Weak$^*$ fixed point property and the space of affine functions

TL;DR

This paper characterizes when the dual of a separable Banach space lacks the weak$^*$ fixed point property for nonexpansive mappings, linking it to the presence of affine function spaces on Choquet simplices.

Contribution

It establishes a precise criterion involving quotients of the space of affine functions on a Choquet simplex for the failure of the weak$^*$ fixed point property in dual spaces.

Findings

Duals of certain Banach spaces lack the weak$^*$ fixed point property.

Presence of an isometric copy of an affine function space characterizes failure.

Quotients, not subspaces, determine the property failure.

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 $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings. Then, we show that the dual of a separable Lindenstrauss space $X$ fails the weak$^*$ 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 $K$.