Are Unitarizable Groups Amenable?
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2005, Birkhuser Verlag Basel/Switzerland. 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 Haagerups 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 such that there are bounded functions f: G B(Hi, Hi1) (Hi Hilbert) with H0 = , Hd = 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, Haagerups theorem implies that M2(G) M4(G).