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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...

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    $\begingroup$ Isn't there something at the end of Woronowicz's original paper, which says that as a Cstar algebra C_q[SU_2] doesn't depend on q (provided q\neq 0) and hence one can choose a particular q that makes calculations easier? Indeed,he shows that C_q[SU_2] is isomorphic as a Cstar algebra to some extension of something abelian by something reasonably nice, if I recall correctly $\endgroup$
    – Yemon Choi
    Jul 7, 2016 at 15:19
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    $\begingroup$ Did you look into arxiv.org/abs/math/0402074 $\endgroup$ Jul 7, 2016 at 23:23
  • $\begingroup$ These are a couple of interesting ideas - I will check them up! $\endgroup$ Jul 8, 2016 at 12:25
  • $\begingroup$ On looking at Woronowicz's construction, the Hilbert space can be modified into a Hilbert C* module, which does indeed give some answers. Thanks! $\endgroup$ Jul 9, 2016 at 6:49

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