Banach spaces whose biduals are $L_{1}$ Let $X$ be a Banach space. If $X^{**}$ is linearly isometric to $L_{1}(\mu)$ for some $\sigma$-finite measue $\mu$, we shall say that $X$ is an $L_{1}$-pre-bidual.
Question 1. What are the examples of $L_{1}$-pre-bidual ?
Question 2. Are there any characterizations or even references about $L_{1}$-pre-biduals ?
Thank you !
 A: If $X$ is an infinite dimensional Banach space such that $X^{**}$ is isomorphic to $L^1(\mu)$ for some $\sigma$-finite measure, then $X^{**}$ is non-reflexive, separable and has DPP (Dunford-Pettis property) since reflexivity, separability and DPP are isomorphic properties. This is not possible (Banach spaces whose second conjugates are separable)
Edit: $L^1(\mu)$ is separable when $\mu$ is a $\sigma$-finite measure and $L^1(\mu)$ is a dual Banach space.
a. If $\mu$ is a $\sigma$-finite measure, then $L^1(\mu)$ is isometrically isomorphic to $L^1(\nu)$ for some probability measure, e.g., see Albiac & Kalton, "Topics in Banach Space Theory", Chapter 5.
b. If $\mu$ is a probability measure, then $L^1(\mu)$ is a dual space if and only if $\mu$ is purely atomic.
Proof. $(\Leftarrow)$ If $\mu$ is purely atomic, then $L^1(\mu)$ is isomorphic to $\ell^1(supp\mu)$, which is a dual space.
$(\Rightarrow)$ $L^1(\mu)$ is weakly compactly generated. Every weakly compactly generated dual Banach space has RNP, and $L^1(\mu)$ has RNP iff $\mu$ is purely atomic; see Diestel & Uhl, "Vector Measures", Section 7.7.7.
c. If $\mu$ is a finite purely atomic measure, then $supp\mu$ is countable. Thus, $\ell^1(supp\mu)$ is separable.
A: If $X$ is isometric to a space $L_1(m)$, then $X^{**}$ is isometric to a (highly nonseparable) $L_1$-space over some measure space $(\Omega, \Sigma, \mu)$, by the duality of abstract $L$- and $M$-spaces. The converse is also true, as proved by Grothendieck, see Theorem II.4.9 in the Lindenstrauss-Tzafriri Springer Lecture Notes.
