An invitation to von Neumann algebras
The contents of this introductory textbook on von Neumann algebras is as follows: Chapter 0 gives the very basics of von Neumann algebra theory: a discussion of the various (weak and strong) topologies, and the double commutant theorem. Chapter 1 reproduces the Murray - von Neumann classification of factors by exhibiting a dimension function. The topics of Chapter 2 are non-commutative integration (weights, Radon-Nikodym, conditional expectations), the GNS-construction, the proof of the Tomita- Takesaki theorem in a special case, and the KMS boundary condition. Chapter 3 explains Connes' classification of type III factors and in Chapter 4 the crossed product construction is used to produce examples of factors of all types, and to elucidate the structure of properly infinite von Neumann algebras. The book is intended for graduate students; as the title already indicates it is not a reference text but an introduction for non-experts. Prerequisite knowledge includes measure theory, elementary function theory (from Cauchy's point of view), and the basic principles of functional analysis. The necessary background from Hilbert space theory is shortly reviewed at the beginning, and some important facts about topological groups are added in an appendix. Exercises are scattered throughout the text. They are not only recommended as tests (at the end of a chapter as usual) but are used to keep the thing going, thereby preparing the ground for further study. The notes at the end give the main references and bibliographical credits for the presented material.
Murray - von Neumann classification of factors
Connes' classification of type III factors
scientific article; zbMATH DE number 3998533