Complemented subspaces in a dual Banach space

Question

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?

Answer 1

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