The strongly Leibniz property and the Gromov--Hausdorff propinquity

TL;DR

This paper introduces a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property, ensuring completeness and convergence for certain classes of quantum metric spaces and C*-algebras.

Contribution

It develops a strongly Leibniz-sensitive Gromov--Hausdorff propinquity and establishes convergence results for inductive limits and AF-algebras using this new metric.

Findings

The new propinquity is complete on strongly Leibniz quantum compact metric spaces.

Provides conditions for inductive limits to converge in the new Gromov--Hausdorff propinquity.

Demonstrates convergence of Effros--Shen algebras with Frobenius--Rieffel norms.

Abstract

We construct a new version of the dual Gromov--Hausdorff propinquity that is sensitive to the strongly Leibniz property. In particular, this new distance is complete on the class of strongly Leibniz quantum compact metric spaces. Then, given an inductive limit of C*-algebras for which each C*-algebra of the inductive limit is equipped with a strongly Leibniz L-seminorm, we provide sufficient conditions for placing a strongly Leibniz L-seminorm on an inductive limit such that the inductive sequence converges to the inductive limit in this new Gromov--Hausdorff propinquity. As an application, we place new strongly Leibniz L-seminorms on AF-algebras using Frobenius--Rieffel norms, for which we have convergence of the Effros--Shen algebras in the Gromov--Hausdorff propinquity with respect to their irrational parameter.