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arXiv · 2104.11519

Projections in Lipschitz-free spaces induced by group actions

Marek Cúth·Michal Doucha
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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.

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Marek Cúth, Michal Doucha. 2021-04-23. Projections in Lipschitz-free spaces induced by group actions. https://doi.org/10.1002/mana.202100222

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