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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Detalles Bibliográficos
Autores principales: Dimant, Verónica, Galicer, Daniel Eric
Publicado: 2012
Materias:
Aron-Berner extension
Extreme points
Integral polynomials
M-ideals
Symmetric tensor products
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221236_v262_n5_p1987_Dimant
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
id paper:paper_00221236_v262_n5_p1987_Dimant
record_format dspace
spelling paper:paper_00221236_v262_n5_p1987_Dimant2023-06-08T14:46:25Z Geometry of integral polynomials, M-ideals and unique norm preserving extensions Dimant, Verónica Galicer, Daniel Eric 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. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221236_v262_n5_p1987_Dimant 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, Verónica
Galicer, Daniel Eric
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.
author Dimant, Verónica
Galicer, Daniel Eric
author_facet Dimant, Verónica
Galicer, Daniel Eric
author_sort Dimant, Verónica
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
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221236_v262_n5_p1987_Dimant
http://hdl.handle.net/20.500.12110/paper_00221236_v262_n5_p1987_Dimant
work_keys_str_mv AT dimantveronica geometryofintegralpolynomialsmidealsanduniquenormpreservingextensions
AT galicerdanieleric geometryofintegralpolynomialsmidealsanduniquenormpreservingextensions
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