Compact quantum metric spaces and ergodic actions of compact quantum groups

Abstract

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T->EA sending each t in T to the isomorphism class of the fibre at t is continuous if and only if the function counting the multiplicity of gamma in each fibre is continuous over T for every equivalence class gamma of irreducible unitary representations of G. Generalizations for arbitrary compact quantum groups are also obtained. In the case G is a compact group, the restriction of this topology on the subset of isomorphism classes of ergodic actions of full multiplicity coincides with the topology coming from the work of Landstad and Wassermann. Podles spheres are shown to be continuous in the natural parameter as ergodic actions of the quantum SU(2) group. When A is separable, we also introduce a notion of regular quantum metric on G, and show how to use it to induce a quantum metric on any ergodic action of G in the sense of Rieffel. Furthermore, we introduce a quantum Gromov-Hausdorff distance between ergodic actions and show that it induces the above topology.