Dixmier's unitarizability problem for discrete groups

Theorem 1.1: a discrete group is amenable if and only if every uniformly bounded representation on every complex Hilbert space is similar to a unitary one. For every nonamenable $G$ and $\varepsilon>0$ there is a representation with $\sup_g\|\pi(g)\|\le1+\varepsilon$ not similar to a unitary one, on a separable space if $G$ is countable. This settles Dixmier's question for all discrete groups. Not shown: anything for nondiscrete locally compact groups, or a bound on similarity degree beyond what the theorem gives.

Theorem 1.1: a discrete group is amenable if and only if every uniformly bounded representation on every complex Hilbert space is similar to a unitary one. For every nonamenable $G$ and $\varepsilon>0$ there is a representation with $\sup_g\|\pi(g)\|\le1+\varepsilon$ not similar to a unitary one, on a separable space if $G$ is countable. This settles Dixmier's question for all discrete groups. Not shown: anything for nondiscrete locally compact groups, or a bound on similarity degree beyond what the theorem gives.