Amenable locally compact groups. (English) Zbl 0621.43001

Pier, Jean-Paul

Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. X, 418 p. £49.80 (1984).

This book is a comprehensive monograph on the theory of amenable locally compact groups. The prerequisites for reading this are collected in the first chapter. Chapter 2 gives various fundamental characterizations of amenable locally compact groups, e.g., the common fixed point properties, the Day’s asymptotical invariance properties, the Reiter’s conditions, the Glicksberg-Reiter property, the Følner’s conditions, the weak containment property, the cohomological characterizations and so forth.
Chapter 3 gives examples of amenable groups and studies the class of amenable groups. Chapters 4 and 5 give further details on necessary and/or sufficient conditions for locally compact groups to be amenable. The final chapter gives some supplementary description of the class of amenable groups and briefly indicates various directions into which amenability properties have been generalized.
This book may facilitate further studies on the phenomenon of amenability, and the notes at the end of some sections and the bibliography may provide meaningful informations on the theory of amenability for us.
[This review replaces the one in Zbl 0597.43001.]

Reviewer: K.Sakai

MSC:

43-02	Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
22-02	Research exposition (monographs, survey articles) pertaining to topological groups
43A07	Means on groups, semigroups, etc.; amenable groups
22D05	General properties and structure of locally compact groups

Keywords:

invariant mean; algebras of functions; amenable locally compact groups; amenable groups; bibliography