Geometric & Functional Analysis GAFA

, Volume 10, Issue 5, pp 1171–1201 | Cite as

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

  • V.G. Pestov


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.


Dynamical System Hilbert Space Unit Sphere Bounded Function Compact Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  • V.G. Pestov
    • 1
  1. 1.School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand, e-mail:

Personalised recommendations