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

TL;DR

This paper extends Jacobi's representation theorem to locally multiplicatively convex topological real algebras, characterizing closures of modules and representing positive linear functionals as integrals with respect to Radon measures.

Contribution

It generalizes the closure description of modules and the representation of positive linear functionals to broader topological algebra settings using Jacobi's theorem.

Findings

Closure of modules characterized by algebra homomorphisms

Positive linear functionals represented as Radon measures

Results hold for any locally multiplicatively convex topology

Abstract

Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $\rho$ be a seminorm on $A$ which satisfies $\rho(ab)\leq\rho(a)\rho(b)$. 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 $\rho$, for any integer $d\ge1$. We show that this closure is exactly the set of all elements $a\in A$ such that $\alpha(a)\ge0$ for every $\rho$-continuous $\mathbb{R}$-algebra homomorphism $\alpha : A \rightarrow \mathbb{R}$ with $\alpha(S)\subseteq[0,\infty)$, and that this result continues to hold when $\rho$ is replaced by any locally multiplicatively convex topology $\tau$ on $A$. We obtain a representation of any linear functional $L : A \rightarrow \reals$ which is continuous with respect to any such $\rho$ or $\tau$ 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.