  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,
i.e. to the uniqueness of a C*-completion of the underlying Hopf *-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.
