Boolean algebras of projections and algebras of spectral operators

We show that, given a weak compactness condition which is always satisfied when the underlying space does not contain an isomorphic copy of c0, all the operators in the weakly closed algebra generated by the real and imaginary parts of a family of commuting scalar-type spectral operators on a Banach space will again be scalar-type spectral operators, provided that (and this is a necessary condition with even only two operators) the Boolean algebra of projections generated by their resolutions of the identity is uniformly bounded