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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!

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Isomorphic according to which topology? The norm? –  Christopher A. Wong Nov 21 '12 at 18:50
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$\ell_1$ has at least $\aleph_1$ many mutually nonisomorphic isometric preduals, e.g. the spaces $C(K)$ with $K$ countable (see my remarks at mathoverflow.net/questions/1380/isomorphisms-of-banach-spaces/…). Dropping the isometric condition, you can get $2^{\aleph_0}$ many isomorphic preduals of $\ell_1$ by considering the $\mathcal{L}_\infty$ spaces of Bourgain and Delbaen (look up `Bourgain-Delbaen space'). –  Philip Brooker Nov 21 '12 at 21:58

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Some things might be said beyond the ones that you mention, but probably not a lot. As an example, look at this answer, where an example is provided of separable $X$, non-separable $Y$, with $X^*=Y^*$.

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