Structure of spectral measures on locally convex spaces

Abstract

0. Introduction. Though spectral measures - i.e. countably additive idempotent-valued set functions defined on a a-algebra of sets with values in ?(E) form the nominal topic of this paper, it can be looked on as having for its twin objects of study the spectral operators and the Boolean algebras of projections on locally convex spaces, since both are intimately connected with spectral measures - the operators with their resolutions of the identity, the Boolean algebras (if complete in a suitable sense) with the measures, defined on the Stone spaces of the algebras, whose values are the projections making up the algebras. Such operators and algebras, acting on Banach spaces, have received a great deal of attention since the operators were introduced and exhaustively investigated in the six papers [1], [2], [7], [11], [19] and [20] and the same service performed for the Boolean algebras in [3] and [4]. The concepts of spectral operator and Boolean algebra of projections are not (at least a priori) connected with normability of the topology of the vector space on which the operator or algebra acts; the definitions usually given for these objects acting on Banach spaces can be extended to general locally convex spaces in a perfectly straightforward way, thus yielding the objects of study of this paper. Presumably the Banach-space techniques which establish the fundamental properties of the associated spectral measures could be imitated and the corresponding fundamental properties of these operators and algebras deduced. In order to derive our principal results, however, we have been obliged to investigate these concepts from a geometrical, topological and order-theoretic point of view, thus continuing the line of research begun in [17]. Aside from our principal results, a consequence of adopting this point of view is a fairly immediate determination of the limits to which many rather deep theorems concerning spectral measures acting on Banach spaces can be generalized for measures acting on locally convex spaces, coupled with immediate and natural proofs of the generalizations and thus a simplification of technique even for Banach spaces. The main theorems of ??2 and 4, however, represent indications of the restrictedness rather than the generality of the concept of spectral measure, and thus the restrictedness