Dixmier's problem and similarity property for locally compact quantum groups

A locally compact group G has the similarity property if every uniformly bounded representation is similar to a unitary representation. A well-known result of Dixmier states that all amenable groups have the similarity property. Moreover, he asked whether this property is equivalent to amenability. In this talk, I will review the history and all the known results about the similarity property and Dixmier's problem for locally compact groups and locally compact quantum groups. I will show that the similarity property holds for all dual quantum groups of QSIN quantum groups which covers all the known results. This talk is based on joint work with Sang-Gyun Youn.