A note about dual representations of group actions on Lipschitz-free spaces

Abstract

Let $\mathcal{F}(M)$ be the Lipschitz-free space of a pointed metric space $M$. For every isometric continuous group action of $G$ we have an induced continuous dual action on the weak-star compact unit ball $B_{\mathcal{F}(M)^*}$ of the dual space $\mathrm{Lip_0} (M)=\mathcal{F}(M)^*$. We pose the question when a given abstract continuous action of $G$ on a topological space $X$ can be represented through a $G$-subspace of $B_{\mathcal{F}(M)^*}$. One of such natural examples is the so-called metric compactification (of isometric $G$-spaces) for a pointed metric space. As well as the Gromov $G$-compactification of a bounded metric $G$-space. Note that there are sufficiently many representations of compact $G$-spaces on Lipschitz-free spaces.