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 $\rho$ 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 $A$ in the topology induced by $\rho$, for any integer $d\ge 1$. We show that this closure is exactly the set of all elements $a\in A$ such that $\alpha (a)\ge 0$ for every $\rho$-continuous $\mathbb{R}$-algebra homomorphism $\alpha :A\to \mathbb{R}$ 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 $L:A\to \mathbb{R}$ 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 $\mathbb{R}$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.

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• The Moment Problem for Continuous Positive Semidefinite Linear functionals
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