2. Non-unitarizable uniformly bounded group representations

Abstract

In this chapter, we introduce the space B(G) of coefficients of unitary representations on a discrete group G and a related space $T_p(G), (1 \leq p \leq \infty)$ of complex valued functions on G. We show that if $G = \mathbb{F}_{N}$, the free group on $N \geq 2$ generators, there are non-unitarizable uniformly bounded representations on G. More generally, this holds whenever G contains a subgroup isomorphic to $\mathbb{F}_{2}$. We give several related characterizations of amenable groups. Then we extend the method to the case of semi-groups. This allows us to produce (this time for $G = \mathbb{N}$) examples of power bounded operators which are not polynomially bounded.