Authors: Farah, Ilijas 
Solecki, Sławomir
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Extreme amenability of L0, a Ramsey theorem, and Lévy groups
Journal: Journal of Functional Analysis
Volume: 255
Issue: 2
First page: 471
Last page: 493
Issue Date: 15-Jul-2008
Rank: M21
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2008.03.016
We show that L0 (φ{symbol}, H) is extremely amenable for any diffused submeasure φ{symbol} and any solvable compact group H. This extends results of Herer-Christensen, and of Glasner and Furstenberg-Weiss. Proofs of these earlier results used spectral theory or concentration of measure. Our argument is based on a new Ramsey theorem proved using ideas coming from combinatorial applications of algebraic topological methods. Using this work, we 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 Pestov. We also show that many Lévy groups have non-Lévy sequences, answering another question of Pestov.
Keywords: Borsuk-Ulam theorem | Extremely amenable groups | L 0 | Ramsey theory | Submeasures
Publisher: Elsevier
Project: NSF, Grant DMS-0400931

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