Noncommutative Solenoids and the Gromov-Hausdorff Propinquity

TL;DR

This paper demonstrates that noncommutative solenoids can be approximated by quantum tori and finite-dimensional spaces, establishing their continuous variation within the framework of quantum metric spaces.

Contribution

It shows noncommutative solenoids are limits of quantum tori in the Gromov-Hausdorff propinquity and can be approximated by finite-dimensional quantum spaces.

Findings

Noncommutative solenoids are limits of quantum tori.

They can be approximated by finite-dimensional quantum spaces.

They form a continuous family over the space of multipliers.

Abstract

We prove that noncommutative solenoids are limits, in the sense of the Gromov-Hausdorff propinquity, of quantum tori. From this observation, we prove that noncommutative solenoids can be approximated by finite dimensional quantum compact metric spaces, and that they form a continuous family of quantum compact metric spaces over the space of multipliers of the solenoid, properly metrized.