Hyperreflexivity of Von Neumann Algebras and Similarity of Finitely Generated C^*-Algebras

Let A A be a C^* C ∗ -algebra. We say that A A satisfies the SP if every bounded homomorphism A→ B(K) A → B ( K ) , with K a Hilbert space, is similar to a $$*$$ ∗ -homomorphism. We introduce three hypotheses that relate to extending hyperreflexive algebras by projections. We prove that our third hypothesis is equivalent to every finitely generated C*-algebra satisfying the SP. We show that to prove that every von Neumann algebra is hyperreflexive it is enough to show that when one extends a hyperreflexive algebra by a single projection it remains hyperreflexive.