If the dual spaces are isometrically isomorphic are the spaces isomorphic?

Solution 1:

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.