Extreme amenability of L0, a Ramsey theorem, and Lévy groups

Abstract

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.