Are Unitarizable Groups Amenable?

We 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 $f\text{\hspace{0.17em}⁣}:G\to \text{bb}C$ such that there are bounded functions ${\xi }_i\text{\hspace{0.17em}⁣}:G\to B(H_i,H_{i-1})$ ($H_i$ Hilbert) with $H_0=\text{bb}C$, $H_d ={\bb C}$suchthat$$f(t_1t_2… t_d) = \xi_1(t_1) \xi_2(t_2)… \xi_d(t_d).\qquad \forall t_i\in GWe prove that if $G$ is a non-commutative free group, for any $d\ge 1$, we haveM_d(G)\not= M_{d+1}(G),$$ 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 $M_2(G)≠M_4(G)\mathrm{.}$