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Isometries of Lipschitz-free Banach spaces

Marek Cúth, Michal Doucha, Tamás Titkos

Abstract

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. 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 this class of metric spaces is surprisingly rich and contains all $3$-connected graphs as well as geometric examples such as non-abelian 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.

Isometries of Lipschitz-free Banach spaces

Abstract

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. 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 this class of metric spaces is surprisingly rich and contains all -connected graphs as well as geometric examples such as non-abelian 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.
Paper Structure (16 sections, 25 theorems, 61 equations)

This paper contains 16 sections, 25 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Lipschitz-free spaces
  4. Graphs
  5. Isometries on Lispchitz-free spaces generated by cyclic bijections
  6. Lipschitz-free rigid spaces
  7. Building new Lipschitz-free rigid spaces
  8. Sums of metric spaces
  9. Disjoint unions of metric spaces
  10. Description of isometry groups of Lipschitz-free spaces over graphs
  11. Examples
  12. Carnot groups as examples of Lipschitz-free rigid Prague spaces
  13. Counter-examples from Section \ref{['sec:rigid']}
  14. Concluding remarks and open problems
  15. Remarks
  16. ...and 1 more sections

Key Result

Proposition 3.3

Let $(\mathcal{M}_1,d_1)$ and $(\mathcal{M}_2,d_2)$ be metric spaces with weakly admissible sets $E_1$ and $E_2$. Let $\sigma: E_1\to E_2$ be a bijection and suppose that one of the following conditions hold Then there exists a surjective isometry $T:\mathcal{F}(\mathcal{M}_1)\to \mathcal{F}(\mathcal{M}_2)$ such that $T(m_e)=m_{\sigma(e)}$, for every $e\in E_1$. Moreover, it:admissibleCond implie

Theorems & Definitions (68)

  • proof
  • Definition 3.1
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Definition 3.6
  • Definition 3.7
  • Lemma 3.8
  • ...and 58 more