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Lipschitz-free spaces and approximating sequences of projections

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Abstract

The Lipschitz-free space \({\mathcal {F}}(M)\) has an F.D.D. when M is a separable \({\mathcal {L}}_1\)-Banach space, or when \(M\subset {\mathbb {R}}^n\) is a somewhat regular subset. The interplay between the existence of extension operators for Lipschitz maps and the \((\pi )\)-property in Lipschitz-free spaces is investigated. If M is an arbitrary metric space, then \({\mathcal {F}}(M)\) has the \((\pi )\)-property up to a universal logarithmic factor. It follows in particular that the \((\pi )\)-property up to a logarithmic factor fails to imply the approximation property. A list of commented open problems is provided.

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Acknowledgements

I am grateful to the referees for their careful reading of this note and for their suggestions and improvements. I am very grateful to Bill Johnson for his tremendous influence on geometry of Banach spaces, and in particular on the interplay between linear and non-linear theory, to which belongs the present work.

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  1. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Paris, France

    Gilles Godefroy

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  1. Gilles Godefroy
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Correspondence to Gilles Godefroy.

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Communicated by Maria Joita.

This paper is dedicated to Professor W. B. Johnson.

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Godefroy, G. Lipschitz-free spaces and approximating sequences of projections. Banach J. Math. Anal. 18, 23 (2024). https://doi.org/10.1007/s43037-024-00332-2

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