Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ||x||_p=τ(|x|^p)^{1/p}, p ≥ 1. The main results include the following. The un...

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Detalles Bibliográficos
Autores: Andruchow, Esteban, Chiumiento, Eduardo Hernan, Larotonda, Gabriel Andrés
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2010
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/111559
Acceso en línea:http://hdl.handle.net/11336/111559
Access Level:acceso abierto
Palabra clave:FINITE VON NEUMANN ALGEBRA
FINSLER METRIC
GEODESIC
HOMOGENEOUS SPACE
PATH METRIC SPACE
P-NORM
QUOTIENT METRIC
UNITARY GROUP
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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spelling Homogeneous manifolds from noncommutative measure spacesAndruchow, EstebanChiumiento, Eduardo HernanLarotonda, Gabriel AndrésFINITE VON NEUMANN ALGEBRAFINSLER METRICGEODESICHOMOGENEOUS SPACEPATH METRIC SPACEP-NORMQUOTIENT METRICUNITARY GROUPhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let M be a finite von Neumann algebra with a faithful trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ||x||_p=τ(|x|^p)^{1/p}, p ≥ 1. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d_{O,p}$,. For p ≥2, we prove that the distances d_p and d_{O , p} coincide. Based on this fact, we show that the metric space (O,d_p) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of UM with the p-norm.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata; ArgentinaFil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; ArgentinaFil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; ArgentinaAcademic Press Inc Elsevier Science2010-05info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/111559Andruchow, Esteban; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; Homogeneous manifolds from noncommutative measure spaces; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 365; 2; 5-2010; 541-5580022-247XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2009.11.024info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0022247X09009640info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T13:53:24Zoai:ri.conicet.gov.ar:11336/111559instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 13:53:25.114CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Homogeneous manifolds from noncommutative measure spaces
title Homogeneous manifolds from noncommutative measure spaces
spellingShingle Homogeneous manifolds from noncommutative measure spaces
Andruchow, Esteban
FINITE VON NEUMANN ALGEBRA
FINSLER METRIC
GEODESIC
HOMOGENEOUS SPACE
PATH METRIC SPACE
P-NORM
QUOTIENT METRIC
UNITARY GROUP
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
title_short Homogeneous manifolds from noncommutative measure spaces
title_full Homogeneous manifolds from noncommutative measure spaces
title_fullStr Homogeneous manifolds from noncommutative measure spaces
title_full_unstemmed Homogeneous manifolds from noncommutative measure spaces
title_sort Homogeneous manifolds from noncommutative measure spaces
dc.creator.none.fl_str_mv Andruchow, Esteban
Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author Andruchow, Esteban
author_facet Andruchow, Esteban
Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author_role author
author2 Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author2_role author
author
dc.subject.none.fl_str_mv FINITE VON NEUMANN ALGEBRA
FINSLER METRIC
GEODESIC
HOMOGENEOUS SPACE
PATH METRIC SPACE
P-NORM
QUOTIENT METRIC
UNITARY GROUP
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic FINITE VON NEUMANN ALGEBRA
FINSLER METRIC
GEODESIC
HOMOGENEOUS SPACE
PATH METRIC SPACE
P-NORM
QUOTIENT METRIC
UNITARY GROUP
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description Let M be a finite von Neumann algebra with a faithful trace τ. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ||x||_p=τ(|x|^p)^{1/p}, p ≥ 1. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d_{O,p}$,. For p ≥2, we prove that the distances d_p and d_{O , p} coincide. Based on this fact, we show that the metric space (O,d_p) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of UM with the p-norm.
publishDate 2010
dc.date.none.fl_str_mv 2010-05
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/111559
Andruchow, Esteban; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; Homogeneous manifolds from noncommutative measure spaces; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 365; 2; 5-2010; 541-558
0022-247X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/111559
identifier_str_mv Andruchow, Esteban; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; Homogeneous manifolds from noncommutative measure spaces; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 365; 2; 5-2010; 541-558
0022-247X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2009.11.024
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0022247X09009640
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15.811543

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