Extreme amenability of \(L_0\), a Ramsey theorem, and Lévy groups
type: Item
instance of: scholarly article
title: Extreme amenability of \(L_0\), a Ramsey theorem, and Lévy groups
zbMATH Open document ID: 1172.22002
author: Ilijas Farah
author: Sławomir Solecki
published in: Journal of Functional Analysis
publication date: 2008-07-31T00:00:00Z
review text: A topological group is called extremely amenable if each of its continuous actions on a compact space has a fixed point. In this article, the authors show that \(L_0(\varphi,H)\) is extremely amenable for any diffused submeasure \(\varphi\) and any solvable compact group \(H\). This extends results of Herer-Christensen, and of Glasner and Furstenberg-Weiss. The argument in the proof is based on a new Ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods. Using this work, the authors give an example of a group which is extremely amenable and contains an increasing sequence of compact subgroups with dense union, but which does not contain a Lévy sequence of compact subgroups with dense union. This answers a question of \textit{V. Pestov} [Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman phenomenon, Univ. Lect. Ser., Vol. 40, Am. Math. Soc., Providence, RI (2006; Zbl 1123.37003)]. The authors also show that many Lévy groups have non-Lévy sequences, what answers another question of Pestov.
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 5306472
zbMATH Keywords: extremely amenable groups
zbMATH Keywords: \(L_{0}\)
zbMATH Keywords: submeasures
zbMATH Keywords: Ramsey theory
zbMATH Keywords: Borsuk-Ulam theorem
reviewed by: Dashan Fan
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.03.016
OpenAlex ID: W2015156320
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DOI: 10.1016/J.JFA.2008.03.016
description: scientific article; zbMATH DE number 5306472