A new approach to the similarity problem

Abstract

We say that a $C^{\ast }$-algebra $\mathcal{A}$ satisfies the similarity property ((SP)) if every bounded homomorphism $u:\mathcal{A}\to \mathcal{B}(H)$, where $H$ is a Hilbert space, is similar to a $\ast$-homomorphism. We introduce the following hypothesis (EP). (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive. We prove that under (EP), all $C^{\ast }$-algebras satisfy (SP).

arXiv commentWe prove that under the hypothesis (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive, all C*-algebras satisfy the similarity property