This question is about the states on the matrix quantum group $C_q[SU_2]$ (generators $a,b,c,d$ with relations...), or possibly about the representations of the $C^*$ algebra $C_q[SU_2]$ - not about its co-representations. I stress this, as there is a vast literature about the co-representations.
Motivation: To generalise the idea of a quantum homogenous space. The quantum Hopf fibration is given by a Hopf algebra map $\pi:C_q[SU_2]\to C[S^1]$, giving a right coaction $\Delta_R=(id\otimes\pi)\Delta:C_q[SU_2]\to C_q[SU_2]\otimes C[S^1]$, and the coinvariants give a quantum sphere.
The problem is to generalise this sort of construction by replacing the Hopf algebra map $\pi$ with a completely positive map with some reasonable behaviour. However CP maps are not that easy to come by - the most obvious being the states. Woronowicz gave a construction of the Haar state on $C_q[SU_2]$, but what other explicit states are known? The obvious source would be star representations of $C_q[SU_2]$, but finding these is not so easy.
Any help on this would be gratefully received - I am sure that I am overlooking some obvious source of information...