TY - JOUR
T1 - Extreme amenability of L0, a Ramsey theorem, and Lévy groups
AU - Farah, Ilijas
AU - Solecki, Sławomir
N1 - Funding Information:
Research supported by NSERC (I. Farah) and NSF grant DMS-0400931 (S. Solecki). Corresponding author. E-mail addresses: [email protected] (I. Farah), [email protected] (S. Solecki).
PY - 2008/7/15
Y1 - 2008/7/15
N2 - 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.
AB - 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.
KW - Borsuk-Ulam theorem
KW - Extremely amenable groups
KW - L
KW - Ramsey theory
KW - Submeasures
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U2 - 10.1016/j.jfa.2008.03.016
DO - 10.1016/j.jfa.2008.03.016
M3 - Article
AN - SCOPUS:44449121814
SN - 0022-1236
VL - 255
SP - 471
EP - 493
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 2
ER -