Operator algebras with a reduction property

Given a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A-module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property. We show that a C*-algebra has the total reduction property if and only if all its representations are similar to *-representations. The question of whether all C*-algebras have this property is the famous 'similarity problem' of Kadison. We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.