Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

In this paper, we study the groups of isometries and the set of bi-Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple (Formula presented.) -algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.