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Transfinite almost square Banach spaces

Volume 271 / 2023

Antonio Avilés, Stefano Ciaci, Johann Langemets, Aleksei Lissitsin, Abraham Rueda Zoca Studia Mathematica 271 (2023), 39-63 MSC: Primary 46B20; Secondary 46B03, 46B04, 46B26. DOI: 10.4064/sm220517-4-11 Published online: 12 January 2023

Abstract

Is is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose of relating them to the containment of isomorphic copies of $c_0(\kappa )$, where $\kappa $ is some uncountable cardinal. We also provide several examples and stability results for the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter 2 property and octahedral norms, we obtain a solution to an open question of Ciaci et al. [Israel J. Math. (online, 2022)].

Authors

  • Antonio AvilésDepartamento de Matemáticas
    Universidad de Murcia
    Campus de Espinardo
    30100 Murcia, Spain
    e-mail
  • Stefano CiaciInstitute of Mathematics and Statistics
    University of Tartu
    51009 Tartu, Estonia
    e-mail
  • Johann LangemetsInstitute of Mathematics and Statistics
    University of Tartu
    51009 Tartu, Estonia
    e-mail
  • Aleksei LissitsinInstitute of Mathematics and Statistics
    University of Tartu
    51009 Tartu, Estonia
    e-mail
  • Abraham Rueda ZocaDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail

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