Injectivity of Lipschitz operators
2022
Online
unknown
Zugriff:
Any Lipschitz map $f\colon M \to N$ between metric spaces can be ``linearised'' in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.
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Injectivity of Lipschitz operators
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Autor/in / Beteiligte Person: | García-Lirola, Luis C. ; Petitjean, Colin ; Procházka, Antonín ; Departamento de matemáticas, Universidad de Zaragoza ; Laboratoire Analyse et Mathématiques Appliquées (LAMA) ; Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel ; Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB) ; Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC) ; Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC) |
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Veröffentlichung: | 2022 |
Medientyp: | unknown |
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