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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Quantum ultrametrics on AF algebras and the Gromov–Hausdorff propinquity

### Tom 231 / 2015

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

#### Streszczenie

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.

#### Autorzy

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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