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Journal of Operator Theory

Volume 90, Issue 1,  Summer 2023  pp. 223-261.

The truncated moment problem for unital commutative $\mathbb{R}$-algebras

Authors:  Raul E. Curto (1), Mehdi Ghasemi (2), Maria Infusino (3), and Salma Kuhlmann (4)
Author institution: (1) Department of Mathematics, University of Iowa, Iowa City, 52246, U.S.A.
(2) Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK, S7N 5E6, Canada
(3) Dipartimento di Matematica e Informatica, Universita degli Studi di Cagliari, Palazzo delle Scienze, Via Ospedale 72, 09124 Cagliari, Italy
(4) Fachbereich Mathematik und Statistik, Universitaet Konstanz, Universitaetstrasse 10, 78457 Konstanz, Germany

Summary:  We investigate when a linear functional $L$ defined on a linear subspace $B$ of a unital commutative real algebra $A$ admits an integral representation with respect to a positive Radon measure supported on a closed subset $K$ of the character space of $A$. We provide a criterion for the existence of such a representation for $L$ when $A$ is equipped with a submultiplicative seminorm. We then build on this result to prove our main theorem for $A$ not necessarily equipped with a topology. This allows us to extend well-known results on truncated moment problems.

Keywords:  Truncated moment problem, full moment problem, measure, integral representation, linear functional.

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