Some isometric preduals of $\ell_1$ are of the form $C_0(K)$ where $K$ is countable. I am wondering whether this is a general rule.
Question: Is there a measure $\mu$ and a (preferably separable) Banach space $X$ without a subspace isomorphic to $c_0$ which has $X^*=L_1(\mu)$ isometrically?
I apologise for three questions in a such short period of time. Now I'll take my time.
EDIT: Corrected according to Philip's remarks.