Compact Quantum Ergodic Systems

We develop theory of multiplicity maps for compact quantum groups, as an application, we obtain a complete classification of right coideal $C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are labeled with Dynkin diagrams, but classification results for positive and negative cases of $q$ are different. Many of the coideals are quantum spheres or quotient spaces by quantum subgroups, but we do have other ones in our classification list.