Geometry of integral polynomials, M-ideals and unique norm preserving extensions

Autores
Dimant, Veronica Isabel; Galicer, Daniel Eric; García, Ricardo
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φk: φ ∈ X∗, φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a nontrivial M-ideal in Y , then k,s εk,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in k,s εk,s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y , it is known that k εkX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in k εkY . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.
Fil: Dimant, Veronica Isabel. Universidad de San Andres; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Galicer, Daniel Eric. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: García, Ricardo. Universidad de Extremadura; España
Materia
Integral Polynomials
Symmetric Tensor Products
M-Ideals
Extreme Points
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:11336/19933
Acceder
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network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Geometry of integral polynomials, M-ideals and unique norm preserving extensionsDimant, Veronica IsabelGalicer, Daniel EricGarcía, RicardoIntegral PolynomialsSymmetric Tensor ProductsM-IdealsExtreme Pointshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φk: φ ∈ X∗, φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a nontrivial M-ideal in Y , then k,s εk,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in k,s εk,s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y , it is known that k εkX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in k εkY . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.Fil: Dimant, Veronica Isabel. Universidad de San Andres; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Galicer, Daniel Eric. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: García, Ricardo. Universidad de Extremadura; EspañaElsevier Inc2012-03info: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/11336/19933Dimant, Veronica Isabel; Galicer, Daniel Eric; García, Ricardo; Geometry of integral polynomials, M-ideals and unique norm preserving extensions; Elsevier Inc; Journal Of Functional Analysis; 262; 5; 3-2012; 1987-20120022-1236CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.jfa.2011.12.021info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022123611004605info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1108.3975info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T15:29:03Zoai:ri.conicet.gov.ar:11336/19933instacron: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:34982025-10-15 15:29:03.309CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Geometry of integral polynomials, M-ideals and unique norm preserving extensions
title Geometry of integral polynomials, M-ideals and unique norm preserving extensions
spellingShingle Geometry of integral polynomials, M-ideals and unique norm preserving extensions
Dimant, Veronica Isabel
Integral Polynomials
Symmetric Tensor Products
M-Ideals
Extreme Points
title_short Geometry of integral polynomials, M-ideals and unique norm preserving extensions
title_full Geometry of integral polynomials, M-ideals and unique norm preserving extensions
title_fullStr Geometry of integral polynomials, M-ideals and unique norm preserving extensions
title_full_unstemmed Geometry of integral polynomials, M-ideals and unique norm preserving extensions
title_sort Geometry of integral polynomials, M-ideals and unique norm preserving extensions
dc.creator.none.fl_str_mv Dimant, Veronica Isabel
Galicer, Daniel Eric
García, Ricardo
author Dimant, Veronica Isabel
author_facet Dimant, Veronica Isabel
Galicer, Daniel Eric
García, Ricardo
author_role author
author2 Galicer, Daniel Eric
García, Ricardo
author2_role author
author
dc.subject.none.fl_str_mv Integral Polynomials
Symmetric Tensor Products
M-Ideals
Extreme Points
topic Integral Polynomials
Symmetric Tensor Products
M-Ideals
Extreme Points
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 use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φk: φ ∈ X∗, φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a nontrivial M-ideal in Y , then k,s εk,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in k,s εk,s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y , it is known that k εkX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in k εkY . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.
Fil: Dimant, Veronica Isabel. Universidad de San Andres; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Galicer, Daniel Eric. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: García, Ricardo. Universidad de Extremadura; España
description We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is {±φk: φ ∈ X∗, φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a nontrivial M-ideal in Y , then k,s εk,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in k,s εk,s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y , it is known that k εkX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in k εkY . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.
publishDate 2012
dc.date.none.fl_str_mv 2012-03
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/11336/19933
Dimant, Veronica Isabel; Galicer, Daniel Eric; García, Ricardo; Geometry of integral polynomials, M-ideals and unique norm preserving extensions; Elsevier Inc; Journal Of Functional Analysis; 262; 5; 3-2012; 1987-2012
0022-1236
CONICET Digital
CONICET
url http://hdl.handle.net/11336/19933
identifier_str_mv Dimant, Veronica Isabel; Galicer, Daniel Eric; García, Ricardo; Geometry of integral polynomials, M-ideals and unique norm preserving extensions; Elsevier Inc; Journal Of Functional Analysis; 262; 5; 3-2012; 1987-2012
0022-1236
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.jfa.2011.12.021
info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022123611004605
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1108.3975
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 Inc
publisher.none.fl_str_mv Elsevier Inc
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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score 13.22299

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