In a recent preprint, Haonan Zhang shows that if (where is a Sekine Finite Quantum Group), then the convolution powers, , converges if

.

The algebra of functions is a multimatrix algebra:

.

As it happens, where , the counit on is given by , that is , dual to .

To help with intuition, making the incorrect assumption that is a classical group (so that is commutative — it’s not), because , the statement , implies that for a real coefficient ,

,

as for classical groups .

That is the condition is a quantum analogue of .

Consider a random walk on a classical (the algebra of functions on is commutative) *finite* group driven by a .

The following is a very non-algebra-of-functions-y proof that implies that the convolution powers of converge.

*Proof: *Let be the smallest subgroup of on which is supported:

.

We claim that the random walk on driven by is *ergordic* (see Theorem 1.3.2).

The driving probability is not supported on any proper subgroup of , by the definition of .

If is supported on a coset of proper normal subgroup , say , then because , this coset must be , but this also contradicts the definition of .

Therefore, converges to the uniform distribution on

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of .

So we have some questions:

- Is there a proof of the classical result (above) in the language of the algebra of functions on , that necessarily bypasses talk of points and of subgroups?
- And can this proof be adapted to the quantum case?
- Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

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