Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras

Abstract

Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $ρ$ be a seminorm on $A$ which satisfies $ρ(ab)\leqρ(a)ρ(b)$. We apply T. Jacobi's representation theorem to determine the closure of a $\sum A^{2d}$-module $S$ of $A$ in the topology induced by $ρ$, for any integer $d\ge1$. We show that this closure is exactly the set of all elements $a\in A$ such that $α(a)\ge0$ for every $ρ$-continuous $\mathbb{R}$-algebra homomorphism $α: A \rightarrow \mathbb{R}$ with $α(S)\subseteq[0,\infty)$, 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 $L : A \rightarrow \reals$ 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 $\reals$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.