Spectral homomorphisms on a locally convex algebra Cb(X)

Let X be a completely regular Hausdorff space and Bo be the σ-algebra of Borel sets in X. Then the space Cb(X) (resp. B(Bo)) of all bounded continuous (resp. bounded Bo-measurable) complex functions on X, equipped with the natural strict topology β is a locally convex algebra with the jointly continuous multiplication. It is shown that every (β,ξ)-continuous homomorphism from Cb(X) to a complex sequentially complete locally convex algebra (A,ξ) that maps 1X to a unit 1 in A is a spectral homomorphism for a unique spectral measure m:Bo→A. As an application, we study continuous algebra homomorphisms from (Cb(X),β) to the algebra L(F) of all bounded linear operators on a Banach space F, equipped with the strong operator topology.