A new approach to the similarity problem

We say that a C^* C ∗ -algebra A A satisfies the similarity property ((SP)) if every bounded homomorphism u: A→ B(H), u : A → B ( H ) , where H is a Hilbert space, is similar to a $$*$$ ∗ -homomorphism and that a von Neumann algebra M M satisfies the weak similarity property ((WSP)) if every w^* w ∗ -continuous, unital and bounded homomorphism π : M→ B(H), π : M → 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^* C ∗ -algebras.