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

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