Commutants of reflexive algebras and classification of completely distributive subspace lattices

Let L be a subspace lattice on a normed space X containing a nontrivial comparable element. If T commutes with all the operators in AlgL, then there exists a scalar lambda such that (T - lambdaI)(2) = 0. Furthermore, we classify the class of completely distributive subspace lattices into subclasses called Type I-(n), Type II(n) and Type III, respectively. It is shown that nontrivial nests or, more generally, completely distributive subspace lattices with a comparable element are Type I-(1), and that nontrivial atomic Boolean subspace lattices are Type II(0).