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Diversity of Lipschitz-free spaces over countable complete discrete metric spaces

Estelle Basset, Gilles Lancien, Antonín Procházka

TL;DR

The paper advances the understanding of Lipschitz-free spaces by showing that $\\\\mathcal{F}(M)$, for countable, complete, discrete $M$, exhibit unprecedented diversity in their isomorphism types, governed by the dentability index $D$ and the weak fragmentability index $\\Phi$. It develops a transfinite construction that yields, for any ordinal $\\xi$, a complete discrete space $D_\\xi$ with $D(\\mathcal{F}(D_\\xi))>\\xi$, proving the existence of uncountably many pairwise non-isomorphic free spaces. It also proves sharp upper bounds for $\\Phi(\\mathcal{F}(M))$ and $D(\\mathcal{F}(M))$ in the uniformly discrete setting, with $\\Phi(\\mathcal{F}(M)) \le \omega^2$ and $D(\\mathcal{F}(M)) \le \omega^3$, and identifies spaces attaining these extremal values via diamond-graph constructions. The results together show that there is a countable complete discrete $M$ whose free space does not embed into any free space of a uniformly discrete space, and that there is no universal separable purely 1-unrectifiable metric space for the class of countable complete discrete spaces, highlighting strong barriers to universality in this setting.

Abstract

We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index $D$ presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of $\{D(\mathcal F(M)): M$ countable, complete, discrete$\}$ is uncountable while $\{D(\mathcal F(M)):M$ infinite, compact, purely 1-unrectifiable$\}=\{ω,ω^2\}$. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: $\{D(\mathcal F(M)):M$ infinite, uniformly discrete$\}=\{ω^2,ω^3\}$.

Diversity of Lipschitz-free spaces over countable complete discrete metric spaces

TL;DR

The paper advances the understanding of Lipschitz-free spaces by showing that , for countable, complete, discrete , exhibit unprecedented diversity in their isomorphism types, governed by the dentability index and the weak fragmentability index . It develops a transfinite construction that yields, for any ordinal , a complete discrete space with , proving the existence of uncountably many pairwise non-isomorphic free spaces. It also proves sharp upper bounds for and in the uniformly discrete setting, with and , and identifies spaces attaining these extremal values via diamond-graph constructions. The results together show that there is a countable complete discrete whose free space does not embed into any free space of a uniformly discrete space, and that there is no universal separable purely 1-unrectifiable metric space for the class of countable complete discrete spaces, highlighting strong barriers to universality in this setting.

Abstract

We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of countable, complete, discrete is uncountable while infinite, compact, purely 1-unrectifiable. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: infinite, uniformly discrete.

Paper Structure

This paper contains 14 sections, 26 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation
  4. Lipschitz-free spaces
  5. Various fragmentability indices of Banach spaces
  6. The Radon-Nikodým and the Point of Continuity properties
  7. The countably branching diamond graphs
  8. Uniformly discrete metric spaces
  9. Weak fragmentability index
  10. Dentability index
  11. Countable complete discrete metric spaces
  12. Construction of the spaces
  13. Properties of the family
  14. Applications

Key Result

Proposition 2.1

Let $(M, d)$ be a pointed metric space. For $k \in \mathbb{Z}$, we set $M_k = \left\{x \in M,\, d(x, 0) \leq 2^k\right\}$. Then $\mathcal{F}(M)$ is linearly isomorphic to a subspace of $(\sum_{k \in \mathbb{Z}} \mathcal{F}(M_k))_{\ell_1}$.

Figures (1)

  • Figure 1: $D_1$ and $D_n$, $n \geq 2$

Theorems & Definitions (47)

  • Proposition 2.1: Kalton's decomposition
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 3.1
  • ...and 37 more