Hyperreflexivity of von Neumann algebras and similarity of finitely generated $C^{\ast }$-algebras

Abstract

Let $A$ be a $C^{\ast }$-algebra. We say that $A$ satisfies the SP if every bounded homomorphism $A\to B(K)$, with $K$ a Hilbert space, is similar to a $\ast$-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.

Several proofs shortened, new material added and extraneous material removed