Isometries of Lipschitz‐free Banach spaces

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometr...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2024-11, Vol.110 (5), p.n/a
Main Authors: Cúth, Marek, Doucha, Michal, Titkos, Tamás
Format: Article
Language:English
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Summary:We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz‐free spaces that includes, for example, Lipschitz‐free spaces over any graph. We define the notion of a Lipschitz‐free rigid metric space whose Lipschitz‐free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3‐connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz‐free rigid space that has only three more points.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms.70000