Gromov-Hausdorff convergence of quantised intervals

Abstract

The Podleś quantum sphere Sq2 admits a natural commutative C⁎-subalgebra Iq with spectrum {0}∪{q2k:k∈N0}, which may therefore be considered as a quantised version of a classical interval. We study here the compact quantum metric space structure on Iq inherited from the corresponding structure on Sq2, 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.