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

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 {±φ k: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, the...

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Autores principales: Dimant, V., Galicer, D., García, R.
Formato: JOUR
Materias:
Aron-Berner extension
Extreme points
Integral polynomials
M-ideals
Symmetric tensor products
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00221236_v262_n5_p1987_Dimant
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00221236_v262_n5_p1987_Dimant2023-10-03T14:27:16Z Geometry of integral polynomials, M-ideals and unique norm preserving extensions Dimant, V. Galicer, D. García, R. Aron-Berner extension Extreme points Integral polynomials M-ideals Symmetric tensor products 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 {±φ k: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 ⊗̂ εk,s k,sX (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ̂εk,s k,sY. 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 ⊗̂ εk kX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗̂ εk kY. 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. © 2011 Elsevier Inc. Fil:Dimant, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Galicer, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00221236_v262_n5_p1987_Dimant
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Aron-Berner extension
Extreme points
Integral polynomials
M-ideals
Symmetric tensor products
spellingShingle Aron-Berner extension
Extreme points
Integral polynomials
M-ideals
Symmetric tensor products
Dimant, V.
Galicer, D.
García, R.
Geometry of integral polynomials, M-ideals and unique norm preserving extensions
topic_facet Aron-Berner extension
Extreme points
Integral polynomials
M-ideals
Symmetric tensor products
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 {±φ k: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 ⊗̂ εk,s k,sX (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ̂εk,s k,sY. 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 ⊗̂ εk kX (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗̂ εk kY. 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. © 2011 Elsevier Inc.
format JOUR
author Dimant, V.
Galicer, D.
García, R.
author_facet Dimant, V.
Galicer, D.
García, R.
author_sort Dimant, V.
title Geometry of integral polynomials, M-ideals and unique norm preserving extensions
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
url http://hdl.handle.net/20.500.12110/paper_00221236_v262_n5_p1987_Dimant
work_keys_str_mv AT dimantv geometryofintegralpolynomialsmidealsanduniquenormpreservingextensions
AT galicerd geometryofintegralpolynomialsmidealsanduniquenormpreservingextensions
AT garciar geometryofintegralpolynomialsmidealsanduniquenormpreservingextensions
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