A renorming characterization of Banach spaces containing ℓ1(κ)

  1. Antonio Avilés
  2. Gonzalo Martínez-Cervantes 2
  3. Rueda Zoca, Abraham 3
  1. 1 Universidad de Murcia
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    Universidad de Murcia

    Murcia, España

    ROR https://ror.org/03p3aeb86

    Localización geográfica de la organización Universidad de Murcia
  2. 2 Universitat d'Alacant
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    Universitat d'Alacant

    Alicante, España

    ROR https://ror.org/05t8bcz72

    Localización geográfica de la organización Universitat d'Alacant
  3. 3 Universidad de Granada
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    Universidad de Granada

    Granada, España

    ROR https://ror.org/04njjy449

    Localización geográfica de la organización Universidad de Granada
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Revista:
Publicacions matematiques

ISSN: 0214-1493

Año de publicación: 2023

Volumen: 67

Número: 0

Páginas: 601-609

Tipo: Artículo

DOI: 10.5565/PUBLMAT6722305 DIALNET GOOGLE SCHOLAR lock_openDDD editor

Otras publicaciones en: Publicacions matematiques

Resumen

A result of G. Godefroy asserts that a Banach space X contains an isomorphic copy of `1 if and only if there is an equivalent norm ||| · ||| such that, for every finite-dimensional subspace Y of X and every ε > 0, there exists x ∈ SX so that |||y+rx||| ≥ (1−ε)(|||y|||+|r|) for every y ∈ Y and every r ∈ R. In this paper we generalise this result to larger cardinals, showing that if κ is an uncountable cardinal, then a Banach space X contains a copy of `1(κ) if and only if there is an equivalent norm ||| · ||| on X such that for every subspace Y of X with dens(Y ) < κ there exists a norm-one vector x so that |||y + rx||| = |||y||| + |r| whenever y ∈ Y and r ∈ R. This result answers a question posed by S. Ciaci, J. Langemets, and A. Lissitsin, where the authors wonder whether the above statement holds for infinite successor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take ε = 0.

Referencias bibliográficas

  • S. Ciaci, J. Langemets, and A. Lissitsin, A characterization of Banach spaces containing `1(κ) via ball-covering properties, Israel J. Math. (2022). DOI: 10.1007/s11856-022-2363-x
  • M. Fabian, P. Habala, P. Hajek, V. Montesinos Santalucía, J. Pelant, and V. Zizler, “Functional Analysis and Infinite-Dimensional Geometry”, CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC 8, SpringerVerlag, New York, 2001. DOI: 10.1007/978-1-4757-3480-5
  • G. Godefroy, Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95(1) (1989), 1–15. DOI: 10.4064/sm-95-1-1-15
  • A. J. Guirao, A. Lissitsin, and V. Montesinos, Some remarks on the ballcovering property, J. Math. Anal. Appl. 479(1) (2019), 608–620. DOI: 10.1016/j.jmaa.2019.06.040
  • V. Kadets, V. Shepelska, and D. Werner, Thickness of the unit sphere, `1-types, and the almost Daugavet property, Houston J. Math. 37(3) (2011), 867–878.
  • J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323–326. DOI: 10.2307/1990758
  • B. Maurey, Types and l1-subspaces, in: “Texas Functional Analysis Seminar 1982–1983” (Austin, Tex.), Longhorn Notes, Univ. Texas Press, Austin, TX, 1983, pp. 123–137
  • J. D. Monk, Continuum cardinals generalized to Boolean algebras, J. Symbolic Logic 66(4) (2001), 1928–1958. DOI: 10.2307/2694986
  • H. P. Rosenthal, A characterization of Banach spaces containing l1, Proc. Nat. Acad. Sci. U.S.A. 71(6) (1974), 2411–2413. DOI: 10.1073/pnas.71.6.2411
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