The Dixmier unitarizability problem for group representations par Gilles Pisier (Texas A&M et UPMC - Institut de Mathématiques de Jussieu UMR 7586 CNRS-UPMC).
A uniformly bounded representation of a group on Hilbert space is called unitarizable if it is similar to a unitary one. A group G is called unitarizable if every uniformly bounded representation on it is unitarizable. In 1950, Dixmier (as well as Day independently) proved that amenable implies unitarizable and then asked whether the converse holds. We will review the history of this problem, describe several partial results and discuss recent progress due to Ozawa and Monod based on the Gaboriau-Lyons result that the free group F2 ”randomly” embeds in any non-amenable group.