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 §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 : G → C such that there are bounded functions ξ i : G → B(H i , H i−1) (H i Hilbert) with H 0 = C, H d = C such that f (t 1 t 2. .. t d) = ξ 1 (t 1)ξ 2 (t 2). .. ξ d (t d). ∀ t i ∈ G We prove that if G is a non-commutative free group, for any d ≥ 1, we have M d (G) = 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).