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Some extremely amenable groups

Thierry Giordano·Vladimir Pestov
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Abstract

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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Thierry Giordano, Vladimir Pestov. 2001-10-27. Some extremely amenable groups. https://arxiv.org/abs/math/0109138

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