Amenable groups and measure concentration on spheres

Abstract

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 $\mathcal{H}$ 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 $U(\mathcal{H})$-equivariant compactification of the sphere in a Hilbert space $\mathcal{H}$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.

17 pages, LaTeX 2e

Cited by in corpus (2)

• Compactifications of topological groups
• Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space