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

Suppose that H is a complex Hilbert space and that 2(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 subset of B(H) is an abelian algebra with the property that given any bounded representation rho : A -> B(H-rho) of A on a Hilbert space H-rho, every invariant subspace of rho(A) is topologically complemented by another invariant subspace of rho(A), then A is similar to an abelian C*-algebra.