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 Latremoliere. In particular we prove that these groups and sets are compact in the automorphism group of the spectral triple C*-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.