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A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces

Ramón J. Aliaga, Eva Pernecká, Richard J. Smith

Abstract

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author.

A solution to the extreme point problem and other applications of Choquet theory to Lipschitz-free spaces

Abstract

We prove that every element of a Lipschitz-free space admits an expression as a convex series of elements with compact support. As a consequence, we conclude that all extreme points of the unit ball of Lipschitz-free spaces are elementary molecules, solving a long-standing problem. We also deduce that all elements of a Lipschitz-free space with the Radon-Nikodým property can be expressed as convex integrals of molecules. Our results are based on a recent theory of integral representation for functionals on Lipschitz spaces which draws on classical Choquet theory, due to the third named author.

Paper Structure

This paper contains 8 sections, 22 theorems, 40 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Preliminaries on Lipschitz-free spaces
  3. De Leeuw representations
  4. Choquet theory of Lipschitz-free spaces
  5. A compact decomposition principle
  6. The extreme point problem
  7. Convex integrals of molecules
  8. Diagonal functionals

Key Result

Theorem 1.1

Let $m \in \mathcal{F}({M})$. Then there exists $\mu \in \mathcal{M}_{\mathrm{op}}(m)$ concentrated on $\mathfrak{p}^{-1}(M\times M)$. Moreover, we can choose it to be concentrated on $\mathfrak{p}^{-1}(A\times A)$, where $A=\mathop{\mathrm{supp}}\nolimits(m)\cup\left\{{0}\right\}$.

Theorems & Definitions (43)

  • Theorem 1.1: Smith; cf. Smith24
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3: Aliaga; cf. Aliaga
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 33 more