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Amenability. (English) Zbl 0648.43001
Mathematical Surveys and Monographs, 29. Providence, RI: American Mathematical Society (AMS). xvii, 452 p. $ 90.00 (1988).
The present volume constitutes a precise and detailed account on all kinds of aspects related to the large field of amenability. The book is organized along the following chapters: 0) Basic concepts and problems of amenability theory. 1) Amenable locally compact groups and amenable semigroups. 2) The algebra of invariant means. 3) Free groups and the amenability of Lie groups. 4) Følner conditions. 5) Ergodic theorems for amenable locally compact groups. 6) Locally compact groups of polynomial growth. 7) Sizes of sets of invariant means. Five appendices give the necessary background: A) Nilpotent, solvable and semidirect product groups. B) Lie groups. C) Existence of Borel cross-sections. D) Mycielski’s theorem. E) On the density of the exponential map.
The volume parallels J.-P. Pier, Amenable locally compact groups (1984; Zbl 0621.43001). The author comments: “As one might expect, there is overlap between Professor Pier’s book and the present one. However, the points of view adopted in the two books are very different, as also are much of the contents”. Whereas, for instance, the book under review provides larger developments on von Neumann’s conjecture and the Banach- Ruziewicz problem, the other one gives more results related to function algebras. A large class of exercises with hints adds to the richness of Paterson’s book. Other recent works relevant to amenability are: S. Wagon, The Banach-Tarski Paradox (1985; Zbl 0569.43001); J.-P. Pier, Amenable Banach Algebras (1988).
Reviewer: J.-P.Pier

43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
43A07 Means on groups, semigroups, etc.; amenable groups
47H10 Fixed-point theorems
22D05 General properties and structure of locally compact groups