Gamma-bounded representations of amenable groups
Functional Analysis
2010-03-09 v1
Abstract
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).
Cite
@article{arxiv.1003.1587,
title = {Gamma-bounded representations of amenable groups},
author = {Christian Le Merdy},
journal= {arXiv preprint arXiv:1003.1587},
year = {2010}
}