On the ranks of single elements of reflexive operator algebras

For any completely distributive subspace lattice L on a real or complex reflexive Banach space and a positive integer n, necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of AlgL of rank n. Similar conditions are given for the existence of single elements of infinite rank. From this follows a relatively simple lattice-theoretic condition which characterises when every non-zero single element has rank one. Slightly stronger results are obtained for the case where L is finite, including the fact that every single element must then be of finite rank.