THE FELL TOPOLOGY AND THE MODULAR GROMOV-HAUSDORFF PROPINQUITY

Given a unital AF-algebra A equipped with a faithful tracial state, we endow each (norm-closed two-sided) ideal of A with a metrized quantum-vector-bundle structure. The ideals are canonically viewed as modules over A, in the sense of Latrémolière, using previous work of Aguilar and Latrémolière. Moreover, we show that the convergence of ideals in the Fell topology implies the 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), we equip each ideal of the commutative C*-algebra C(X) with a metrized quantum-vector-bundle structure, and show that convergence in the Fell topology implies the convergence in the modular Gromov-Hausdorff propinquity. (This is joint work with Jiahui Yu: arXiv:2211.11107.)