Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity

• Autorzy: Konrad Aguilar , Frédéric Latrémolière
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 58B34: Noncommutative geometry (\`a la Connes)
• 46L89: Other ``noncommutative'' mathematics based on C * -algebra theory
• 46L30: States

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 231
• Numer: 2
• Strony: 149-193

Daty

wydano

2015

Twórcy

autor

Konrad Aguilar

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

autor

Frédéric Latrémolière

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

Identyfikatory

DOI

10.4064/sm8478-2-2016