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Invariant means on Abelian groups capture complementability of Banach spaces in their second duals

Volume 260 / 2021

Adam P. Goucher, Tomasz Kania Studia Mathematica 260 (2021), 91-101 MSC: Primary 43A07; Secondary 46B50. DOI: 10.4064/sm200706-15-1 Published online: 25 February 2021


Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $\ell _\infty (G, X)\to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq (2002) and the second-named author (2017), where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J. M. F. Castillo (private communication) that was also considered by Kania (2017). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.


  • Adam P. GoucherDepartment of Pure Mathematics
    and Mathematical Statistics
    University of Cambridge
    Cambridge, UK
  • Tomasz KaniaInstitute of Mathematics
    Czech Academy of Sciences
    Žitná 25
    115~67 Praha 1, Czech Republic
    Institute of Mathematics
    Jagiellonian University
    {Ł}ojasiewicza 6
    30-348 Kraków, Poland

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