*Taken from Random Walks on Finite Quantum Groups by Franz & Gohm.*

## Theorem

*Let be a state on a finite quantum group . Then the Cesaro mean*

,

*converges to an idempotent state on , i.e. to a state such that .*

*Proof *: Let be an accumulation point of , this exists since the states on form a compact set. We have

.

I have no idea where the equality comes from.

Choose sequence such that , we get and similarly . By linearity this implies . If is another accumulation point of and a sequence such that , then we get and thus by symmetry (??). Therefore the sequence has a unique accumulation point, i.e. it converges

### Remark

If is faithful, then the Cesaro limit is the Haar state on (prove this).

### Remark

Due to *cyclicity *the sequence does not converge in general. Take, for example, the state (p.28) on the Kac-Paljutkin quantum group , then we have

,

but

.

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