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Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$.

My question reads as follows:

Does there exist $\kappa$ for which there is $X\in\mathcal{B}(\kappa)$ such that for every $Y\in\mathcal{B}(\kappa)$ the dual space $X^\ast$ contains a complemented copy of the dual space $Y^\ast$?

What about $\kappa=\omega$ or $\kappa=2^\omega$?

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  • $\begingroup$ Did you consider $X=C(\{-1,1\}^\kappa)$? $\endgroup$
    – Onur Oktay
    Commented May 24, 2023 at 20:37
  • $\begingroup$ @OnurOktay, the dual $C(\{-1,1\}^\kappa)^\ast$ is weakly sequentially complete, so it cannot contain $\ell_\infty(\kappa)$, which is the dual of $\ell_1(\kappa)$ $\endgroup$
    – Damian Sobota
    Commented May 25, 2023 at 8:47

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The answer is positive for $\kappa = \omega$. The space $X$ is the $\ell_1$ sum of a sequence $(E_n)$ of finite dimensional spaces such that for every $\epsilon>0$ and every finite dimensional $E$, $E$ is $1+\epsilon$-isomorphic to one of the $E_n$. Basically the same argument works for every $\kappa$--use the $\ell_1$ sum of $\kappa$ copies of $X$. See my old paper "A complementably universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 nos. 3 and 4 (1972), 301–310".

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  • $\begingroup$ Thank you, Bill! $\endgroup$
    – Damian Sobota
    Commented Jun 2, 2023 at 16:58

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