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 $\kappa=2^\omega$?



Complemented subspaces of a dual Banach space
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".


