Similarity Problems: Which Groups Are Unitarizable?

Abstract

This thesis covers some theory on similarity of group representations to unitary representations. We discuss the notion of amenability and give some classes of groups that are amenable. We then prove the Dixmier-Day theorem, that states that a locally compact group G is unitarizable if it is amenable. We also investigate the converse of this statement, which is still an open problem. We will give some statements where we make some assumptions on the similarity that are equivalent to amenability. We will also investigate when bounded algebra homomorphism A → B(H), where A is a C∗-algebra, are similar to a *-homomorphism. We will present connections between the unitarizability of groups and unitarizability of group C∗-algebras, and this will be useful for some results about the converse of the Dixmier-Day theorem. We will also investigate the notions of completely positive and completely bounded maps and prove Stinespring’s theorem for completely positive maps followed by Wittstock’s theorem for completely bounded maps. We then prove Haagerup’s theorem that states that unitarizability of homomorphisms is equivalent to the property of being completely bounded.