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 ϱ: A → B(H_ϱ) of A on a Hilbert space H_ϱ, every invariant subspace of ϱ(A) is topologically complemented by another invariant subspace of ϱ(A), then A is similar to an abelian C^*-algebra.