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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∗   ‐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.

On C ∗ ‐completions of discrete quantum group rings