  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).
