  In this paper we study the reflexivity of a unital strongly closed algebra of
operators with complemented invariant subspace lattice on a Banach space. We
prove that if such an algebra contains a complete Boolean algebra of
projections of finite uniform multiplicity and with the direct sum property,
then it is reflexive, i.e. it contains every operator that leaves invariant
every closed subspace in the invariant subspace lattice of the algebra. In
particular, such algebras coincide with their bicommutant.
