Gromov-Hausdorff convergence of quantised intervals

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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.
Translated title of the contributionGromov-Hausdorff convergence of quantised intervals
Original languageEnglish
Journalarxiv.org
Number of pages12
Publication statusSubmitted - Jun 2020

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