Weak${}^{\ast }$ fixed point property and the space of affine functions

Abstract

First we prove that if a separable Banach space $X$ contains an isometric copy of an infinite-dimensional space $A(S)$ of affine continuous functions on a Choquet simplex $S$, then its dual $X^{\ast }$ lacks the weak${}^{\ast }$ fixed point property for nonexpansive mappings. Then, we show that the dual of a separable Lindenstrauss space $X$ fails the weak${}^{\ast }$ fixed point property for nonexpansive mappings if and only if $X$ has a quotient isometric to some space $A(S)$. Moreover, we provide an example showing that "quotient" cannot be replaced by "subspace". Finally, it is worth to be mentioned that in our characterization the space $A(S)$ cannot be substituted by any space $\mathcal{C}(K)$ of continuous functions on a compact Hausdorff $K$.