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

TL;DR

This paper characterizes the closure of sums of 2d-powers in real topological algebras, showing it equals the set of elements nonnegative on a certain set, and explores implications for positive linear functionals and measure representation.

Contribution

It establishes the closure of the cone of sums of 2d-powers in specific topologies and links positive linear functionals to measures supported on K, extending classical results.

Findings

Closure of sums of 2d-powers equals nonnegative elements on K.

Positive linear functionals correspond to measures supported on K.

Provides conditions for continuity of linear functionals in these topologies.

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 $\|a\|_K:=\sup_{\alpha\in K}|\alpha(a)|$ 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 $Psd(K):=\{a\in R:\alpha(a)\geq 0,\textrm{for all}\alpha\in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{R}[X_1,...,X_n]$ with $L(h^{2d})\ge0$ 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.