Strict topologies and spectral measures

Abstract

Let X be a completely regular Hausdorff space. Then the space of all bounded continuous complex functions on X, equipped with the natural strict topology β is a locally convex algebra with the jointly continuous multiplication. Let be a barreled quasicomplete locally convex Hausdorff space and denote the space of all continuous operators of into itself, equipped with the topology τs of simple convergence. We establish the integral representation (with respect to τs -Radon spectral measures -continuous unital algebra homomorphisms . In particular, if λ is a positive Radon measure on X and Lϱ is a Banach function space in we study a homomorphism given by the equality , where Mu (f ) = uf for f ∈ Lϱ.