Some extremely amenable groups

Details Some extremely amenable groups Some extremely amenable groups

2001-09-19

arxiv.org/abs/math/0109138v3

C.r. Acad. Sci. Paris, Ser. I 334 (2002), No. 4, 273-278.

Mathematics

Group Theory

7 pages, English with abridged French version

Scientific paper

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, 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.

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