The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices.

Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. Must $X$ be isomorphic to $C(K)$ for some countable $K$?

Well, the classical Bourgain-Delbaen spaces are not good candidates because they have no copies of $c_0$, hence they are not isomorphic to a Banach lattice (a Banach lattice without a copy of $c_0$ is weakly sequentially complete and weakly sequentially Banach lattices are complemented in their biduals).

`$\ell_1$`

. – Bill Johnson Jun 29 '12 at 16:14