Reflexivity and approximate reflexivity for bounded Boolean algebras of projections

Abstract

Let K be a compact Hausdorff space. It is proven that any bounded unital representation m of C( K) on a Banach space X has the property that the closure of m( C( K)) in the weak operator topology is a reflexive operator algebra. As a consequence, it is shown that if B is an arbitrary bounded Boolean algebra of bounded projections on a Banach space X, then AlgLat( B ) is the weak operator topology closure of the linear span of B . These generalize the work of several authors. As a corollary, an alternate proof of a theorem of Bade is obtained. In addition, approximate reflexivity results are obtained for the norm closures of m( C( K)) and span( B ).