Boolean algebras of commuting projections

Abstract

In this note we give a simple characterization of completeness (in the sense of Bade [3]) of a Boolean algebra of projections on a Banach space (Theorem 1). The ideas we use are present in [3,4, 6, 18] and [17]. However the simplifications this characterization brings to some of the results seems to have escaped notice. Theorem2 is a strengthening of Bade's result that the uniformly closed algebra generated by a complete Boolean algebra of projections on X is closed with respect to the weak operator topology in L(X). In Theorem 3 we show that the range of a homomorphism of B = C(K) into L(X) is contained in the uniformly closed algebra generated by a complete Boolean algebra of projections, provided the action of B in X is weakly compact for each xEX. As a consequence of Theorem 1 the converse is always true. We also consider the relationship of the above to the regularity of the Arens extension of the module multiplication on a Banach module over B = C(K) (Remark 4). Lemma 1 needed for Theorem 2 answers a question of Kaijser [11] in the case of a Banach lattice with order continuous norm. For basic information on Boolean algebras, Banach spaces of continuous functions and Banach lattices we use [20] and [17].