Some remarks on commutative algebras of operators on Banach spaces

Abstract

Introduction. In this paper a series of propositions are given concerning commutative algebras of operators on a Banach space and more especially commutative algebras of scalar operators. A number of the results are known and due to W. G. Bade [1; 2] but different proofs are given here. The inspiration for this paper, both in the choice of the subject matter and of method, has largely been derived from [2; 7] and [10]. The material presented here is divided into four paragraphs. The first, which is introductory, contains various results on spectral families of measures. The principal propositions are contained in paragraphs 2 and 3. Theorem 1, proved in paragraph 2, is a generalisation of a theorem due to W. G. Bade [2], and almost all the other results of this same paragraph are more or less consequences of it. In paragraph 3 it is shown that, under certain conditions, an algebra of scalar operators can be identified, in a sense made precise below, with a von Neumann algebra. This fact makes it possible to reduce many results concerning algebras of scalar operators or a-complete boolean algebras of projections, in Banach spaces, to corresponding results in Hilbert spaces. Various remarks on spectral families of measures are made in paragraph 4.