Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space

We establish a close link between the amenability of a unitary representation $\pi$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_\pi,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that $\pi$ is amenable if and only if either $\pi$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_\pi$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_\pi,G)$ has the concentration property: every finite cover of the sphere $\s_\pi$ contains a set $A$ such that for every $\e>0$ the $\e$-neighbourhoods of the translations of $A$ by finitely many elements of $G$ always intersect. As a corollary, amenability of $\pi$ is equivalent to the existence of a $G$-invariant mean on the uniformly continuous bounded functions on $\s_\pi$. As another corollary, a locally compact group $G$ is amenable if and only if for every strongly continuous unitary representation of $G$ in an infinite-dimensional Hilbert space $\mathcal H$ the system $(\s_\H,G)$ has the property of concentration.