The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable. (See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem.)
Definition: A dual Banach space $X^*$ is said to be $w^*$-HI if $\operatorname{dist} (S_Y,S_Z )= 0$ for every $Y$, $Z$ infinite dimensional $w^*$-closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$-indecomposable if it is not the direct sum of two infinite dimensional $w^*$-closed subspaces.
Question I: Does there exist non separable $X$ such that $X^*$ is $w^*$-HI?
Question II: For $X$ HI is the second dual $X^{**}$ $w^*$-inecomposable?
There exists $X$ HI such that $X^*$ is non separable and $X^{**}= X \oplus l^2(2^N)$.
$X^{**} = X \oplus l^2(2^N)$, not $X^{**}$= $X$ $\oplus$ $l^2(2^N)$$X^{**}$= $X$ $\oplus$ $l^2(2^N)$. (Notice the different appearance of symbols, and the difference in spacing.) I have edited accordingly. $\endgroup$