An ergodic action of a compact quantum group $G$ on an operator algebra $A$ can be interpreted as a quantum homogeneous space for $G$. Such an action gives rise to the category of finite equivariant Hilbert modules over $A$, which has a module structure over the tensor category $Rep(G)$ of finite-dimensional representations of $G$. We show that there is a one-to-one correspondence between the quantum $G$-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module $C^*$-categories over $Rep(G)$ up to natural equivalence. This gives a global approach to the duality theory for ergodic actions as developed by C. Pinzari and J. Roberts.
Keywords: compact quantum groups; $C^*$-algebras; Hilbert modules; ergodic actions; module categories
2010 MSC: 17B37; 20G42; 46L08
Theory and Applications of Categories, Vol. 28, 2013, No. 31, pp 1099-1138.