Some Problems concerning Reflexive Operator Algebras

We discuss below some problems concerning a certain class of algebras of operators on complex Banach space. Each algebra of the class arises from a lattice of subspaces of the underlying space (in a way that will soon be made precise) and most of the problems are of the fonn: find conditions, additional to those specified a priori, on the lattice of subspaces, which are both necessary and sufficient for the corresponding algebra of operators to have a certain, specified, algebraic and/or topological property. It is more accurate to say, then, that these problems concern lattices of subspaces of certain types. Naturally, all the problems have partial solutions some of which will be described. While the interested reader can find complete proofs elsewhere, some brief proofs are included for the purpose of illustration. Most of these problems have arisen (in some cases, have re-surfaced) in joint work with S. Argyros and M.S. Lambrou.