A new approach to the similarity problem

Abstract

We say that a \(C^*\)-algebra \({\mathcal {A}}\) satisfies the similarity property ((SP)) if every bounded homomorphism \(u: {\mathcal {A}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\)-homomorphism and that a von Neumann algebra \({\mathcal {M}}\) satisfies the weak similarity property ((WSP)) if every \(\textrm{w}^*\)-continuous, unital and bounded homomorphism \(\pi : {\mathcal {M}}\rightarrow {\mathcal {B}}(H),\) where H is a Hilbert space, is similar to a \(*\)-homomorphism. The similarity problem is known to be equivalent to the question of whether every von Neumann algebra is hyperreflexive. We improve on that by introducing the following hypothesis (EP): Every separably acting von Neumann algebra with a cyclic vector is hyperreflexive. We prove that under (EP), every separably acting von Neumann algebra satisfies (WSP) and we pass from the case of separably acting von Neumann algebras to all \(C^*\)-algebras.