Reflexive operator algebras on Banach spaces

Abstract

In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e., it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant. Let A B.X/ denote a strongly closed algebra of operators on the Banach space X. Suppose that A has the property that each of its invariant subspaces has an invariant complement. If A contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property as defined below, we prove that A is reflexive in the sense that it contains all the operators which leave its closed invariant subspaces invariant (Theorem 15). In particular such an algebra is equal to its bicommutant A 00 (Corollary 22). The problem of whether a strongly closed algebra of operators with complemented invariant subspace lattice is reflexive started to be studied in the sixties. This problem is a generalization of the invariant subspace problem in operator theory. Arveson [1967] introduced a technique for studying the particular case of transitive algebras on Hilbert spaces, namely the strongly closed algebras of operators on Hilbert spaces that have no nontrivial closed invariant subspaces. He proved that every transitive algebra that contains a maximal abelian von Neumann algebra coincides with the full algebra B.X/ if X is a complex Hilbert space. Douglas and Pearcy [1972] extended the result of Arveson to the case of transitive operator algebras containing an abelian von Neumann algebra of finite multiplicity. Hoover [1973] extended the result of Douglas and Pearcy to the case of reductive operator algebras on Hilbert spaces that contain C. Peligrad and M. Peligrad were supported in part by a Charles Phelps Taft Memorial Fund grant. M. Peligrad was also supported by NSF grant DMS-1208237. MSC2010: primary 47A15, 47B48; secondary 47C05.