On C∗-completions of discrete quantum group rings

We discuss just infiniteness of C^*-algebras associated to discrete quantum groups and relate it to the C^*-uniqueness of the quantum groups in question, that is, to the uniqueness of a C^*-completion of the underlying Hopf C^*-algebra. It is shown that duals of q-deformations of simply connected semisimple compact Lie groups are never C^*-unique. On the other hand, we present an example of a discrete quantum group which is not locally finite and yet is C^*-unique.