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RETRACTED - Compact reduction in Lipschitz-free spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Spaces with richer structures

Published online by Cambridge University Press:  08 December 2021

Ramón J. Aliaga Open the ORCID record for Ramón J. Aliaga [Opens in a new window] ,
Camille Noûs ,
Colin Petitjean Open the ORCID record for Colin Petitjean [Opens in a new window]  and
Antonín Procházka Open the ORCID record for Antonín Procházka [Opens in a new window]
Ramón J. Aliaga
Affiliation:
Universitat Politècnica de València, Instituto Universitario de Matemática Pura y Aplicada, Camino de Vera S/N, 46022, Valencia, Spain (raalva@upvnet.upv.es)
Camille Noûs
Affiliation:
Laboratoire Cogitamus, 16, route de Gray, 25030, Besançon, France (camille.nous@cogitamus.fr)
Colin Petitjean
Affiliation:
LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vallée, France (colin.petitjean@univ-eiffel.fr)
Antonín Procházka
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR-6623, 16, route de Gray, 25030, Besançon Cedex, France (antonin.prochazka@univ-fcomte.fr)
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Abstract

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

Keywords

Lipschitz-free spaceLipschitz functionLipschitz lifting propertySchur propertyapproximation propertyweak sequential completenessDunford–Pettis property

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 54E50: Complete metric spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1683 - 1699
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Albiac, F., Ansorena, J. L., Cúth, M. and Doucha, M.. Lipschitz free spaces isomorphic to their infinite sums and geometric applications, arXiv:2005.06555.Google Scholar
Aliaga, R. J. and Pernecká, E.. Supports and extreme points in Lipschitz-free spaces. Rev. Mat. Iberoam. (2020), doi: 10.4171/rmi/1191 (to appear in print).CrossRefGoogle Scholar
Aliaga, R. J.. Pernecká, E. Petitjean, C. and Procházka, A.. Supports in Lipschitz-free spaces and applications to extremal structure. J. Math. Anal. Appl. 489 (2020), 124128.10.1016/j.jmaa.2020.124128CrossRefGoogle Scholar
Bogachev, V. I.. Measure theory, Vol. II (Berlin: Springer-Verlag, 2007).10.1007/978-3-540-34514-5CrossRefGoogle Scholar
Cúth, M., Doucha, M. and Wojtaszczyk, P.. On the structure of Lipschitz-free spaces. Proc. Am. Math. Soc. 144 (2016), 3833–3846.10.1090/proc/13019CrossRefGoogle Scholar
Dalet, A.. Free spaces over countable compact metric spaces. Proc. Am. Math. Soc. 143 (2015), 3537–3546.10.1090/S0002-9939-2015-12518-XCrossRefGoogle Scholar
Deville, R., Godefroy, G. and Zizler, V.. Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64 (Harlow: Longman Scientific & Technical, 1993).Google Scholar
Enflo, P.. A counterexample to the approximation property for Banach spaces. Acta Math. 130 (1973), 309–317.10.1007/BF02392270CrossRefGoogle Scholar
Fetter, H. and Gamboa de Buen, B.. The James forest. London Mathematical Society Lecture Note Series, vol. 236 (Cambridge: Cambridge University Press, 1997).10.1017/CBO9780511662379CrossRefGoogle Scholar
Gartland, C. Lipschitz free spaces over locally compact metric spaces, preprint, arXiv:2004.11951 (2020).10.4064/sm200511-10-10CrossRefGoogle Scholar
Godefroy, G. and Kalton, N. J.. Lipschitz-free Banach spaces. Studia Math. 159 (2003), 121–141.10.4064/sm159-1-6CrossRefGoogle Scholar
Godefroy, G. and Ozawa, N.. Free Banach spaces and the approximation properties. Proc. Am. Math. Soc. 142 (2014), 1681–1687.10.1090/S0002-9939-2014-11933-2CrossRefGoogle Scholar
Hájek, P., Lancien, G. and Pernecká, E.. Approximation and Schur properties for Lipschitz free spaces over compact metric spaces. Bull. Belg. Math. Soc. Simon Stevin 23 (2016), 63–72.10.36045/bbms/1457560854CrossRefGoogle Scholar
Kalton, N. J.. Spaces of Lipschitz and Hölder functions and their applications. Collect. Math. 55 (2004), 171–217.Google Scholar
Kalton, N. J.. Lipschitz and uniform embeddings into $\ell _\infty$. Fund. Math. 212 (2011), 53–69.10.4064/fm212-1-4CrossRefGoogle Scholar
Kochanek, T. and Pernecká, E.. Lipschitz-free spaces over compact subsets of superreflexive spaces are weakly sequentially complete. Bull. Lond. Math. Soc. 50 (2018), 680–696.10.1112/blms.12179CrossRefGoogle Scholar
Kuratowski, K.. Sur les espaces complets. Fund. Math. 15 (1930), 301–309.10.4064/fm-15-1-301-309CrossRefGoogle Scholar
Petitjean, C.. Lipschitz-free spaces and Schur properties. J. Math. Anal. Appl. 453 (2017), 894–907.10.1016/j.jmaa.2017.04.047CrossRefGoogle Scholar
Weaver, N.. Lipschitz algebras, 2nd edn (River Edge, NJ: World Scientific Publishing Co., 2018).10.1142/9911CrossRefGoogle Scholar