I thought I had a bit of a breakthrough. So, consider the algebra of a functions on the dual (quantum) group . Consider the projection:

.

Define by:

.

Note

.

Note so is a partition of unity.

I know that corresponds to a quasi-subgroup but not a quantum subgroup because is not normal.

This was supposed to say that the result I proved a few days ago that (in context), that corresponded to a quasi-subgroup, was as far as we could go.

For , note

,

is a projection, in fact a group like projection, in .

Alas note:

That is the group like projection associated to is subharmonic. This *should* imply that nearby there exists a projection such that for all … also is subharmonic.

This really should be enough and I should be looking perhaps at the standard representation, or the permutation representation, or … but I want to find the projection…

Indeed …and .

The punchline… the result of Fagnola and Pellicer holds when the random walk is is irreducible. This walk is not… back to the drawing board.

I have constructed the following example. The question will be does it have periodicity.

Where is the permutation representation, , and , is given by:

.

This has (duh), , and otherwise .

The above is still a cyclic partition of unity… but is the walk irreducible?

The easiest way might be to look for a subharmonic . This is way easier… with it is easy to construct non-trivial subharmonics… not with this . It is straightforward to show there are no non-trivial subharmonics and so is irreducible, periodic, but is not a quantum subgroup.

It also means, in conjunction with work I’ve done already, that I have my result:

**Definition** Let be a finite quantum group. A state is *concentrated on a cyclic coset of a proper quasi-subgroup* if there exists a pair of projections, , such that , is a group-like projection, and there exists () such that .

## (Finally) The Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group is ergodic if and only if the driving probability is not concentrated on a proper quasi-subgroup, nor on a cyclic coset of a proper quasi-subgroup.

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