Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity

Konrad Aguilar, and Frédéric Latrémolière. "Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity." Studia Mathematica 231.2 (2015): 149-193. <http://eudml.org/doc/285389>.

@article{KonradAguilar2015,
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.},
author = {Konrad Aguilar, Frédéric Latrémolière},
journal = {Studia Mathematica},
keywords = {noncommutative metric geometry; Gromov-Hausdorff convergence; Monge-Kantorovich distance; quantum metric spaces; Lip-norm; AF algebras; C*-algebra},
language = {eng},
number = {2},
pages = {149-193},
title = {Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity},
url = {http://eudml.org/doc/285389},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Konrad Aguilar
AU - Frédéric Latrémolière
TI - Quantum ultrametrics on AF algebras and the Gromov-Hausdorff propinquity
JO - Studia Mathematica
PY - 2015
VL - 231
IS - 2
SP - 149
EP - 193
AB - 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.
LA - eng
KW - noncommutative metric geometry; Gromov-Hausdorff convergence; Monge-Kantorovich distance; quantum metric spaces; Lip-norm; AF algebras; C*-algebra
UR - http://eudml.org/doc/285389
ER -