Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space (Q1596305)

Let \(\pi\) be a unitary representation of a locally compact group \(G\) in a Hilbert space \(\mathcal H\). Then \(G\) acts on the unit sphere \({\mathcal S}_{\mathcal H}\) of the Hilbert space. The author establishes the link between the amenability of \(\pi\) (in the sense of Bekka) and the properties of the corresponding dynamical system \(({\mathcal S}_{\mathcal H},G,\pi)\). It is proved in this paper that the following statements are equivalent: (i) \(\pi\) is amenable. (ii) Either \(\pi\) contains a finite-dimensional subrepresentation or the maximal uniform compactification of \({\mathcal S}_{\mathcal H}\) has a \(G\)-fixed point. (iii) The dynamical system \(({\mathcal S}_{\mathcal H},G,\pi)\) has the concentration property: every finite cover of the sphere \({\mathcal S}_{\mathcal H}\) contains a set \(A\) such that for every \(\varepsilon > 0\) the \(\varepsilon\)-neighborhoods of the translations of \(A\) by finitely many elements of \(G\) always intersect. (iv) There exists a \(G\)-invariant mean on the uniformly continuous bounded functions on \({\mathcal S}_{\mathcal H}\). As a 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 dynamical system \(({\mathcal S}_{\mathcal H},G,\pi)\) has the concentration property.