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arXiv · 1911.02872

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

Emanuele Casini·Enrico Miglierina·Łukasz Piasecki
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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 $K$.

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Emanuele Casini, Enrico Miglierina, Łukasz Piasecki. 2019-11-07. Weak$^*$ fixed point property and the space of affine functions. https://arxiv.org/abs/1911.02872

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