Projections in Lipschitz-free spaces induced by group actions

TL;DR

This paper studies how group actions on metric spaces influence the structure of Lipschitz-free spaces, showing conditions under which these spaces are complemented and exploring implications for Lipschitz functions.

Contribution

It introduces new conditions for the complementability of Lipschitz-free spaces under group actions, extending previous results to broader classes of groups and actions.

Findings

Lipschitz-free space over the orbit space is complemented in the original space under certain group actions.

Lipschitz function spaces over the orbit space are complemented in the original Lipschitz function spaces.

Provides sufficient conditions for the complementability of Lipschitz-free spaces based on properties of the original space.

Abstract

We show that given a compact group $G$ acting continuously on a metric space $M$ by bi-Lipschitz bijections with uniformly bounded norms, the Lipschitz-free space over the space of orbits $M/G$ (endowed with Hausdorff distance) is complemented in the Lipschitz-free space over $M$. We also investigate the more general case when $G$ is amenable, locally compact or SIN and its action has bounded orbits. Then we get that the space of Lipschitz functions $Lip_0(M/G)$ is complemented in $Lip_0(M)$. Moreover, if the Lipschitz-free space over $M$, $F(M)$, is complemented in its bidual, several sufficient conditions on when $F(M/G)$ is complemented in $F(M)$ are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces.