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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.

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The answer to your first question is no: take $X=\ell_2$ and $Y = \ell_1^\ast$. I haven't really thought much about your second question. –  Philip Brooker Jun 18 '12 at 14:04
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The arrows seem to "go the wrong way" - an embedding from $X^{**}$ into $Y$ would, if weak-star continuous, give a quotient map from $Y_*$ onto $X^*$ –  user16299 Jun 18 '12 at 18:41
    
@Yemon: that crossed my mind when I read the question. –  Philip Brooker Jun 18 '12 at 19:39

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