TITLE: Unique preduals up to (nonisometric) isomorphism?
QUESTION [6 upvotes]: It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra.  As other questions on here (for example, Isomorphisms of Banach Spaces  ) have shown, there are also Banach spaces $E$ with non-isomorphic (either isometric, or not) $E_1$ and $E_2$ with $E_1^*$ and $E_2^*$ both (isometrically) isomorphic to $E$.
However, I don't know the answer to the following:

Is there a Banach space $E$ such that, if $F$ is another Banach space with $T:F^*\rightarrow E^*$ a (not necessarily isometric) isomorphism, then necessarily $T$ is weak$^*$-continuous?

This is clearly true if $E$ is finite-dimensional, or more generally, reflexive.  But I cannot think of a non-reflexive example.  (I sort of think that the James space might be an example, but I can't see how to show this).

REPLY [9 votes]: No.  Take $E=F$ and perturb the identity on $E^*$ by a rank one operator that is not weak$^*$ continuous; such an operator existing is equivalent to $E$ being non reflexive.