On the infinite-dimensional moment problem

TL;DR

This paper extends the moment problem to infinite-dimensional commutative algebras, providing integral representations for positive functionals and solving specific cases like symmetric algebras of vector spaces.

Contribution

It introduces a generalized Haviland theorem and new methods for representing positive functionals on infinite-dimensional algebras.

Findings

Integral representations for positive linear functionals established.

Solved the moment problem for symmetric algebras of vector spaces.

Developed new approaches for nuclear spaces and topological algebras.

Abstract

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra $A$. We define moment functionals on $A$ as linear functionals which can be written as integrals over characters of $A$ with respect to cylinder measures. Our main results provide such integral representations for $A_+$--positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application we solve the moment problem for the symmetric algebra $S(V)$ of a real vector space $V$. As a byproduct we obtain a new approaches to the moment problem on $S(V)$ for a nuclear space $V$ and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra $A$.