[Paper Review] Extremal structure and Duality of Lipschitz free spaces

Luis C. García‐Lirola, Colin Petitjean+2 more|arXiv (Cornell University)|Jul 28, 2017

Advanced Banach Space Theory28 references42 citations

TL;DR

This paper investigates the extremal structure and duality of Lipschitz free spaces, proving that every preserved extreme point of the unit ball is a denting point in general. It establishes that in spaces with a natural predual or under uniform discreteness, extreme points coincide with molecules and that molecules are extreme precisely when the metric segment between their defining points contains no other points—resolving key open questions in special cases.

ABSTRACT

We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, say $x$ and $y$, which define it satisfy that the metric segment $[x, y]$ only contains $x$ and $y$. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm-attainment in spaces of vector valued Lipschitz functions.

Motivation & Objective

• To clarify the relationship between extremal notions—strongly exposed, denting, preserved extreme, and extreme points—in Lipschitz free spaces.
• To resolve two open problems: (a) whether every extreme point is a molecule, and (b) whether a molecule is extreme when its defining points form a metric segment with no intermediate points.
• To establish conditions under which the set of extreme points coincides with the set of strongly exposed points or molecules.
• To apply the results to norm attainment in spaces of vector-valued Lipschitz functions.
• To study the existence and properties of isometric preduals for Lipschitz free spaces, particularly natural preduals.

Proposed method

• Prove that every preserved extreme point of the unit ball in a Lipschitz free space is a denting point using a characterization of preserved extreme points via weak convergence of sequences.
• Provide a new proof of the metric characterization of preserved extreme points, showing that they are molecules when the metric segment between x and y contains only x and y.
• Introduce and study 'natural preduals' of Lipschitz free spaces, which are isometric preduals with additional structural properties.
• Use the existence of a natural predual to show that extreme points coincide with strongly exposed points under an additional assumption.
• Analyze the case of uniformly discrete and bounded metric spaces to show that molecules are extreme points when the metric segment between x and y contains no other points.
• Apply the results to norm attainment by showing that norm-attaining Lipschitz functions strongly attain their norm under certain geometric and structural conditions on M and F(M).

Experimental results

Research questions

• RQ1 Is every extreme point of the unit ball in a Lipschitz free space a molecule?
• RQ2 Is a molecule mxy an extreme point of the unit ball whenever the metric segment [x, y] contains no other points of M?
• RQ3 Under what conditions does the set of extreme points coincide with the set of strongly exposed points in F(M)?
• RQ4 When does the norm-attainment of a Lipschitz function on F(M) imply its strong norm attainment?
• RQ5 What are the structural and geometric conditions under which F(M) admits an isometric predual, particularly a natural predual?

Key findings

• Every preserved extreme point of the unit ball in a Lipschitz free space is a denting point, and this holds in full generality.
• The set of molecules V is weakly closed in F(M), and the canonical embedding δ(M) is also weakly closed.
• In uniformly discrete and bounded metric spaces, every molecule mxy is an extreme point if the metric segment [x, y] contains no other points of M.
• If F(M) admits a natural predual and M is uniformly discrete and bounded, then every extreme point of BF(M) is a molecule.
• For compact metric spaces where lip0(M) has the 1-SPU property, every extreme point of BF(M) is a denting point.
• Under the Krein-Milman property and the condition that ext(BF(M)) ⊆ V, every norm-attaining function in Lip0(M, R) also strongly attains its norm, i.e., NA(F(M), R) = LipSNA(M, R).