic representation
zbMATH Keywords: Plancherel weight
zbMATH Keywords: Fourier algebra
zbMATH Keywords: modular automorphism group
zbMATH Keywords: centralizer of a weight
zbMATH Keywords: The Connes cocycle derivative
zbMATH Keywords: standard form of a von Neumann algebra
zbMATH Keywords: measurable operator
zbMATH Keywords: integral for a trace
zbMATH Keywords: bimodule
zbMATH Keywords: spatial derivative
zbMATH Keywords: relative tensor product
zbMATH Keywords: conditional expectation
zbMATH Keywords: operator valued weight
zbMATH Keywords: crossed product
zbMATH Keywords: dual weight
zbMATH Keywords: duality for crossed products by abelian groups
zbMATH Keywords: equivariant disintegration
zbMATH Keywords: induced covariant system
zbMATH Keywords: abelian automorphism group
zbMATH Keywords: spectral analysis
zbMATH Keywords: Connes spectrum
zbMATH Keywords: derivations
zbMATH Keywords: inner automorphism
zbMATH Keywords: von Neumann algebra of type III
zbMATH Keywords: factor of types III\(_{\lambda}, 0<\lambda<1\)
zbMATH Keywords: factor of type III\(_{\infty}\)
zbMATH Keywords: the flow of weights
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1834431