Projections in Lipschitz‐free spaces induced by group actions

Abstract

We show that given a compact group G acting continuously on a metric space by bi‐Lipschitz bijections with uniformly bounded norms, the Lipschitz‐free space over the space of orbits (endowed with Hausdorff distance) is complemented in the Lipschitz‐free space over . 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 is complemented in . Moreover, if the Lipschitz‐free space over , , is complemented in its bidual, several sufficient conditions on when is complemented in are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces.