Let $X$ and $Y$ be Banach spaces and suppose moreover that there is an isometric embedding of $X^{**}$ into $Y$. Assume moreover that $Y$ has the unique predual $Y_*$ up to isometry (like von Neumann algebras do have but this follows from some algebraic stuff).
Can we conclude that $X^*$ embeds isomorphically into $Y_*$? I guess not but this is the case for $X, Y$ being von Neumann algebras. What conditions should we impose on $X$ and $Y$ to obtain such a claim?
EDIT: I agree that there might be no good answer to this vaguely posed question. Feel free to delete it.