Coarse quotients by group actions and the maximal Roe algebra

For a discrete metric space (or more generally a large scale space) $X$ and an action of a group $G$ on $X$ by coarse equivalences, we define a type of coarse quotient space $X_G$, which agrees up to coarse equivalence with the orbit space $X/G$ when $G$ is finite. We then restrict our attention to what we call coarsely discontinuous actions and show that for such actions the group $G$ can be recovered as an appropriately defined automorphism group $\mathsf{Aut}(X/X_G)$ when $X$ satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group $G$ on a discrete bounded geometry metric space $X$ there is a relation between the maximal Roe algebras of $X$ and $X_G$, namely that there is a $\ast$-isomorphism $C^\ast_{\max}(X_G)/\mathcal{K} \cong C^\ast_{\max}(X)/\mathcal{K} \rtimes G$, where $\mathcal{K}$ is the ideal of compact operators. If $X$ has Property A and $G$ is amenable, then we show that $X_G$ has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.