Abstract:
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.
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Documento: |
Artículo
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Título: | Geometry of integral polynomials, M-ideals and unique norm preserving extensions |
Autor: | Dimant, V.; Galicer, D.; García, R. |
Filiación: | Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires, Argentina CONICET, Argentina Departamento de Matemática Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, (C1428EGA) Buenos Aires, Argentina Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, (ES-06071) Badajoz, Spain
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Palabras clave: | Aron-Berner extension; Extreme points; Integral polynomials; M-ideals; Symmetric tensor products |
Año: | 2012
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Volumen: | 262
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Número: | 5
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Página de inicio: | 1987
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Página de fin: | 2012
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DOI: |
http://dx.doi.org/10.1016/j.jfa.2011.12.021 |
Título revista: | Journal of Functional Analysis
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Título revista abreviado: | J. Funct. Anal.
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ISSN: | 00221236
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CODEN: | JFUAA
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00221236_v262_n5_p1987_Dimant |
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Citas:
---------- APA ----------
Dimant, V., Galicer, D. & García, R.
(2012)
. Geometry of integral polynomials, M-ideals and unique norm preserving extensions. Journal of Functional Analysis, 262(5), 1987-2012.
http://dx.doi.org/10.1016/j.jfa.2011.12.021---------- CHICAGO ----------
Dimant, V., Galicer, D., García, R.
"Geometry of integral polynomials, M-ideals and unique norm preserving extensions"
. Journal of Functional Analysis 262, no. 5
(2012) : 1987-2012.
http://dx.doi.org/10.1016/j.jfa.2011.12.021---------- MLA ----------
Dimant, V., Galicer, D., García, R.
"Geometry of integral polynomials, M-ideals and unique norm preserving extensions"
. Journal of Functional Analysis, vol. 262, no. 5, 2012, pp. 1987-2012.
http://dx.doi.org/10.1016/j.jfa.2011.12.021---------- VANCOUVER ----------
Dimant, V., Galicer, D., García, R. Geometry of integral polynomials, M-ideals and unique norm preserving extensions. J. Funct. Anal. 2012;262(5):1987-2012.
http://dx.doi.org/10.1016/j.jfa.2011.12.021