Inductive limits of C*-algebras and compact quantum metrics spaces

TL;DR

This paper develops methods to construct quantum metrics on inductive limits of C*-algebras, especially AF algebras, using the dual Gromov-Hausdorff propinquity to analyze their metric topology.

Contribution

It provides sufficient conditions for inducing quantum metrics on inductive limits, extending previous work on AF algebras with faithful traces, and relates these to the Fell topology.

Findings

Constructed quantum metrics on inductive limits of C*-algebras.

Extended quantum metric structures to all unital AF algebras.

Connected the ideal space topology with the dual Gromov-Hausdorff propinquity.

Abstract

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a compact quantum metric of Rieffel, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere on compact quantum metric spaces. This allows us to place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov-Hausdorff propinquity topology.