Closure of the cone of sums of 2d-powers in real topological algebras

Abstract

Let $R$ be a unitary commutative real algebra and $K\subseteq Hom(R,\mathbb{R})$, closed with respect to the product topology. We consider $R$ endowed with the topology ${\mathcal{T}}_K$, induced by the family of seminorms ${\rho }_{\alpha }(a):=|\alpha (a)|$, for $\alpha \in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\Vert a{\Vert }_K:={\sup }_{\alpha \in K}|\alpha (a)|$ for $a\in R$. If $K$ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, $\sum R^{2d}$, with respect to those two topologies is equal to $Psd(K):=\{a\in R:\alpha (a)\ge 0,\text{for all}\alpha \in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{R}[X_1,...,X_n]$ with $L(h^{2d})\ge 0$ for each $h\in R$ is integration with respect to a positive Borel measure supported on $K$. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.

References in corpus (3)

• The Moment Problem for Continuous Positive Semidefinite Linear functionals
• Closure of the cone of sums of 2d-powers in certain weighted ${\ell }_1$-seminorm topologies
• The K-moment problem for continuous linear functionals

Cited by in corpus (7)

• Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras
• Infinite dimensional moment problem: open questions and applications
• Moment problem for symmetric algebras of locally convex spaces
• An intrinsic characterization of moment functionals in the compact case
• On the determinacy of the moment problem for symmetric algebras of a locally convex space
• Projective limits techniques for the infinite dimensional moment problem
• On the topologies induced by a cone