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.



Separable $L_1$-predual
Zippin proved that every isometric $L\_1$ predual contains $c\_0$ isometrically. 


Zippin, M.
On some subspaces of Banach spaces whose duals are L1 spaces. 
Proc. Amer. Math. Soc. 23 1969 378–385.


