Convergence of Quotients of AF Algebras in Quantum Propinquity by Convergence of Ideals

TL;DR

This paper develops a topology on the ideal space of inductive limits of C*-algebras, establishing conditions under which ideal convergence implies convergence of quotients in the quantum propinquity, thus linking ideal topology to quantum metric convergence.

Contribution

It introduces a new topology on ideal spaces of AF algebras and connects ideal convergence to quantum Gromov-Hausdorff propinquity convergence of quotients.

Findings

Established a topology on ideal spaces compatible with inductive limits.

Provided criteria for ideal convergence implying quotient convergence in quantum propinquity.

Constructed a continuous map from ideal classes to quotient spaces with the propinquity topology.

Abstract

We introduce a topology on the ideal space of inductive limits of C*-algebras built by a topological inverse limit of the Fell topologies on the C*-algebras of the given inductive sequence and we produce conditions for when this topology agrees with the Fell topology of the inductive limit. With this topology, we impart criteria for when convergence of ideals of an AF algebra can provide convergence of quotients in the quantum Gromov-Hausdorff propinquity building off previous joint work with Latremoliere. These findings bestow a continuous map from a class of ideals of the Boca-Mundici AF algebra equipped with various topologies including Jacobson and Fell topologies to the space of quotients equipped with the propinquity topology.