Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras

Ghasemi, Mehdi; Kuhlmann, Salma; Marshall, Murray

Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras

Dateien

ghasemi_212648.pdfGröße: 413.07 KBDownloads: 136

Datum

2012

Autor:innen

Ghasemi, Mehdi

Kuhlmann, Salma

Marshall, Murray

Open Access-Veröffentlichung

Open Access Green

Sammlungen

Mathematik und Statistik

Publikationstyp

Preprint

Publikationsstatus

Published

Zusammenfassung

Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $\rho$ be a seminorm on $A$ which satisfies $\rho(ab)\leq\rho(a)\rho(b)$. We apply T. Jacobi's representation theorem to determine the closure of a $\sum A^{2d}$-module $S$ of $A$ in the topology induced by $\rho$, for any integer $d\ge1$. We show that this closure is exactly the set of all elements $a\in A$ such that $\alpha(a)\ge0$ for every $\rho$-continuous $\mathbb{R}$-algebra homomorphism $\alpha : A \rightarrow \mathbb{R}$ with $\alpha(S)\subseteq[0,\infty)$, and that this result continues to hold when $\rho$ is replaced by any locally multiplicatively convex topology $\tau$ on $A$. We obtain a representation of any linear functional $L : A \rightarrow \reals$ which is continuous with respect to any such $\rho$ or $\tau$ and non-negative on $S$ as integration with respect to a unique Radon measure on the space of all real valued $\reals$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.

Fachgebiet (DDC)

510 Mathematik

Zitieren

ISO 690

GHASEMI, Mehdi, Salma KUHLMANN, Murray MARSHALL, 2012. Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras

BibTex

@unpublished{Ghasemi2012Appli-21264,
  year={2012},
  title={Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras},
  author={Ghasemi, Mehdi and Kuhlmann, Salma and Marshall, Murray}
}

RDF

<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/21264">
    <dc:rights>terms-of-use</dc:rights>
    <dcterms:title>Application of Jacobi's Representation Theorem to locally multiplicatively convex topological real Algebras</dcterms:title>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-01-28T10:16:55Z</dc:date>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:creator>Kuhlmann, Salma</dc:creator>
    <dc:creator>Marshall, Murray</dc:creator>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <dcterms:issued>2012</dcterms:issued>
    <dc:contributor>Marshall, Murray</dc:contributor>
    <dspace:hasBitstream rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/21264/1/ghasemi_212648.pdf"/>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dcterms:abstract xml:lang="eng">Let $A$ be a commutative unital $\mathbb{R}$-algebra and let $\rho$ be a seminorm on $A$ which satisfies $\rho(ab)\leq\rho(a)\rho(b)$. We apply T. Jacobi's representation theorem to determine the closure of a $\sum A^{2d}$-module $S$ of $A$ in the topology induced by $\rho$, for any integer $d\ge1$. We show that this closure is exactly the set of all elements $a\in A$ such that $\alpha(a)\ge0$ for every $\rho$-continuous $\mathbb{R}$-algebra homomorphism $\alpha : A \rightarrow \mathbb{R}$ with $\alpha(S)\subseteq[0,\infty)$, and that this result continues to hold when $\rho$ is replaced by any locally multiplicatively convex topology $\tau$ on $A$. We obtain a representation of any linear functional $L : A \rightarrow \reals$ which is continuous with respect to any such $\rho$ or $\tau$ and non-negative on $S$ as integration with respect to a unique Radon measure on the space of all real valued $\reals$-algebra homomorphisms on $A$, and we characterize the support of the measure obtained in this way.</dcterms:abstract>
    <dcterms:hasPart rdf:resource="https://kops.uni-konstanz.de/bitstream/123456789/21264/1/ghasemi_212648.pdf"/>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2013-01-28T10:16:55Z</dcterms:available>
    <dc:creator>Ghasemi, Mehdi</dc:creator>
    <dc:language>eng</dc:language>
    <bibo:uri rdf:resource="http://kops.uni-konstanz.de/handle/123456789/21264"/>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dc:contributor>Ghasemi, Mehdi</dc:contributor>
    <dc:contributor>Kuhlmann, Salma</dc:contributor>
  </rdf:Description>
</rdf:RDF>

Universitätsbibliographie

Ja