Hyperreflexivity of von Neumann algebras and similarity of finitely generated $C^*$-algebras

TL;DR

This paper investigates conditions under which finitely generated C*-algebras satisfy the similarity property, linking hyperreflexivity of von Neumann algebras to extensions by projections.

Contribution

It introduces three hypotheses related to hyperreflexive algebra extensions and proves their equivalence to finitely generated C*-algebras satisfying the SP.

Findings

Third hypothesis is equivalent to all finitely generated C*-algebras satisfying SP.

Hyperreflexivity of von Neumann algebras reduces to hyperreflexivity preservation under single projection extensions.

Provides a new approach to proving hyperreflexivity for von Neumann algebras.

Abstract

Let $A$ be a $C^*$-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 $*$-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.