= Weak-star topologies from different preduals can be incomparable
{title2=$\sigma(\ell^1,c_0),\quad\sigma(\ell^1,c)$}
The same <Banach space> can acquire incomparable <weak-star topologies> from different isometric preduals. Pair $\ell^1$ with $c_0$ by $\sum\alpha_n\beta_n$ and with the space $c$ of convergent sequences by $\alpha_1\gamma_\infty+\sum_{n\geq2}\alpha_n\gamma_{n-1}$. The functional $\alpha\mapsto\alpha_1$ is continuous only for the first topology: $e_1-e_n\to0$ for the second but its values remain one. Conversely $\alpha\mapsto\sum\alpha_n$ is continuous only for the second: $e_n\to0$ for the first but its values remain one. More generally, comparable weak-star topologies arising from isometric preduals agree on the common closed <unit ball>, since its identity map is a <continuous bijection> from a <compact space> to a <Hausdorff space> by the <Banach-Alaoglu theorem>.
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