Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space
| dc.creator | Pestov, Vladimir G. | |
| dc.date | 1999-03-15 | |
| dc.date | 1999-08-25 | |
| dc.date.accessioned | 2026-07-07T08:27:01Z | |
| dc.date.available | 2026-07-07T08:27:01Z | |
| dc.description | We establish a close link between the amenability of a unitary representation $π$ of a group $G$ (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 $π$ is amenable if and only if either $π$ contains a finite-dimensional subrepresentation or the maximal uniform compactification of $\s_π$ has a $G$-fixed point. Equivalently, the latter means that the $G$-space $(\s_π,G)$ has the concentration property: every finite cover of the sphere $\s_π$ contains a set $A$ such that for every $\e>0$ the $\e$-neighbourhoods 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 $G$-invariant mean on the uniformly continuous bounded functions on $\s_π$. 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 $\mathcal H$ the system $(\s_\H,G)$ has the property of concentration. | |
| dc.description | 24 pages, LaTeX 2e | |
| dc.identifier | https://arxiv.org/abs/math/9903085 | |
| dc.identifier | http://arxiv.org/abs/math/9903085 | |
| dc.identifier | Geom. Funct. Anal. 10 (2000), 1171-1201. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137141 | |
| dc.subject | Functional Analysis | |
| dc.subject | Representation Theory | |
| dc.subject | 22D10, 43A07, 43A65, 46C05, 54H20 | |
| dc.title | Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space | |
| dc.type | text |