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A Choquet theory of Lipschitz-free spaces

Richard J. Smith

TL;DR

The paper develops a Choquet-like framework for Lipschitz-free spaces by leveraging De Leeuw representations on $\beta\widetilde{M}$ and introducing a Choquet-type quasi-order $\preccurlyeq$ on Radon measures. It builds a function cone $G$, constructs a corresponding compactification $\widetilde{M}^G$, and analyzes minimal measures (instead of the classical focus on maximal ones) to obtain well-behaved representations and shadow/extended-support structure. Key results include a characterization of minimality via a Choquet-type test, the possibility of mutually singular marginals for optimal/minimal representations, and a plan to decompose free-space elements into convex integrals of molecules. These tools culminate in establishing that every free-space element has optimal representations concentrated on $\mathfrak{p}^{-1}(M\times M)$, enabling a robust extreme-point analysis and laying groundwork for a complete description of extreme points in $B_{\mathcal{F}(M)}$. The framework thus provides a deeper structural understanding of Lipschitz-free spaces with potential applications to convex decompositions and point-extremality questions.

Abstract

Let $(M,d)$ be a complete metric space and let $\mathcal{F}(M)$ denote the Lipschitz-free space over $M$. We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of $\mathcal{F}(M)$ (and its bidual) by positive Radon measures on $β\widetilde{M}$, where $\widetilde{M}$ is the space of pairs $(x,y) \in M \times M$, $x \neq y$. We define a quasi-order $\preccurlyeq$ on the positive Radon measures on $β\widetilde{M}$ that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the $\preccurlyeq$-minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces.

A Choquet theory of Lipschitz-free spaces

TL;DR

The paper develops a Choquet-like framework for Lipschitz-free spaces by leveraging De Leeuw representations on and introducing a Choquet-type quasi-order on Radon measures. It builds a function cone , constructs a corresponding compactification , and analyzes minimal measures (instead of the classical focus on maximal ones) to obtain well-behaved representations and shadow/extended-support structure. Key results include a characterization of minimality via a Choquet-type test, the possibility of mutually singular marginals for optimal/minimal representations, and a plan to decompose free-space elements into convex integrals of molecules. These tools culminate in establishing that every free-space element has optimal representations concentrated on , enabling a robust extreme-point analysis and laying groundwork for a complete description of extreme points in . The framework thus provides a deeper structural understanding of Lipschitz-free spaces with potential applications to convex decompositions and point-extremality questions.

Abstract

Let be a complete metric space and let denote the Lipschitz-free space over . We develop a ``Choquet theory of Lipschitz-free spaces'' that draws from the classical Choquet theory and the De Leeuw representation of elements of (and its bidual) by positive Radon measures on , where is the space of pairs , . We define a quasi-order on the positive Radon measures on that is analogous to the classical Choquet order. Rather than in the classical case where the focus lies on maximal measures, we study the -minimal measures and show that they have a host of desirable properties. Among the applications of this theory is a solution (given elsewhere) to the extreme point problem for Lipschitz-free spaces.

Paper Structure

This paper contains 23 sections, 74 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. The De Leeuw transform and optimal representations
  4. Topological and metric preliminaries
  5. Supports, shadows, extended supports and functionals that avoid infinity
  6. Relationship with classical Choquet theory
  7. Plan of the paper
  8. The function cone G and the associated compact space
  9. Definition and fundamental properties of G
  10. A compactification associated with G
  11. The connection between the G- and uniform compactifications of tilde M
  12. A Choquet-like quasi-order on the Radon measures on the De Leeuw domain
  13. Definition and basic results
  14. Minimal measures
  15. The quasi-order and sets on which measures are concentrated
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $M$ be a complete metric space and let $m$ be a preserved extreme point of $B_{\mathcal{F}({M})}$. Then $m \in \mathop{\mathrm{Mol}}\nolimits$.

Theorems & Definitions (146)

  • Theorem 1.1: cf. Weaver
  • Theorem 1.3: AP_rmi
  • Theorem 1.4: APS24c
  • Proposition 1.5: APS24a
  • Proposition 1.6: Woods
  • Proposition 1.7
  • proof
  • Example 1.8
  • Proposition 1.9: APS24b
  • Proposition 1.10: APS24b
  • ...and 136 more