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

Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $\rho$ be a seminorm on $A$ which satisfies $\rho (ab)\le \rho (a)\rho (b)$. We apply T. Jacobi’s representation theorem to determine the closure of a $\sum A^{2d}$-module$S$of$A$inthetopologyinducedby$\rho$,foranyinteger$d\ge1$\mathrm{.}Weshowthatthisclosureisexactlythesetofallelements$a\in A$suchthat$\alpha(a)\ge0$forevery$\rho$-continuous$$\mathbb{R}$$-algebra homomorphism $\alpha :A\to \mathbb{R}$ with $\alpha (S)\subseteq [0,\mathrm{\infty })$, 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 $L : A \rightarrow \reals$whichiscontinuouswithrespecttoanysuch$\rho$or$\tau$ and non-negative on $S$ as integration with respect to a unique Radon measure on the space of all real valued $\mathbb{R}$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.