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

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 Latrémolière. 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. CONTENTS 1. Introduction 1 Acknowledgments 5 2. The Isometry Groups of a Metric Spectral Triple 5 2.1. The Isometry Group of a Quantum Compact Metric Space 6 2.2. The Isometry Group of a Spectral Triple 12 3. The Propinquity and Isometry Groups 17 3.1. The Covariant Propinquity 17 3.2. Covariant Bridge Builders 19 4. Applications 29 4.1. The Identity as a Bridge Builder 30 4.2. Almost Inners 31 4.3. AF Algebras with Faithful Traces 32 4.4. Dual Actions and Inductive Limits 35 References 37