Are unitarizable groups amenable?

Abstract

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 Md(G) which is formed of all the functions f: G → ℂ such that there are bounded functions f: G → B(Hi, Hi−1) (Hi Hilbert) with H0 = ℂ, Hd = ℂ such that (formula presented) We prove that if G is a non-commutative free group, for any d ≥ 1, we have (formula presented) and hence there are elements of Md(G) which are not coefficients of uniformly bounded representations. In the case d = 2, Haagerup’s theorem implies that M2(G) ≠ M4(G).