Gamma-bounded representations of amenable groups

Details Gamma-bounded representations of amenable groups Gamma-bounded representations of amenable groups

2010-03-08

arxiv.org/abs/1003.1587v1

Mathematics

Functional Analysis

Scientific paper

Let G be an amenable group, let X be a Banach space and let \pi : G --> B(X) be a bounded representation. We show that if the set {\pi(t) : t \in G} is gamma-bounded then \pi extends to a bounded homomorphism w : C*(G) --> B(X) on the group C*-algebra of G. Moreover w is necessarily gamma-bounded. This extends to the Banach space setting a theorem of Day and Dixmier saying that any bounded representation of an amenable group on Hilbert space is unitarizable. We obtain additional results and complements when G is equal to either the real numbers, the integers or the unit circle, and/or when X has property (\alpha).

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