Reflexive Operator Algebras on Banach Spaces

TL;DR

This paper investigates conditions under which certain operator algebras on Banach spaces are reflexive, focusing on algebras containing specific Boolean algebras of projections with finite multiplicity.

Contribution

It establishes that unital strongly closed operator algebras with complemented invariant subspace lattices containing a finite uniform multiplicity Boolean algebra are reflexive.

Findings

Algebras with the specified Boolean algebra are reflexive.

Such algebras coincide with their bicommutant.

The results extend understanding of operator algebra reflexivity.

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.