Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces

TL;DR

This paper introduces an explicit, functorial method to convert unbounded metric spaces into bounded ones, preserving the structure of their Lipschitz-free spaces, enabling the transfer of results between bounded and unbounded cases.

Contribution

It provides a novel, explicit construction linking Lipschitz-free spaces over unbounded and bounded metric spaces, with applications in analysis and Banach algebra structures.

Findings

Isomorphism between Lipschitz-free spaces over unbounded and bounded metric spaces

Transferability of analytical arguments from bounded to unbounded spaces

Lipschitz algebra $ m{Lip}_0( ext{space})$ forms a Banach algebra with modified multiplication

Abstract

We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into a bounded metric space $\mathcal{B}$ in such a way that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are isomorphic. The construction we provide is functorial in a weak sense and has the advantage of being explicit. Apart from its intrinsic theoretical interest, it has many applications in that it allows to transfer many arguments valid for Lipschitz-free spaces over bounded spaces to Lipschitz-free spaces over unbounded spaces. Furthermore, we show that with a slightly modified point-wise multiplication, the space $\rm{Lip}_0(\mathcal{M})$ of scalar-valued Lipschitz functions vanishing at zero over any (unbounded) pointed metric space is a Banach algebra with its canonical Lipschitz norm.