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Injectivity of Lipschitz operators

Abstract : 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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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03659971
Contributor : Colin PETITJEAN Connect in order to contact the contributor
Submitted on : Thursday, May 5, 2022 - 2:40:58 PM
Last modification on : Wednesday, May 25, 2022 - 3:51:16 PM

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GPP22_v1arXiv.pdf
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  • HAL Id : hal-03659971, version 1
  • ARXIV : 2205.03093

Citation

Luis García-Lirola, Colin Petitjean, Antonin Prochazka. Injectivity of Lipschitz operators. 2022. ⟨hal-03659971⟩

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