Gromov-Hausdorff convergence of quantised intervals

TL;DR

This paper investigates the quantum metric space structure of a subalgebra of the Podles quantum sphere, demonstrating its continuous Gromov-Hausdorff convergence to a classical interval as the deformation parameter approaches 1.

Contribution

It provides an explicit formula for the induced metric on the spectrum and proves the Gromov-Hausdorff convergence to a classical interval in the quantum deformation setting.

Findings

Explicit metric formula on the spectrum of I_q

Continuous Gromov-Hausdorff convergence in q

Limit of quantum spaces as q approaches 1 is a classical interval

Abstract

The Podles quantum sphere S^2_q admits a natural commutative C*-subalgebra I_q with spectrum {0} \cup {q^{2k}: k = 0,1,2,...}, which may therefore be considered as a quantised version of a classical interval. We study here the compact quantum metric space structure on I_q inherited from the corresponding structure on S^2_q, and provide an explicit formula for the metric induced on the spectrum. Moreover, we show that the resulting metric spaces vary continuously in the deformation parameter q with respect to the Gromov-Hausdorff distance, and that they converge to a classical interval of length pi as q tends to 1.