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.
- 17B37 (secondary)
- 46L05 (primary)