The similarity problem and hyperreflexivity of von Neumann algebras

Abstract

The similarity problem is one of the most famous open problems in the theory of $C^*$-algebras. We say that a $C^*$-algebra $\cl A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\colon \cl A\to \cl B(H)$ is similar to a $*$-homomorphism and that a von Neumann algebra $\cl A$ satisfies the weak similarity property ((WSP) for short) if every $\mathrm{w}^*$-conitnuous unital and bounded homomorphism $u\colon \cl A\to \cl B(H),$ where $H$ is a Hilbert space, is similar to a $*$-homomorphism. We prove that a von Neumann algebra $\cl A$ satisfies (WSP) if and only if the algebras $\cl A^{\prime}\bar \otimes \cl B(\ell^2(I))$ are hyperreflexive for all cardinals $I.$ In the case in which $\cl A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\cl A^\prime \bar \otimes \cl B(\ell^2(\bb{N}))$ is hyperreflexive. We also introduce the hypothesis {\bf (CHH)}: Every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under {\bf (CHH)}, all $C^*$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\cl A\bar \otimes \cl B,$ where $\cl A$ is an injective von Neumann algebra and $\cl B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the $\text{w}^*$-similarity degree $d_{*}(\cl A\bar \otimes \cl B).$