Weak-star topologies from different preduals can be incomparable ($\sigma ({\ell }^1,c_0),\sigma ({\ell }^1,c)$)

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 ${\alpha }_1{\gamma }_{\infty }+{\sum }_{n\ge 2}{\alpha }_n{\gamma }_{n-1}$. The functional $\alpha \mapsto {\alpha }_1$ is continuous only for the first topology: $e_1-e_n\to 0$ for the second but its values remain one. Conversely $\alpha \mapsto \sum {\alpha }_n$ is continuous only for the second: $e_n\to 0$ 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.