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

TL;DR

This paper investigates the structure and convergence of isometry groups in spectral triples within the noncommutative metric framework, providing conditions for their limits and illustrating with examples like AF algebras.

Contribution

It establishes compactness of isometry groups in the propinquity framework and characterizes their convergence in inductive limits of quantum metric spaces.

Findings

Isometry groups are compact in the Monge-Kantorovich metric topology.

Provides necessary and sufficient conditions for convergence of isometry group actions.

Examples include AF algebras and noncommutative solenoids.

Abstract

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.