Amenable groups and measure concentration on spheres

Details Amenable groups and measure concentration on spheres Amenable groups and measure concentration on spheres

1998-10-29

arxiv.org/abs/math/9810168v2

C.r. Acad. Sc. Paris, Ser. I 328 (1999), 669--674.

Mathematics

Functional Analysis

17 pages, LaTeX 2e

Scientific paper

10.1016/S0764-4442(99)80232-8

It is proved that a discrete group $G$ is amenable if and only if for every unitary representation of $G$ in an infinite-dimensional Hilbert space $\cal H$ the maximal uniform compactification of the unit sphere $\s_{\cal H}$ has a $G$-fixed point, that is, the pair $(\s_{\cal H},G)$ has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariant compactification of the sphere in a Hilbert space $\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.

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