Quantum Ultrametrics on AF Algebras and The Gromov-Hausdorff Propinquity

TL;DR

This paper develops quantum metric structures on AF algebras, demonstrating their limits under the quantum propinquity, and explores the geometric properties of various classes of AF algebras within noncommutative metric geometry.

Contribution

It introduces quantum metrics on AF algebras and analyzes their geometric properties using the quantum propinquity, connecting AF algebras with noncommutative metric geometry.

Findings

AF algebras are limits of finite dimensional C*-algebras under the quantum propinquity.

Quantum metrics recover classical ultrametrics on the Cantor space.

Continuity of Lip-norms on AF algebras is established.

Abstract

We construct quantum metric structures on unital AF algebras with a faithful tracial state, and prove that for such metrics, AF algebras are limits of their defining inductive sequences of finite dimensional C*-algebras for the quantum propinquity. We then study the geometry, for the quantum propinquity, of three natural classes of AF algebras equipped with our quantum metrics: the UHF algebras, the Effros-Shen AF algebras associated with continued fraction expansions of irrationals, and the Cantor space, on which our construction recovers traditional ultrametrics. We also exhibit several compact classes of AF algebras for the quantum propinquity and show continuity of our family of Lip-norms on a fixed AF algebra. Our work thus brings AF algebras into the realm of noncommutative metric geometry.