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

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