Projections in Lipschitz‐free spaces induced by group actions

We show that given a compact group G acting continuously on a metric space M$\mathcal {M}$ by bi‐Lipschitz bijections with uniformly bounded norms, the Lipschitz‐free space over the space of orbits M/G$\mathcal {M}/G$ (endowed with Hausdorff distance) is complemented in the Lipschitz‐free space over M$\mathcal {M}$. 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 Lip0(M/G)$\mathrm{Lip}_0(\mathcal {M}/G)$ is complemented in Lip0(M)$\mathrm{Lip}_0(\mathcal {M})$. Moreover, if the Lipschitz‐free space over M$\mathcal {M}$, F(M)$\mathcal {F}(\mathcal {M})$, is complemented in its bidual, several sufficient conditions on when F(M/G)$\mathcal {F}(\mathcal {M}/G)$ is complemented in F(M)$\mathcal {F}(\mathcal {M})$ are given. Some applications are discussed. The paper contains preliminaries on projections induced by actions of amenable groups on general Banach spaces.