Some extremely amenable groups
Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 273-278.

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.

Un groupe topologique G est extrêmement moyennable si toute action continue de G sur un espace compact possède un point fixe. En utilisant les techniques de concentration de mesure développées par Gromov et Milman, nous démontrons que le groupe des automorphismes d'un espace de Lebesgue avec une mesure diffuse est extrêmement moyennable s'il est muni de la topologie faible, mais ne l'est pas avec la topologie uniforme. Si M est une algèbre de von Neumann, nous montrons en utilisant un résultat de P. de la Harpe que M est approximativement de dimension finie si et seulement si son groupe unitaire (muni de la topologie forte) est le produit d'un groupe compact et d'un groupe extrêmement moyennable.

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DOI: 10.1016/S1631-073X(02)02218-5
Giordano, Thierry 1; Pestov, Vladimir 2

1 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada
2 School of Mathematical and Computing Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand
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Giordano, Thierry; Pestov, Vladimir. Some extremely amenable groups. Comptes Rendus. Mathématique, Volume 334 (2002) no. 4, pp. 273-278. doi : 10.1016/S1631-073X(02)02218-5. http://www.numdam.org/articles/10.1016/S1631-073X(02)02218-5/

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