Are Unitarizable Groups Amenable?

Minor corrections and clarificationsWe give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group $G$ are similar to unitary ones, is the group amenable? In \S 5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prove a refinement of this result involving a generalization of the class of Herz-Schur multipliers, namely the class $M_d(G)$ which is formed of all the functions $f\colon G\to {\bb C}$ such that there are bounded functions $\xi_i\colon G\to B(H_i, H_{i-1})$ ($H_i$ Hilbert) with $H_0 = {\bb C}$, $H_d ={\bb C}$ such that $$f(t_1t_2... t_d) = \xi_1(t_1) \xi_2(t_2)... \xi_d(t_d).\qquad \forall t_i\in G$$ We prove that if $G$ is a non-commutative free group, for any $d\ge 1$, we have $$M_d(G)\not= M_{d+1}(G),$$ and hence there are elements of $M_d(G)$ which are not coefficients of uniformly bounded representations. In the case $d=2$, Haagerup's theorem implies that $M_2(G)\not= M_{4}(G).