Weak* fixed point property and the space of affine functions

Casini, Emanuele; Miglierina, Enrico; Piasecki, Łukasz

doi:10.1090/proc/15327

Weak$^*$ fixed point property and the space of affine functions
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by Emanuele Casini, Enrico Miglierina and Łukasz Piasecki PDF

Proc. Amer. Math. Soc. 149 (2021), 1613-1620 Request permission

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 $L_1$-predual $X$ fails the weak$^*$ fixed point property for nonexpansive mappings if and only if $X$ has a quotient isometric to some infinite-dimensional space $A(S)$. Moreover, we provide an example showing that “quotient” cannot be replaced by “subspace”. Finally, it is worth mentioning 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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