  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
$C^*$-algebra.
