On Boolean Algebras of Projections and Prespectral Operators

Abstract

Complete and σ-complete Boolean algebras of projections in a complex Banach space were studied first by Bade [1] (see also [3; XVII.3]). The purpose of this paper is to find the appropriate extensions of several of his results to the more general case of G-complete and G-σ-complete Boolean algebras of projections, where G is a total linear manifold in the dual of the underlying Banach space. We shall prove e.g. that a Boolean algebra of projections is G-σ-complete if and only if it coincides with the range of a spectral measure of class G (Theorem 2), and we shall give a sufficient condition ensuring that the uniformly closed operator algebra generated by a G-σ-complete Boolean algebra B of projections coincides with the first commutant of B (Theorem 3). The new techniques will include the application of certain weak topologies and some of the duality theory of paired linear spaces as well as an idea due to Palmer [5].