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

We show that L-0(phi, H) is extremely amenable for any diffused submeasure phi 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 Levy sequence of compact subgroups with dense union. This answers a question of Pestov. We also show that many Levy groups have non-Levy sequences, answering another question of Pestov. (c) 2008 Elsevier Inc. All rights reserved.