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Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity

Volume 231 / 2015

Konrad Aguilar, Frédéric Latrémolière Studia Mathematica 231 (2015), 149-193 MSC: Primary 46L89, 46L30, 58B34. DOI: 10.4064/sm8478-2-2016 Published online: 18 February 2016

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 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.

Authors

  • Konrad AguilarDepartment of Mathematics
    University of Denver
    2199 S. University Blvd
    Denver, CO 80208, U.S.A.
    e-mail
  • Frédéric LatrémolièreDepartment of Mathematics
    University of Denver
    2199 S. University Blvd
    Denver, CO 80208, U.S.A.
    e-mail

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