γ-Bounded representations of amenable groups

摘要

Let G be an amenable group, let X be a Banach space and let πG→B(X) be a bounded representation. We show that if the set {π(t)t∈G} is γ-bounded then π extends to a bounded homomorphism w:C*(G)→B(X) on the group C*-algebra of G. Moreover w is necessarily γ-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=Z, R or T, and/or when X has property (α).