Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space
arXiv:math/9903085
Abstract
We establish a close link between the amenability of a unitary representation of a group (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_π,G)$, where $\s_\H$ is the unit sphere the Hilbert space of representation. We prove that is amenable if and only if either contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a -fixed point. Equivalently, the latter means that the -space $(\s_π,G)$ has the concentration property: every finite cover of the sphere $\s_π$ contains a set such that for every $\e>0$ the $\e$-neighbourhoods of the translations of by finitely many elements of always intersect. As a corollary, amenability of is equivalent to the existence of a -invariant mean on the uniformly continuous bounded functions on $\s_π$. As another corollary, a locally compact group is amenable if and only if for every strongly continuous unitary representation of in an infinite-dimensional Hilbert space the system $(\s_\H,G)$ has the property of concentration.
24 pages, LaTeX 2e