Bunce-Deddens algebras as quantum Gromov-Hausorff distance limits of circle algebras

TL;DR

This paper demonstrates that Bunce-Deddens algebras can be viewed as limits of circle algebras within the framework of quantum Gromov-Hausdorff distance, establishing a continuous family of such limits.

Contribution

It introduces a quantum metric structure on Bunce-Deddens algebras and connects inductive limits with metric limits in the quantum setting.

Findings

Bunce-Deddens algebras are limits of circle algebras in quantum Gromov-Hausdorff distance.

Bunce-Deddens algebras form a continuous family indexed by the Baire space.

Reconciliation of different quantum metric constructions was achieved.

Abstract

We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow Bunce-Deddens algebras with a quantum metric structure, a step which requires that we reconcile the constructions of the Latremoliere's Gromov-Hausdorff propinquity and Rieffel's quantum Gromov-Hausdorff distance when working on order-unit quantum metric spaces. This work thus continues the study of the connection between inductive limits and metric limits.