The authors introduce a two-parameter family of Dirac operators on quantum SU(2) and analyse their properties from the point of view of non-commutative metric geometry. It is shown that these Dirac operators give rise to compact quantum metric structures and that the corresponding two-parameter family of compact quantum metric spaces varies continuously in Rieffel’s quantum Gromov–Hausdorff distance.

This continuity result includes the classical case where the authors recover the round 3-sphere up to a global scaling factor on the metric. Their main technical tool is a quantum SU(2) analogue of the Berezin transform, together with its associated fuzzy approximations, the analysis of which also leads to a systematic way of approximating Lipschitz operators by means of polynomial expressions in the generators.