INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov-Hausdorff propinquity topology.