The Truncated Moment Problem for Unital Commutative R-Algebras

DOI10.7900/JOT.2021NOV26.2392arXiv2009.05115MaRDI QIDQ6507505

Mehdi Ghasemi, Salma Kuhlmann, Maria Infusino, Raúl E. Curto

Abstract: Let $A$ be a unital commutative $mathbbR-$algebra, $K$ a closed subset of the character space of $A$, and $B$ a linear subspace of $A$. For a linear functional $L:BomathbbR$, we investigate conditions under which $L$ admits an integral representation with respect to a positive Radon measure supported in $K$. When $A$ is equipped with a submultiplicative seminorm, we employ techniques from the theory of positive extensions of linear functionals to prove a criterion for the existence of such an integral representation for $L$. When no topology is prescribed on $A$, we identify suitable assumptions on $A$, $K$, $B$ and $L$ which allow us to construct a seminormed structure on $A$, so as to exploit our previous result to get an integral representation for $L$. Our main theorems allow us to extend some well-known results on the Classical Truncated Moment Problem, the Truncated Moment Problem for point processes, and the Subnormal Completion Problem for $2$--variable weighted shifts. We also analyze the relation between the Full and the Truncated Moment Problem in our general setting; we obtain a suitable generalization of Stochel's Theorem which readily applies to Full Moment Problems for localized algebras.

Mathematics Subject Classification ID

General theory of commutative topological algebras (46J05) Linear operator methods in interpolation, moment and extension problems (47A57) Random measures (60G57) Moment problems (44A60) Miscellaneous topics in measure theory (28E99) Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures (28C05) Polynomials and matrices (11C99)

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