Let $X$, $Y$ be Banach spaces such that the duals $X^\ast$ and $Y^\ast$ are isometrically isomorphic. Are $X$ and $Y$ necessarily isomorphic?

The answer to the question whether $X$ and $Y$ are automatically also isometrically isomorphic is no as the example $c$, $c_0$ which both have dual $\ell^1$ but are not isometrically isomorphic shows.

  • $\begingroup$ See this. $\endgroup$ – David Mitra Jan 22 '14 at 10:02
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    $\begingroup$ Let me point out that if you only demand $X^*$ and $Y^*$ to be isomorphic, a relatively easy counter example is at hand. $L_\infty[0,1]$ and $\ell_\infty$ are isomorphic. But $L_1[0,1]$ and $\ell_1$ are not. $\endgroup$ – David Mitra Jan 22 '14 at 10:47

Remark 4.5.3 in Kalton and Albiac's Topics in Banach Space Theory states:

" ... since $C(K)^*$ is isometric to $\ell_1$ for every countable compact metric space $K$, the Banach space $\ell_1$ is isometric to the dual of many nonisomorphic Banach spaces."

With regard to the above, from H. P. Rosenthal's article in The Handbook of the Geometry of Banach Spaces, Vol 2, a result of Bessaga and Pełczyński is given:

Let $K$ be an infinite countable compact metric space.

$\ \ \ $(a) $C(K)$ is isomorphic to $C(\omega^{\omega^\alpha}+)$ for some countable ordinal $\alpha\ge0$.

$\ \ \ $(b) If $0\le\alpha<\beta<\omega_1$, then $C(\omega^{\omega^\alpha}+)$ is not isomorphic to $C(\omega^{\omega^\beta}+)$

See also this post at MathOverflow which gives an example of a separable Banach space $X$ and a non-separable Banach space $Y$ such that $X^*$ and $Y^*$ are isometrically isomorphic.


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