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Grothendieck $C(K)$-spaces and the Josefson–Nissenzweig theorem

Volume 263 / 2023

Jerzy Kąkol, Damian Sobota, Lyubomyr Zdomskyy Fundamenta Mathematicae 263 (2023), 105-131 MSC: Primary 46E15; Secondary 28A33, 28C15, 46E27. DOI: 10.4064/fm218-6-2023 Published online: 20 November 2023


For a compact space $K$, the Banach space $C(K)$ is said to have the $\ell _1$-Grothendieck property if every weak$^*$ convergent sequence $\langle \mu _n\colon n\in \omega \rangle $ of functionals on $C(K)$ such that $\mu _n\in \ell _1(K)$ for every $n\in \omega $ is weakly convergent. Thus, the $\ell _1$-Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that $C(K)$ has the $\ell _1$-Grothendieck property if and only if there does not exist any sequence of functionals $\langle \mu _n\colon n\in \omega \rangle $ on $C(K)$, with $\mu _n\in \ell _1(K)$ for every $n\in \omega $, satisfying the conclusion of the classical Josefson–Nissenzweig theorem. We construct an example of a separable compact space $K$ such that $C(K)$ has the $\ell _1$-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces $K$ their Banach spaces $C(K)$ do not have the $\ell _1$-Grothendieck property.


  • Jerzy KąkolFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    61-614 Poznań, Poland
    Institute of Mathematics
    Czech Academy of Sciences
    115 67 Praha 1, Czech Republic
  • Damian SobotaKurt Gödel Research Center
    Faculty of Mathematics
    University of Vienna
    1090 Wien, Austria
  • Lyubomyr ZdomskyyInstitute of Discrete Mathematics and Geometry
    TU Wien
    1040 Wien, Austria

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