Gromov-Hausdorff convergence of quantised intervals

The Podleś quantum sphere S _q ² admits a natural commutative C ^⁎-subalgebra I _q with spectrum {0}∪{q ^2k:k∈N ₀}, 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 _q ², 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 π as q tends to 1.