The Fell topology and the modular Gromov-Hausdorff propinquity

TL;DR

This paper establishes a connection between the Fell topology on ideals of certain algebras and convergence in the modular Gromov-Hausdorff propinquity, linking algebraic and metric convergence in noncommutative geometry.

Contribution

It introduces a framework to view ideals as metrized quantum vector bundles and proves that Fell topology convergence implies convergence in the modular Gromov-Hausdorff propinquity.

Findings

Convergence of ideals in the Fell topology implies convergence of associated quantum bundles.

The framework applies to AF-algebras and continuous functions on compact metric spaces.

Provides a new perspective on the relationship between algebraic and metric convergence.

Abstract

Given a unital AF-algebra $A$ equipped with a faithful tracial state, we equip each (norm-closed two-sided) ideal of $A$ with a metrized quantum vector bundle structure, when canonically viewed as a module over $A$, in the sense of Latr\'emoli\`ere using previous work of the first author and Latr\'emoli\`ere. 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 Latr\'emoli\`ere. In a similar vein but requiring a different approach, given a compact metric space $(X,d)$, we equip each ideal of $C(X)$ with a metrized quantum vector bundle structure, and show that convergence in the Fell topology implies convergence in the modular Gromov-Hausdorff propinquity.