Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity

Konrad Aguilar¹ ; Frédéric Latrémolière¹

¹ Department of Mathematics University of Denver 2199 S. University Blvd Denver, CO 80208, U.S.A.

Studia Mathematica, Tome 231 (2015) no. 2, pp. 149-193

Résumé

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 Effrös–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.

Détail

DOI : 10.4064/sm8478-2-2016

Keywords: construct quantum metric structures unital algebras faithful tracial state prove metrics algebras limits their defining inductive sequences finite dimensional * algebras quantum propinquity study geometry quantum propinquity three natural classes algebras equipped quantum metrics uhf algebras effr shen algebras associated continued fraction expansions irrationals cantor space which construction recovers traditional ultrametrics exhibit several compact classes algebras quantum propinquity continuity family lip norms fixed algebra work brings algebras realm noncommutative metric geometry