The similarity problem and hyperreflexivity of von Neumann algebras

The similarity problem is one of the most famous open problems in the theory of $C^{\ast }$-algebras. We say that a $C^{\ast }$-algebra $\text{cl}A$ satisfies the similarity property ((SP) for short) if every bounded homomorphism $u\text{\hspace{0.17em}⁣}:\text{cl}A\to \text{cl}B(H)$ is similar to a $\ast$-homomorphism and that a von Neumann algebra $\text{cl}A$ satisfies the weak similarity property ((WSP) for short) if every ${\mathrm{w}}^{\ast }$-conitnuous unital and bounded homomorphism $u\text{\hspace{0.17em}⁣}:\text{cl}A\to \text{cl}B(H),$ where $H$ is a Hilbert space, is similar to a $\ast$-homomorphism. We prove that a von Neumann algebra $\text{cl}A$ satisfies (WSP) if and only if the algebras $\text{cl}{A}^{\mathrm{\prime }}\bar{\otimes }\text{cl}B({\mathrm{\ell }}^2(I))$ are hyperreflexive for all cardinals $I\mathrm{.}$ In the case in which $\text{cl}A$ is a separably acting von Neumann algebra we prove that it satisfies (WSP) if and only if the algebra $\text{cl}{A}^{\mathrm{\prime }}\bar{\otimes }\text{cl}B({\mathrm{\ell }}^2(\text{bb}N))$ 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 $C^{\ast }$-algebras satisfy (SP). Finally, we prove that the spatial tensor product $\text{cl}A\bar{\otimes }\text{cl}B,$ where $\text{cl}A$ is an injective von Neumann algebra and $\text{cl}B$ is a von Neumann algebra satisfying (WSP), also satisfies (WSP) and we provide an upper bound for the ${\text{w}}^{\ast }$-similarity degree $d_{\ast }(\text{cl}A\bar{\otimes }\text{cl}B)\mathrm{.}$