Amenable Representations and Dynamics of the Unit Sphere in an Infinite-dimensional Hilbert Space

We establish a close link between the amenability property of a unitary representation π of a groupG (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system (Sπ , G), where SH is the unit sphere the Hilbert space of representation. We prove that π is amenable if and only if either π contains a finite-dimensional subrepresentation or the maximal uniform compactification of Sπ has a G-fixed point. Equivalently, the latter means that the G-space (Sπ , G) has the concentration property: every finite cover of the sphere Sπ contains a set A such that for every ǫ > 0 the ǫneighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of π is equivalent to the existence of a Ginvariant mean on the uniformly continuous bounded functions on Sπ. 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 H the system (SH, G) has the property of concentration.