We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.

Casini, E., Miglierina, E., Piasecki, Ł., Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1, <<TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS>>, 2024; 63 (1): 39-51. [doi:10.12775/tmna.2023.009] [https://hdl.handle.net/10807/272908]

Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1

Miglierina, Enrico;
2024

Abstract

We provide a concrete isometric description of all the preduals of $\ell_1$ for which the standard basis in $\ell_1$ has a finite number of $w^*$-limit points. Then, we apply this result to give an example of an $\ell_1$-predual $X$ such that its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings (briefly, $w^*$-FPP), but $X$ does not contain an isometric copy of any hyperplane $W_{\alpha}$ of the space $c$ of convergent sequences such that $W_\alpha$ is a predual of $\ell_1$ and $W_\alpha^*$ lacks the $w^*$-FPP. This answers a question left open in the 2017 paper of the present authors.
2024
Inglese
Casini, E., Miglierina, E., Piasecki, Ł., Explicit models of ℓ_1-preduals and the weak* fixed point property in ℓ_1, <<TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS>>, 2024; 63 (1): 39-51. [doi:10.12775/tmna.2023.009] [https://hdl.handle.net/10807/272908]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/272908
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