Gamma-bounded representations of amenable groups

TL;DR

This paper proves that gamma-bounded representations of amenable groups on Banach spaces extend to bounded homomorphisms on the group C*-algebra, generalizing unitarizability results from Hilbert spaces.

Contribution

It extends the unitarizability theorem for bounded representations of amenable groups from Hilbert spaces to Banach spaces under gamma-boundedness conditions.

Findings

Gamma-boundedness implies extension to C*(G)

Extension preserves gamma-boundedness

Results specialized for G = real numbers, integers, or circle

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).