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A forgotten theorem of Pełczyński: $(\lambda +)$-injective spaces need not be $\lambda $-injective—the case $\lambda \in (1,2]$

Volume 268 / 2023

Tomasz Kania, Grzegorz Lewicki Studia Mathematica 268 (2023), 311-317 MSC: Primary 46B04; Secondary 46B25, 46E15, 54G05. DOI: 10.4064/sm220119-25-6 Published online: 8 September 2022

Abstract

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane that is $(2+\varepsilon )$-injective for every $\varepsilon \gt 0$, yet is not $2$-injective, and remarked in a footnote that Pełczyński had proved for every $\lambda \gt 1$ the existence of a $(\lambda + \varepsilon )$-injective space ($\varepsilon \gt 0$) that is not $\lambda $-injective. Unfortunately, no trace of the proof of Pełczyński’s result has been preserved. In the present paper, we establish that result for $\lambda \in (1,2]$ by constructing an appropriate renorming of $\ell _\infty $. This contrasts (at least for real scalars) with the case $\lambda = 1$ for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.

Authors

  • Tomasz KaniaMathematical Institute
    Czech Academy of Sciences
    Žitná 25
    115 67 Praha 1, Czech Republic
    and
    Institute of Mathematics
    and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail
    e-mail
  • Grzegorz LewickiInstitute of Mathematics
    and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail

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