The Fell topology and the modular Gromov-Hausdorff propinquity

Abstract

We show that any (norm-closed two-sided) ideal of a unital C*-algebra that comes equipped with certain natural quantum metric structure is a metrized quantum vector bundle, when canonically viewed as a module over A A . Next, given a unital AF-algebra A A equipped with a faithful tracial state, we equip each ideal of A A with a metrized quantum vector bundle structure using previous work of the first author and Latrémolière. Moreover, we show that convergence of ideals in the Fell topology implies convergence of the associated metrized quantum vector bundles in the modular Gromov-Hausdorff propinquity of Latrémolière. In a similar vein but requiring a different approach, given a compact metric space ( X , d ) (X,d) , we equip each ideal of C ( X ) C(X) with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity.