On the preserved extremal structure of Lipschitz-free spaces

Abstract

We characterize preserved extreme points of Lipschitzfree spaces F(X) in terms of simple geometric conditions on the underlying metric space (X, d). Namely, each preserved extreme point corresponds to a pair of points p, q in X such that the triangle inequality d(p, q) ≤ d(p, r) + d(q, r) is uniformly strict for r away from p, q. For compact X, this condition reduces to the triangle inequality being strict. This result gives an affirmative answer to a conjecture of N. Weaver that compact spaces are concave if and only if they have no triple of metrically aligned points.2010 Mathematics Subject Classification. Primary 46B20; Secondary 46E15, 54E45.

The Bishop–Phelps–Bollobás property for Lipschitz maps

“…A pointed metric space M is said to be concave if every molecule of M is a preserved extreme point, that is, an extreme point of the unit ball of F(M ) * * . This property has been recently characterized in [5] and, for a boundedly compact pointed metric space M it is shown in [ A strengthening of the concept of concavity is provided when we require all the molecules to be strongly exposed points of the unit ball of F(M ). By the characterization given in [17,Theorem 5.4], the property can be written in terms of the metric space and we may also introduce a uniform version of it.…”

Supports and extreme points in Lipschitz-free spaces

“…In [10], García-Lirola, Procházka and Rueda Zoca gave a complete geometric characterization of the strongly exposed points of B F (M ) (see Theorem 2.2(b)). In [1], the first author and Guirao gave a similar geometric characterization of preserved extreme points (see Theorem 2.2(a)), and asked whether extreme points could be described analogously. In particular, they asked if it is true that u pq is extreme if and only if [p, q] = {p, q} [1, Question 1].…”

“…In particular, they asked if it is true that u pq is extreme if and only if [p, q] = {p, q} [1, Question 1]. The answer is positive when M is compact by [1,Theorem 4.2]. Concurrently, García-Lirola, Petitjean, Procházka and Rueda Zoca proved in [9] that all preserved extreme points of B F (M ) are denting points, and gave a positive answer to [1,Question 1] for bounded, uniformly discrete M .…”

“…Our main result in this note is that the answer to [1, Question 1] is positive for any complete metric space M , that is: The proof relies on a refinement of the methods used in [1] for obtaining the characterization of preserved extreme points, but an additional key observation is required: the fact that the class of Lipschitz-free spaces over closed subsets of M , considered as subspaces of F(M ), is closed under intersections when M is bounded (Theorem 3.3). We prove this by considering the algebra structure of Lip 0 (M ) -we show that the w * -closure of any ideal in Lip 0 (M ) is also an ideal and appeal to results in [18,Chapter 4].…”

Isometric Composition Operators on Lipschitz Spaces

“…Let us recall that F (X) is the canonical predual of Lip 0 (X). By Theorems 3.39 in [8] and 4.1 in [1], X is uniformly concave if and only if every molecule…”