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 τ_s of simple convergence. We establish the integral representation (with respect to τ_s-Radon spectral measures -continuous unital algebra homomorphisms . In particular, if λ is a positive Radon measure on X and L_ϱ is a Banach function space in we study a homomorphism given by the equality , where M_u(f ) = uf for f ∈ L_ϱ.

Mathematics Subject Classification (2010):

• 46G10
• 47C5
• 47A67

Key words:

• Spaces of bounded continuous functions
• locally convex algebras
• strict topologies
• algebra homomorphisms
• spectral measures
• Banach function spaces
• scalar type operators