On spectral characterizations of amenability
We show that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satisfied: the associated unitary representationπ X ofG intoL 2(X, μ) is weakly contained into the regular representationλ G and there exists aG-equivariant norm one projection fromL∞(X×X) ontoL∞(X). We give examples of ergodic discrete group actions which are not amenable, althoughπ X is weakly contained intoλ G.
KeywordsNormal Subgroup Compact Group Unitary Representation Closed Subgroup Regular Representation
Unable to display preview. Download preview PDF.
- [ADR]C. Anantharaman-Delaroche and J. Renault,Amenable Groupoids, Monographie de L’Enseignement Mathématique No36, Genève, 2000.Google Scholar
- [Di1]J. Dixmier,Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964.Google Scholar
- [Gri]R. I. Grigorchuk,Symmetrical random walks on discrete groups, inMulticomponent Random Systems (R. L. Dobrushin and Ya. G. Sinai, eds.), Advances in Probability and Related Topics, Vol.6, Marcel Dekker, New York, 1980, pp. 285–325.Google Scholar
- [Ne]A. Nevo,The spectral theory of amenable actions and invariants of discrete groups, in preparation.Google Scholar