Mathematics – Functional Analysis
Scientific paper
2000-04-03
Israel J. Math. 127 (2002), 317-358; Corrigendum, Israel J. Math. 145 (2005), 375-379
Mathematics
Functional Analysis
The present version includes a Corrigendum (to the proof of Th. 3.2)
Scientific paper
In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of isometries of Urysohn's universal complete separable metric space $\mathbb U$, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The same is true if $\U$ is replaced with any generalized Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent theorem by Uspenskij that every topological group embeds into a topological group of the form $\Iso(U)$, our result implies that every topological group embeds into an extremely amenable group (one admitting an invariant multiplicative mean on bounded right uniformly continuous functions). By way of the proof, we show that every topological group is approximated by finite groups in a certain weak sense. Our technique also results in a new proof of the extreme amenability (fixed point on compacta property) for infinite orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type theorems for metric subspaces of Hilbert spaces and for spherical metric spaces from existing results on extreme amenability of infinite unitary groups and groups of isometries of Hilbert spaces.
No associations
LandOfFree
Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-222348