An intrinsic characterization of moment functionals in the compact case

TL;DR

This paper provides a new intrinsic growth condition characterization for moment functionals on unital commutative real algebras, precisely identifying the compact support of their representing measures and offering formulas for singleton measures.

Contribution

It introduces a growth condition-based characterization of moment functionals that exactly determines the support of the representing measure in the compact case.

Findings

Exact identification of the compact support of the measure.

A formula for singleton measures in the support.

Necessary and sufficient conditions for finite support.

Abstract

We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is non-negative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.