Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras

Abstract

Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.