Explicit models of ${\ell }_1$-preduals and the weak${}^{\ast }$ fixed point property in ${\ell }_1$

Abstract

We provide a concrete isometric description of all the preduals of ${\ell }_1$ for which the standard basis in ${\ell }_1$ has a finite number of $w^{\ast }$-limit points. Then, we apply this result to give an example of an ${\ell }_1$-predual $X$ such that its dual $X^{\ast }$ lacks the weak${}^{\ast }$ fixed point property for nonexpansive mappings (briefly, $w^{\ast }$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha }$ of the space $c$ of convergent sequences such that $W_{\alpha }$ is a predual of ${\ell }_1$ and $W_{\alpha }^{\ast }$ lacks the $w^{\ast }$-FPP. This answers a question left open in the 2017 paper of the present authors.

Dedicated to the Memory of Professor Kazimierz Goebel