Lipschitz-Free Spaces: A Topometric Approach and Group Actions

We introduce a topometric version of Lipschitz-free spaces and study its universal property. Another aim of this paper is to investigate actions of topological groups $G$ on Lipschitz-free spaces $\mathcal{F}(M)$, induced by isometric actions on pointed metric spaces $M$. In particular, we study the associated dynamical $G$-systems under the weak-star topology, focusing on the dual action on ${\mathrm{L}\mathrm{i}\mathrm{p}}_0(M)=\mathcal{F}(M{)}^{\ast }$ and the bidual $\mathcal{F}(M{)}^{\ast \ast }$.