The similarity problem and hyperreflexivity of von Neumann algebras

We say that a C∗-algebra A satisfies the similarity property ((SP)) if every bounded homomorphism u:A→B(H), where H is a Hilbert space, is similar to a ∗-homomorphism and that a von Neumann algebra A satisfies the weak similarity property ((WSP)) if every w∗-continuous, unital and bounded homomorphism u:A→B(H), where H is a Hilbert space, is similar to a ∗-homomorphism. We prove that a von Neumann algebra A satisfies (WSP) if and only if the algebras A′¯¯¯¯⊗B(ℓ2(J)) are hyperreflexive for all cardinals J. We also introduce the hypothesis (CHH): every hyperreflexive separably acting von Neumann algebra is completely hyperreflexive. We show that under (CHH), all C∗-algebras satisfy (SP). Finally, we prove that the spatial tensor product of an injective von Neumann algebra and a von Neumann algebra satisfying (WSP) also satisfies (WSP).