Application of Jacobi’s 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. Jacobi’s representation theorem [10] to determine the closure of a ∑A-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 ∶ AÐ→ 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 Ð→ R which is continuous with respect to any such ρ or τ and non-negative 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.