Application of Jacobiʼs representation theorem to locally multiplicatively convex topological R-algebras

Let A be a commutative unital R-algebra and let ρ be a seminorm on A which satisfies ρ(ab)⩽ρ(a)ρ(b). We apply T. Jacobiʼs representation theorem [10] to determine the closure of a ∑A2d-module S of A in the topology induced by ρ, for any integer d⩾1. We show that this closure is exactly the set of all elements a∈A such that α(a)⩾0 for every ρ-continuous R-algebra homomorphism α:A→R with α(S)⊆[0,∞), and that this result continues to hold when ρ is replaced by any locally multiplicatively convex topology τ on A. We obtain a representation of any linear functional L:A→R which is continuous with respect to any such ρ or τ and nonnegative on S as integration with respect to a unique Radon measure on the space of all real-valued R-algebra homomorphisms on A, and we characterize the support of the measure obtained in this way.