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

Abstract.

We establish a close link between the amenability property 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 $({\Bbb S}_{\pi}, G)$, where ${\Bbb S}_{\cal H}$ 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 $({\Bbb S}_{\pi}$ has a G-fixed point. Equivalently, the latter means that the G-space $({\Bbb S}_{\pi}, G)$ has the concentration property: every finite cover of the sphere ${\Bbb S}_{\pi}$ contains a set A such that for every $\epsilon > 0$ the $\epsilon$-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 ${\Bbb S}_{\pi}$. 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 ${\cal H}$ the system $({\Bbb S}_{\cal H}, G)$ has the property of concentration.