Some extremely amenable groups related to operator algebras and ergodic theory
| dc.creator | Giordano, Thierry | |
| dc.creator | Pestov, Vladimir | |
| dc.date | 2004-05-14 | |
| dc.date | 2006-02-06 | |
| dc.date.accessioned | 2026-07-07T08:26:59Z | |
| dc.date.available | 2026-07-07T08:26:59Z | |
| dc.description | A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal R$ on a standard Borel space is amenable if and only if the full group $[{\mathcal R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a nontrivial space in the sense of Gromov occuring in the automorphism groups of injective factors of type $III$. | |
| dc.description | 37 pages, LaTeX 2e, final version, to appear in J. Inst. Math. Jussieu | |
| dc.identifier | https://arxiv.org/abs/math/0405288 | |
| dc.identifier | http://arxiv.org/abs/math/0405288 | |
| dc.identifier | J. Inst. Math. Jussieu 6 (2007), 279-315. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137131 | |
| dc.subject | Operator Algebras | |
| dc.subject | 22A05; 22F50; 37A15; 43A07; 46L05; 46L10 | |
| dc.title | Some extremely amenable groups related to operator algebras and ergodic theory | |
| dc.type | text |