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

Type of Publication: | Journal article |

Author: | Ghasemi, Mehdi; Kuhlmann, Salma; Marshall, Murray |

Year of publication: | 2014 |

Published in: | Journal of Functional Analysis ; 266 (2014), 2. - pp. 1041-1049. - ISSN 0022-1236. - eISSN 1096-0783 |

DOI (citable link): | https://dx.doi.org/10.1016/j.jfa.2013.09.001 |

Summary: |
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
^{2d}-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. |

Subject (DDC): | 510 Mathematics |

Keywords: | Positivity, sums of squares, sums of 2d-powers, locally multiplicatively convex topologies, norms and seminorms, continuous linear functionals, moment problem |

Bibliography of Konstanz: | Yes |

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GHASEMI, Mehdi, Salma KUHLMANN, Murray MARSHALL, 2014. Application of Jacobi's representation theorem to locally multiplicatively convex topological R-algebras. In: Journal of Functional Analysis. 266(2), pp. 1041-1049. ISSN 0022-1236. eISSN 1096-0783. Available under: doi: 10.1016/j.jfa.2013.09.001

@article{Ghasemi2014Appli-26398, title={Application of Jacobi's representation theorem to locally multiplicatively convex topological R-algebras}, year={2014}, doi={10.1016/j.jfa.2013.09.001}, number={2}, volume={266}, issn={0022-1236}, journal={Journal of Functional Analysis}, pages={1041--1049}, author={Ghasemi, Mehdi and Kuhlmann, Salma and Marshall, Murray} }