Slides of a talk given to the Functional Analysis seminar in Besancon.

Some of these problems have since been solved.

### “e in support” implies convergence

Consider a on a *finite* quantum group such that where

,

with . This has a positive density of trace one (with respect to the Haar state ), say

,

where is the Haar element.

An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so . Assume that (if , then for all and we have trivial convergence)

Therefore let

be the density of .

Now we can explicitly write

.

This has stochastic operator

.

Let be an eigenvalue of of eigenvector . This yields

and thus

.

Therefore, as is also an eigenvector for , and is a stochastic operator (if is an eigenvector of eigenvalue , then , contradiction), we have

.

This means that the eigenvalues of lie in the ball and thus the only eigenvalue of magnitude one is , which has (left)-eigenvector the stationary distribution of , say .

If is symmetric/reversible in the sense that , then is self-adjoint and has a basis of (left)-eigenvectors and we have, if we write ,

,

which converges to (so that ).

If is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers converge and thus so do the

### Total Variation Decrasing

Uses Simeng Wang’s . Result holds for compact Kac if the state has a density.

### Periodic is concentrated on a coset of a proper normal subgroup of

is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by

### Supported on Subgroup implies Reducible

I believe I have a full proof that reducible is equivalent to supported on a pre-subgroup.

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