Moment problem for algebras generated by a nuclear space

TL;DR

This paper provides a criterion for the existence of Radon measures representing linear functionals on certain algebras generated by nuclear spaces, with applications to symmetric tensor algebras.

Contribution

It introduces a general criterion for representing measures on algebras generated by nuclear spaces, extending the moment problem to this setting.

Findings

Established a criterion for Radon measure existence on algebras generated by nuclear spaces.

Applied the criterion to symmetric tensor algebras of nuclear spaces.

Proved support containment in the space of characters with $q$-continuity.

Abstract

We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra $A$, which we assume to be generated by a vector space $V$ endowed with a Hilbertian seminorm $q$. Such a general criterion provides representing measures with support contained in the space of characters of $A$ whose restrictions to $V$ are $q-$continuous. This allows us in turn to prove existence results for the case when $V$ is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.