On Musielak-Orlicz Sequence Spaces with an Asymptotic $\ell_\infty$ dual

We investigate Mushielak-Orlicz sequence spaces $\ell_\Phi$ with a dual $\ell_\Phi^{*}$, which is stabilized asymptotic $\ell_\infty$ with respect to the unit vector basis. We give a complete characterization of the bounded relatively weakly compact subsets $K\subset\ell_\Phi$. We prove that $\ell_\Phi$ is saturated with asymptotically isometric copies of $\ell_1$ and thus $\ell_\Phi$ fails the fixed point property for closed, bounded convex sets and non--expansive (or contractive) maps on them.