Our dissertation focuses on bringing approximately finite-dimensional (AF) algebras into the realm of noncommutative metric geometry. We construct quantum metric structures on unital AF algebras equipped 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 Gromov-Hausdorff 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 Effros-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. Next, given a C*-algebra, the ideal space may be equipped with natural topologies. Motivated by this, we impart criteria for when convergence of ideals of an AF algebra can provide convergence of quotients in quantum propinquity, while introducing a metric on the ideal space of a C*-algebra. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies including the Jacobson and Fell topologies to the space of quotients with the quantum propinquity topology.
","abstract_html":"<p>Our dissertation focuses on bringing approximately finite-dimensional (AF) algebras into the realm of noncommutative metric geometry. We construct quantum metric structures on unital AF algebras equipped 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 Gromov-Hausdorff 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 Effros-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. Next, given a C*-algebra, the ideal space may be equipped with natural topologies. Motivated by this, we impart criteria for when convergence of ideals of an AF algebra can provide convergence of quotients in quantum propinquity, while introducing a metric on the ideal space of a C*-algebra. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies including the Jacobson and Fell topologies to the space of quotients with the quantum propinquity topology.</p>","abstract_has_math":false,"creators":["Aguilar, Konrad"],"institution":null,"degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Frédéric Latrémolière, Ph.D.","Jennifer L. Hoffman, Ph.D.","Alvaro Arias","Nicholas Ormes","Judith Packer"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-01-01T08:00:00Z","date_published":"2017-01-01T08:00:00Z","updated_at":"2026-07-24T02:03:03Z","subjects":["AF algebras","Gromov-Hausdorff convergence","Noncommutative metric geometry","Quantum metric spaces","Approximately finite-dimensional algebras","Mathematics"],"languages":["en"],"rights":["Copyright is held by the author. User is responsible for all copyright compliance.
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"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Quantum Metrics on Approximately Finite-Dimensional Algebras"]}]}],"canonical_facts":{"dc:contributor":["Frédéric Latrémolière, Ph.D.","Jennifer L. Hoffman, Ph.D.","Alvaro Arias","Nicholas Ormes","Judith Packer"],"dc:creator":["Aguilar, Konrad"],"dc:date.available":["2017-07-25T07:00:00Z"],"dc:description.abstract":["Our dissertation focuses on bringing approximately finite-dimensional (AF) algebras into the realm of noncommutative metric geometry. We construct quantum metric structures on unital AF algebras equipped 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 Gromov-Hausdorff 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 Effros-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. Next, given a C*-algebra, the ideal space may be equipped with natural topologies. Motivated by this, we impart criteria for when convergence of ideals of an AF algebra can provide convergence of quotients in quantum propinquity, while introducing a metric on the ideal space of a C*-algebra. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies including the Jacobson and Fell topologies to the space of quotients with the quantum propinquity topology.
"],"dc:format":["application/pdf"],"dc:identifier":["https://digitalcommons.du.edu/etd/1277"],"dc:language":["en"],"dc:rights":["Copyright is held by the author. User is responsible for all copyright compliance.
"],"dc:subject":["AF algebras","Gromov-Hausdorff convergence","Noncommutative metric geometry","Quantum metric spaces","Approximately finite-dimensional algebras","Mathematics"],"dc:title":["Quantum Metrics on Approximately Finite-Dimensional Algebras"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T02:03:03Z"}