Suppose $X$ and $Y$ are Banach spaces with duals $X^*$ and $Y^*$. If we know $X^*$ and $Y^*$ are isomorphic, what can we say about $X$ and $Y$?
One trivial thing is that they can be isometrically embedded into the same Banach space $X^{**}$. Another is that they can be isometrically embedded into the same $C(S)$, where $S$ is a compact Hausdorff space.
But I am guessing much more can be implied about $X$ and $Y$? Am I right?
Thanks!