Amenable groups and measure concentration on spheres

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 ${\text{s}}_{\mathcal{H}}$ has a $G$-fixed point, that is, the pair $({\text{s}}_{\mathcal{H}},G)$ has the concentration property in the sense of Milman. Consequently, the maximal $U({\cal H})$-equivariantcompactificationofthesphereinaHilbertspace$\cal H$ has no fixed points, which answers a 1987 question by Milman. This is a version as of November 19, 1998, incorporating some revisions.