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Amenable groupoids. (English) Zbl 0960.43003
Monographies de l’Enseignement Mathématique. 36. Genève: L’Enseignement Mathématique; Université de Genève. 196 p. (2000).
More than twenty years ago the study of the subject was inaugurated by Zimmer’s examinations of discrete ergodic group actions and measured countable equivalence relations [R. J. Zimmer, J. Funct. Anal. 27, 350-372 (1978; Zbl 0391.28011)]. The next breakthrough was the characterization of such amenable equivalence relations by Connes, Feldman, and Weiss [A. Connes, J. Feldman and B. Weiss, J. Ergodic Theory Dyn. Syst. 1, 431-450 (1981; Zbl 0491.28018)]. Adams, Elliott and Giordano established new characterizations of Zimmer’s basic fixed point property in the case of a locally compact group [S. Adams, G. A. Elliott and T. Giordano, Trans. Am. Math. Soc. 344, 803-822 (1994; Zbl 0814.47009)].
The present monograph is a far reaching extension dealing with measured groupoids and locally compact groups, establishing equivalence for many characterizations modeled after the classical equivalent definitions of amenability. The authors mention that results concerning the Baum-Connes conjecture have been transposed to amenable groupoids.
The first chapter provides functional analytic preliminaries. The next one introduces the definition of amenability for locally compact groupoids, largely extending situations described by Greenleaf [F. P. Greenleaf, J. Funct. Anal. 4, 295-315 (1969; Zbl 0195.42301)] and Eymard [P. Eymard, Moyennes et représentations unitaires. Lect. Notes Math. 300. (Berlin 1972; Zbl 0249.43004)]. For locally compact groupoids, amenability is invariant with respect to equivalence of groupoids. Chapter 3 deals with amenability for measured groupoids. Classical properties, due to Day, Reiter, Godement, and Følner are carried over. Chapter 4 is about adaptations of Zimmer’s fixed point property. Chapter 5 establishes a large collection of properties of amenable groupoids, in particular combinatorial stabilities. Chapter 6 is about generalizations of fundamental amenability properties for operator algebras, e.g., weak containment of representations, injectivity, Leptin’s property concerning the Fourier algebra.
The volume testifies vitality and progress in harmonic analysis and operator theory; the exposition is clear, even in its most technical parts, and reasonably self-contained. We quote from Skandalis’ introduction: “Comprehensive, up-to-date, well written, elegant, this monograph will undoubtedly become a chief reference for the subject”.

43A07 Means on groups, semigroups, etc.; amenable groups
22D40 Ergodic theory on groups