When do two quasi-Banach spaces with identical dual spaces have equivalent norms?

Question

Let $X$ and $Y$ be two quasi-Banach spaces such that the dual spaces satisfy $X^*=Y^*$.

I want to know if there are some conditions that imply $X=Y$ (in the sense of equivalent norms).

Answer 1

I suspect that there are no good conditions. For a quasi-Banach space $X$ the dual $X^*$ is a dual of a Banach envelope $\hat X$ of $X$ which is a Banach space whose unit ball is a convex hull of the unit ball of $X$. So for every quasi-Banach space $X$ there exist two different quasi-Banach spaces $X_1$ and $X_2$ (and many more) such that $X_1^*=X_2^*=X^*$ isometrically. Assuming $X^*\neq 0$ it suffices to put $X_1=\hat X$ and $X_2=\hat X\oplus L_{1/2}$. Possibly there always exists a $Z$ so one can take $X_1=X$ and $X_2=X\oplus Z$ where $Z$ is a quasi-Banach space with $Z^*=0$ and $X\oplus Z$ not isomorphic to $X$.