Spectral measures and the Bade reflexivity theorem

Let $\text{B}$ be a strongly equicontinuous Boolean algebra of projections on the quasi-complete locally convex space X and assume that the space L(X) of continuous linear operators on X is sequentially complete for the strong operator topology. Methods of integration with respect to spectral measures are used to show that the closed algebra generated by $\text{B}$ in L(X) consists precisely of those continuous linear operators on X which leave invariant each closed $\text{B}$-invariant subspace of X.