Metric geometry of partial isometries in a finite von Neumann algebra

Autores
Andruchow, Esteban
Año de publicación
2008
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).
Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina
Materia
Partial Isometry
Finite Algebra
Homogeneous Spaces
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/19448

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network_name_str CONICET Digital (CONICET)
spelling Metric geometry of partial isometries in a finite von Neumann algebraAndruchow, EstebanPartial IsometryFinite AlgebraHomogeneous Spaceshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; ArgentinaElsevier2008-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/19448Andruchow, Esteban; Metric geometry of partial isometries in a finite von Neumann algebra; Elsevier; Journal Of Mathematical Analysis And Applications; 337; 2; 12-2008; 1226-12370022-247XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022247X07005124info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2007.04.056info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:47:36Zoai:ri.conicet.gov.ar:11336/19448instacron: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:34982025-09-29 10:47:36.571CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Metric geometry of partial isometries in a finite von Neumann algebra
title Metric geometry of partial isometries in a finite von Neumann algebra
spellingShingle Metric geometry of partial isometries in a finite von Neumann algebra
Andruchow, Esteban
Partial Isometry
Finite Algebra
Homogeneous Spaces
title_short Metric geometry of partial isometries in a finite von Neumann algebra
title_full Metric geometry of partial isometries in a finite von Neumann algebra
title_fullStr Metric geometry of partial isometries in a finite von Neumann algebra
title_full_unstemmed Metric geometry of partial isometries in a finite von Neumann algebra
title_sort Metric geometry of partial isometries in a finite von Neumann algebra
dc.creator.none.fl_str_mv Andruchow, Esteban
author Andruchow, Esteban
author_facet Andruchow, Esteban
author_role author
dc.subject.none.fl_str_mv Partial Isometry
Finite Algebra
Homogeneous Spaces
topic Partial Isometry
Finite Algebra
Homogeneous Spaces
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).
Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina
description We study the geometry of the set Ip = v ∈ M: v∗v = p of partial isometries of a finite von Neumann algebra M, with initial space p (p is a projection of the algebra). This set is a C∞ submanifold of M in the norm topology of M. However, we study it in the strong operator topology, in which it does not have a smooth structure. This topology allows for the introduction of inner products on the tangent spaces by means of a fixed trace τ in M. The quadratic norms do not define a Hilbert–Riemann metric, for they are not complete. Nevertheless certain facts can be established: a restricted result on minimality of geodesics of the Levi-Civita connection, and uniqueness of these as the only possible minimal curves. We prove also that (Ip,dg) is a complete metric space, where dg is the geodesic distance of the manifold (or the metric given by the infima of lengths of piecewise smooth curves).
publishDate 2008
dc.date.none.fl_str_mv 2008-12
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/11336/19448
Andruchow, Esteban; Metric geometry of partial isometries in a finite von Neumann algebra; Elsevier; Journal Of Mathematical Analysis And Applications; 337; 2; 12-2008; 1226-1237
0022-247X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/19448
identifier_str_mv Andruchow, Esteban; Metric geometry of partial isometries in a finite von Neumann algebra; Elsevier; Journal Of Mathematical Analysis And Applications; 337; 2; 12-2008; 1226-1237
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/url/http://www.sciencedirect.com/science/article/pii/S0022247X07005124
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2007.04.056
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
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
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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