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

Ilijas Farah, Sławomir Solecki

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)471-493
Number of pages23
JournalJournal of Functional Analysis
Issue number2
StatePublished - Jul 15 2008
Externally publishedYes


  • Borsuk-Ulam theorem
  • Extremely amenable groups
  • L
  • Ramsey theory
  • Submeasures

ASJC Scopus subject areas

  • Analysis


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