Abstract
We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all \(\mathbb {R}\)-trees has the Plichko property witnessed by molecules and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of \(\ell _\infty \) containing \(c_0\) is an r-Lipschitz retract for \(r<2\), which implies in particular that \(\mathcal {F}(\ell _\infty )\) is not r-Plichko for \(r<2\).
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Acknowledgements
The authors are immensely grateful to Marek Cúth for his remarks and suggestions regarding the first version of this article, which have been key to improving the overall exposition. The authors would also like to thank Abraham Rueda-Zoca, Antonin Procházka and the anonymous referee for their very useful comments. This research was supported by grant PID2021-122126NB-C33 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. The second author’s research has been supported by PAID-01-19, by GA23-04776S and by the project SGS22/053/OHK3/1T/13.
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Guirao, A.J., Montesinos, V. & Quilis, A. On Projectional Skeletons and the Plichko Property in Lipschitz-Free Banach Spaces. Mediterr. J. Math. 20, 305 (2023). https://doi.org/10.1007/s00009-023-02505-z
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DOI: https://doi.org/10.1007/s00009-023-02505-z