On Reflexivity of Closed Unital Subalgebras in Locally Convex Spaces

Abstract

IntroductionIn this paper, we are interested in Bade's reflexivity theorem indicated in [2] in barrelled locally convex Hausdorff spaces. The Bade theorem says that a continuous linear operator T on a Banach space X belongs to the strongly closed algebra generated by a a-complete Boolean algebra B of projections if and only if T leaves invariant each B-invariant subspace of X. As a consequence, it is shown that each unital closed subalgebra of the span closure of an equicontinuous Boolean algebra of projections in the barrelled spaces is reflective. It extends Proposition 5.8 in [2].For the concepts and notations not explained in this paper we refer to the books [1], [7]. Throughout, X will denote a quasicomplete locally convex Hausdorff topological vector space. By L(X) we denote the space of continuous linear operators from X into X. The identity operator in X is denoted by I,X' will denote the continuous dual of X. The adjoint of an operator T G L(X) is denoted by T". Let B be an equicontinuous Boolean algebra of projections in the barrelled locally convex Hausdorff space X. It is known that if X is a barrelled space, then X' is cr(X', X)-quasicomplete, [7, page 148]. Let p be the family of all continuous semi-norms in X, then the topology in X is equal to the topology generated by p.