Let $Y$ be a complemented subspace in a dual Banach space $X$. *Is it true that $Y$ is itself isomorphic to a dual?*


This is the case of a $w^\*$-closed subspace $Y$, but a complemented subspace of $X^\*$ need not be $w^\*$-closed (for instance $Z^\*\subset Z^{\*\*\*}$ is complemented but never $w^\*$-closed unless $Z$ is reflexive). I think it is not true, but is there a simple counterexample?



Complemented subspaces in a dual Banach space
$L^1$ is complemented in the measure space $M([0,1])$, $L^1$ is not a dual space.


