# Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra

## TL;DR

This paper develops quantum metrics on the tensor product of a commutative C*-algebra and an AF algebra, proving convergence and compatibility properties within the framework of noncommutative geometry.

## Contribution

It introduces a new quantum metric on the tensor product of C(X) and A, demonstrating its compatibility with tensor products and establishing convergence in the Gromov-Hausdorff propinquity.

## Key findings

- Quantum metric on tensor product is compatible with tensor structure.
- The inductive limit of tensor products converges in the Gromov-Hausdorff propinquity.
- Extended results to continuous families of tensor products with AF algebras.

## Abstract

Given a compact metric space X and a unital AF algebra A equipped with a faithful tracial state, we place quantum metrics on the tensor product of C(X) and A given established quantum metrics on C(X) and A from work with Bice and Latr\'emoli\`ere. We prove the inductive limit of C(X) tensor A given by A is a metric limit in the Gromov-Hausdorff propinquity. We show that our quantum metric is compatible with the tensor product by producing a Leibniz rule on elementary tensors and showing the diameter of our quantum metric on the tensor product is bounded above the diameter of the Cartesian product of the quantum metric spaces. We provide continuous families of C(X) tensor A which extends our previous results with Latr\'emoli\`ere on UHF algebras.