Abelian, amenable operator algebras are similar to C∗-algebras

Abstract

Suppose that $H$ is a complex Hilbert space and that $\mathcal{B}\left(H\right)$ denotes the bounded linear operators on $H$. We show that every abelian, amenable operator algebra is similar to a $C^{\ast }$-algebra. We do this by showing that if $\mathcal{A}\subseteq \mathcal{B}\left(H\right)$ is an abelian algebra with the property that given any bounded representation $\varrho :\mathcal{A}\to \mathcal{B}\left(H_{\varrho }\right)$ of $\mathcal{A}$ on a Hilbert space $H_{\varrho }$, every invariant subspace of $\varrho \left(\mathcal{A}\right)$ is topologically complemented by another invariant subspace of $\varrho \left(\mathcal{A}\right)$, then $\mathcal{A}$ is similar to an abelian $C^{\ast }$-algebra.