Ergodic theorems and idempotents on compact quantum groups

Recent results of M. Junge and Q. Xu on the ergodic properties of the averages of kernels in noncommutative L^p-spaces are applied to the
analysis of the almost uniform convergence of operators induced by the convolutions on compact quantum groups. The limit corresponds to an
idempotent state, which, if it is faithful, coincides with the Haar state. To gain a better understanding of the general case, we then try
to characterise all idempotent states on compact quantum groups.