Approximation and Schur properties for Lipschitz free spaces over compact metric spaces

We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to th...

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Bibliographic Details
Published inBulletin of the Belgian Mathematical Society, Simon Stevin Vol. 23; no. 1; pp. 63 - 72
Main Authors Hájek, P., Lancien, G., Pernecká, E.
Format Journal Article
LanguageEnglish
Published Belgian Mathematical Society 01.01.2016
The Belgian Mathematical Society
Subjects
46B20
46B80
approximation property
Cantor space
Functional Analysis
Lipschitz free spaces
Mathematics
Schur property
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Summary:We prove that for any separable Banach space $X$, there exists a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space contains a complemented subspace isomorphic to $X$. As a consequence we give an example of a compact metric space which is homeomorphic to the Cantor space and whose Lipschitz-free space fails the approximation property and we prove that there exists an uncountable family of topologically equivalent distances on the Cantor space whose free spaces are pairwise non isomorphic. We also prove that the free space over a countable compact metric space has the Schur property. These results answer questions by G. Godefroy.
ISSN:1370-1444
2034-1970
DOI:10.36045/bbms/1457560854