Actions of Categories by Lipschitz morphisms on limits for the Gromov-Hausdorff Propinquity

TL;DR

This paper establishes a compactness theorem for actions of small categories on quantum metric spaces via Lipschitz maps, enabling new methods for constructing group actions and analyzing their limits in noncommutative geometry.

Contribution

It introduces a novel compactness result for category actions on quantum spaces under the covariant Gromov-Hausdorff propinquity, including applications to group actions and limits.

Findings

Proves a compactness theorem for Lipschitz actions on quantum metric spaces.

Provides a new method to construct group actions on C*-algebras.

Demonstrates that certain classes of quantum metric spaces are closed under limits.

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 {\gQqcms s} for the propinquity.