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.

istrue is that when $C_0(K)^\ast$ is isometric to $\ell_1$ it is necessarily the case that $K$ is countable. There are isometric preduals of $\ell_1$ that are not isomorphic to a space $C_0(K)$; the first example is due to Benyamini and Lindenstrauss,A predual of $\ell_1$ which is not isomorphic to a $C(K)$ space, Israel J. Math.13(1972), 246-254. Other constructions have since been given, see Gasparis' preprint arxiv.org/pdf/1205.4317.pdf for a brief survey. Gasparis' paper contains a new approach to constructing an $\ell_$ $\endgroup$ – Philip Brooker Jun 24 '12 at 9:49