Strict topologies and spectral measures

Abstract

Let X be a completely regular Hausdorff space. Then the space Cb(X) of all bounded continuous complex functions on X, equipped with the natural  strict topology is a locally convex algebra with the jointly continuous multiplication. Let Y be a barreled quasicomplete locally convex Hausdorff space and L(Y) denote the space of all continuous operators of Y into itself, equipped with the topology s of simple convergence. We establish the integral representation (with respect to s-Radon spectral measures m : Bo→ L(Y)) of (β, T_s)-continuous unital algebra homomorphisms T λ : C_b(χ)→  L(Y). In particular, if is a positive Radon measure on X and Lϱ is a Banach function space in L⁰(λ) we study a homomorphism Tλ : Cb(χ) → L(Y) given by the equality T_λ(u) := M_u for u ∈ C_b(X), where M_u (f) = uf for f ∈ Lϱ.

Key words: Spaces of bounded continuous functions, locally convex algebras, strict topologies, algebra homomorphisms, spectral measures, Banach function spaces, scalar type operators.