[Quelques groupes extrêmement moyennables]
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.
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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@article{CRMATH_2002__334_4_273_0,
author = {Giordano, Thierry and Pestov, Vladimir},
title = {Some extremely amenable groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {273--278},
publisher = {Elsevier},
volume = {334},
number = {4},
year = {2002},
doi = {10.1016/S1631-073X(02)02218-5},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02218-5/}
}
TY - JOUR AU - Giordano, Thierry AU - Pestov, Vladimir TI - Some extremely amenable groups JO - Comptes Rendus. Mathématique PY - 2002 SP - 273 EP - 278 VL - 334 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02218-5/ DO - 10.1016/S1631-073X(02)02218-5 LA - en ID - CRMATH_2002__334_4_273_0 ER -
%0 Journal Article %A Giordano, Thierry %A Pestov, Vladimir %T Some extremely amenable groups %J Comptes Rendus. Mathématique %D 2002 %P 273-278 %V 334 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02218-5/ %R 10.1016/S1631-073X(02)02218-5 %G en %F CRMATH_2002__334_4_273_0
Giordano, Thierry; Pestov, Vladimir. Some extremely amenable groups. Comptes Rendus. Mathématique, Tome 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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