Application of Jacobi's representation theorem to locally multiplicatively convex topological R-algebras

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2014
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Ghasemi, Mehdi
Marshall, Murray
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Journal of Functional Analysis. 2014, 266(2), pp. 1041-1049. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2013.09.001
Zusammenfassung

Let A be a commutative unital RR-algebra and let ρ be a seminorm on A which satisfies ρ(ab)⩽ρ(a)ρ(b)ρ(ab)⩽ρ(a)ρ(b). We apply T. Jacobi's representation theorem to determine the closure of a ∑A2d-module S of A in the topology induced by ρ, for any integer d⩾1d⩾1. We show that this closure is exactly the set of all elements a∈Aa∈A such that α(a)⩾0α(a)⩾0 for every ρ-continuous RR-algebra homomorphism α:A→Rα:A→R with α(S)⊆[0,∞)α(S)⊆[0,∞), and that this result continues to hold when ρ is replaced by any locally multiplicatively convex topology τ on A. We obtain a representation of any linear functional L:A→RL:A→R which is continuous with respect to any such ρ or τ and nonnegative on S as integration with respect to a unique Radon measure on the space of all real-valued RR-algebra homomorphisms on A, and we characterize the support of the measure obtained in this way.

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Fachgebiet (DDC)
510 Mathematik
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Positivity, sums of squares, sums of 2d-powers, locally multiplicatively convex topologies, norms and seminorms, continuous linear functionals, moment problem
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ISO 690GHASEMI, Mehdi, Salma KUHLMANN, Murray MARSHALL, 2014. Application of Jacobi's representation theorem to locally multiplicatively convex topological R-algebras. In: Journal of Functional Analysis. 2014, 266(2), pp. 1041-1049. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2013.09.001
BibTex
@article{Ghasemi2014Appli-26398,
  year={2014},
  doi={10.1016/j.jfa.2013.09.001},
  title={Application of Jacobi's representation theorem to locally multiplicatively convex topological R-algebras},
  number={2},
  volume={266},
  issn={0022-1236},
  journal={Journal of Functional Analysis},
  pages={1041--1049},
  author={Ghasemi, Mehdi and Kuhlmann, Salma and Marshall, Murray}
}
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    <dcterms:abstract xml:lang="eng">Let A be a commutative unital RR-algebra and let ρ be a seminorm on A which satisfies ρ(ab)⩽ρ(a)ρ(b)ρ(ab)⩽ρ(a)ρ(b). We apply T. Jacobi's representation theorem to determine the closure of a ∑A&lt;sup&gt;2d&lt;/sup&gt;-module S of A in the topology induced by ρ, for any integer d⩾1d⩾1. We show that this closure is exactly the set of all elements a∈Aa∈A such that α(a)⩾0α(a)⩾0 for every ρ-continuous RR-algebra homomorphism α:A→Rα:A→R with α(S)⊆[0,∞)α(S)⊆[0,∞), and that this result continues to hold when ρ is replaced by any locally multiplicatively convex topology τ on A. We obtain a representation of any linear functional L:A→RL:A→R which is continuous with respect to any such ρ or τ and nonnegative on S as integration with respect to a unique Radon measure on the space of all real-valued RR-algebra homomorphisms on A, and we characterize the support of the measure obtained in this way.</dcterms:abstract>
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