An intrinsic characterization of moment functionals in the compact case
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929638395084800 |
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| author | Infusino, Maria Kuhlmann, Salma Kuna, Tobias Michalski, Patrick |
| author_facet | Infusino, Maria Kuhlmann, Salma Kuna, Tobias Michalski, Patrick |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05630 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | An intrinsic characterization of moment functionals in the compact case Infusino, Maria Kuhlmann, Salma Kuna, Tobias Michalski, Patrick Functional Analysis 44A60, 28C20, 28C05 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. |
| title | An intrinsic characterization of moment functionals in the compact case |
| topic | Functional Analysis 44A60, 28C20, 28C05 |
| url | https://arxiv.org/abs/2204.05630 |