Isometries of Lipschitz-free Banach spaces

Abstract

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes e.g. Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose Lipschitz-free space only admits surjective linear isometries coming from surjective dilations (i.e. rescaled isometries) of the metric space itself. We show this class of metric spaces is surprisingly rich and contains all $3$-connected graphs as well as geometric examples such as non-abelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz-free rigid space that has only three more points.

References in corpus (9)

• Supports in Lipschitz-free spaces and applications to extremal structure
• On the preserved extremal structure of Lipschitz-free spaces
• A geometric study of Wasserstein spaces: Hadamard spaces
• Supports and extreme points in Lipschitz-free spaces
• Isometric embedding of ${\ell }_1$ into Lipschitz-free spaces and ${\ell }_{\infty }$ into their duals
• The isometry group of Wasserstein spaces: the Hilbertian case
• Isometric Embeddings into Heisenberg Groups
• Light groups of isomorphisms of Banach spaces and invariant LUR renormings
• Convex integrals of molecules in Lipschitz-free spaces