Separable Lindenstrauss spaces whose duals lack the weak${}^{\ast }$ fixed point property for nonexpansive mappings

In this paper we study the $w^{\ast }$-fixed point property for nonexpansive mappings. First we show that the dual space $X^{\ast }$ lacks the $w^{\ast }$-fixed point property whenever $X$ contains an isometric copy of the space $c$. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in ${\mathrm{\ell }}_1$ space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces $X$ inducing the failure of $w^{\ast }$-fixed point property in $X^{\ast }$.