Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space |
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Authors: | VG Pestov |
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Institution: | (1) Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1 Edmonton, AB, Canada |
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Abstract: | We establish a close link between the amenability property of a unitary representation p \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 Sp, G) ({\Bbb S}_{\pi}, G) , where \Bbb SH {\Bbb S}_{\cal H} is the unit sphere the Hilbert space of representation. We prove that p \pi is amenable if and only if either p \pi contains a finite-dimensional subrepresentation or the maximal uniform compactification of (\Bbb Sp ({\Bbb S}_{\pi} has a G-fixed point. Equivalently, the latter means that the G-space (\Bbb Sp, G) ({\Bbb S}_{\pi}, G) has the concentration property: every finite cover of the sphere \Bbb Sp {\Bbb S}_{\pi} contains a set A such that for every e > 0 \epsilon > 0 the e \epsilon -neighbourhoods of the translations of A by finitely many elements of G always intersect. As a corollary, amenability of p \pi is equivalent to the existence of a G-invariant mean on the uniformly continuous bounded functions on \Bbb Sp {\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 H {\cal H} the system (\Bbb SH, G) ({\Bbb S}_{\cal H}, G) has the property of concentration. |
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