Projections in Lipschitz-free spaces induced by group actions

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 $Li{p}_0(M/G)$ is complemented in $Li{p}_0(M)$. 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.

References in corpus (8)

• Introduction to Arithmetic Groups
• On the structure of Lipschitz-free spaces
• On quotients of spaces with Ricci curvature bounded below
• Lipschitz free spaces isomorphic to their infinite sums and geometric applications
• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• Embeddings of Lipschitz-free spaces into ${\ell }_1$
• Isomorphisms between spaces of Lipschitz functions
• Approximation properties in Lipschitz-free spaces over groups

Cited by in corpus (1)

• Approximation properties in Lipschitz-free spaces over groups