Herz Schur multipliers and uniformly bounded representations of discrete groups (Q803861)

On a discrete group we consider the Herz-Schur multiplier algebra and we show that any of its elements is a coefficient of a representation of the group by bounded operators acting on some Hilbert space. In general this representation will not be uniformly bounded, but for Littlewood functions we construct a representation with a uniform bound. In fact in this case the uniform bound can be choosen as close to one as one likes. The characteristic function of the set of words of length one in a free group on countably many generators is an example of a Littlewood function not belonging to the Fourier-Stieltjes algebra and the corresponding representation is not unitarizable.