Reflexive operator algebras on Banach spaces

Florence Merlevède, Costel Peligrad and Magda Peligrad

Vol. 267 (2014), No. 2, 451–464

DOI: 10.2140/pjm.2014.267.451

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.

Keywords

operator algebras, invariant subspace lattice, Boolean algebra of projections, spectral operator

Mathematical Subject Classification 2010

Primary: 47A15, 47B48

Secondary: 47C05

Milestones

Received: 3 April 2012

Accepted: 8 October 2013

Published: 11 May 2014