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

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2012
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Ghasemi, Mehdi
Marshall, Murray
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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.

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ISO 690GHASEMI, Mehdi, Salma KUHLMANN, Murray MARSHALL, 2012. Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras
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@unpublished{Ghasemi2012Appli-21264,
  year={2012},
  title={Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras},
  author={Ghasemi, Mehdi and Kuhlmann, Salma and Marshall, Murray}
}
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