Representation Theory on Locally Compact Groups

In this project we try to reach one of the famous problems, namely similarity problems. Itcan be stated as follows: Is every uniformly bounded continuous representation of locally compact group G is unitarizable? This problem is open for a while until, in 1955, Ehrenpreeis and Mautner gave counterexample for G = SL2(R). In the positive direction, the most general result seems to be Dixmiers theorem which says that if G is amenable then the above problem is affirmative. But the converse remains an open problem. It can be stated as, if all uniformly bounded continuous representations of a group G are uniterizable is G necessarily amenable