A Limit Theorem for Discrete Quantum Groups

We consider the {\it concentration functions problem} for discrete quantum groups; we prove that if $\mathbb{G}$ is a discrete quantum group, and $\mu$ is an irreducible state in $l^1(\mathbb{G})$, then the convolution powers $\mu^n$, considered as completely positive maps on $c_0(\mathbb{G})$, converge to zero in strong operator topology.