Recall that a closed subsapce subspace $Y$ of a Banach space $X$ is weakly complemented if the set $$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$ is a complemented subsapce subspace of $ X^*$. For example, $c_0$ is a weakly complemented subsapce subspace of $l_{\infty}$.

Question: Is there a Banach space $X$ such that there is a weak$^*$-closed weak${}^*$-closed subspace $Y$ which is weakly complemented but not complemented in $X$.

Recall that a closed subsapce $Y$ of a Banach space $X$ is weakly complemented if the set $$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$ is a complemented subsapce of $ X^*$. For example, $c_0$ is a weakly complemented subsapce of $l_{\infty}$.

Question: Is there a Banach space $X$ such that there is a weak$^*$-closed subspace $Y$ which is weakly complemented but not complemented in $X$.