Actions of categories by Lipschitz morphisms on limits for the Gromov–Hausdorff propinquity

Abstract

We prove a compactness result for classes of actions of many small categories on quantum compact metric spaces by Lipschitz linear maps, for the topology of the covariant Gromov–Hausdorff propinquity. In particular, our result applies to actions of proper groups by Lipschitz isomorphisms on quantum compact spaces. Our result provides a first example of a structure which passes to the limit of quantum metric spaces for the propinquity, as well as a new method to construct group actions, including from non-locally compact groups seen as inductive limits of compact groups, on unital C*-algebras. We apply our techniques to obtain some properties of closure of certain classes of quasi-Leibniz quantum compact metric spaces for the propinquity.