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What started about ten months ago as a technical question to an expert, led to a talk, and led to me producing this weird production here.

Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

Finally cracked this egg.

Preprint here.

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.

The end of the previous Research Log suggested a way towards showing that can be associated to an idempotent state . Over night I thought of another way.

Using the Pierce decomposition with respect to (where ),

.

The corner is a hereditary -subalgebra of . This implies that if and for , .

Let . We know from Fagnola and Pellicer that and .

By assumption in the background here we have an irreducible and periodic random walk driven by . This means that for all projections , there exists such that .

Define:

.

Define:

.

The claim is that the support of , is equal to .

We probably need to write down that:

.

Consider for any . Note

that is each is supported on . This means furthermore that .

Suppose that the support . A question arises… is ? This follows from the fact that and is hereditary.

Consider a projection . We know that there exists a such that

.

This implies that , say (note ):

By assumption . Consider

.

For this to equal one every must equal one but .

Therefore is the support of .

Let . We have shown above that for all . This is an idempotent state such that is its support (a similar argument to above shows this). Therefore is a group like projection and so we denote it by and !

Today, for finite quantum groups, I want to explore some properties of the relationship between a state , its density (), and the support of , .

I also want to learn about the interaction between these object, the stochastic operator

,

and the result

,

where is defined as (where by ).

.

An obvious thing to note is that

.

Also, because

That doesn’t say much. We are possibly hoping to say that .

In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

## Introduction

Consider a random walk on a finite group driven by . The state of the random walk after steps is given by , defined inductively (on the algebra of functions level) by the associative

.

The convolution is also implemented by right multiplication by the stochastic operator:

,

where has entries, with respect to a basis . Furthermore, therefore

,

and so the stochastic operator describes the random walk just as well as the driving probabilty .

The random walk driven by is said to be *irreducible *if for all , there exists such that (if ) .

The *period *of the random walk is defined by:

.

The random walk is said to be *aperiodic *if the period of the random walk is one.

These statements have counterparts on the set level.

If is not irreducible, there exists a proper subset of , say , such that the set of functions supported on are -invariant. It turns out that is a proper subgroup of .

Moreover, when is irreducible, the period is the greatest common divisor of all the natural numbers such that there exists a partition of such that the subalgebras of functions supported in satisfy:

and (slight typo in the paper here).

In fact, in this case it is necessarily the case that is concentrated on a coset of a proper normal subgroup , say . Then .

Suppose that is supported on . We want to show that for . Recall that

.

This shows how the stochastic operator reduces the index .

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

,

specifically the maps can speak to the irreducibility and periodicity of the random walk given by . I am not convinced that I need these results (even though I show how they are applicable).

## Stochastic Operators and Operator Algebras

Let be a -algebra (so that is in general a virtual object). A -subalgebra is *hereditary *if whenever and , and , then .

It can be shown that if is a hereditary subalgebra of that there exists a projection such that:

.

All hereditary subalgebras are of this form.

*This sandbox is going to take from a variety of sources, mostly Shuzhou Wang.*

## C*-Ideals

Let be a closed (two-sided) ideal in a non-commutative unital -algebra . Such an ideal is self-adjoint and so a non-commutative -algebra . The quotient map is given by , , where is the equivalence class of under the equivalence relation:

.

Where we have the product

,

and the norm is given by:

,

the quotient is a -algebra.

Consider now elements and . Consider

.

The tensor product . Now note that

,

by the nature of the Tensor Product (). Therefore .

### Definition

A WC*-ideal (W for *Woronowicz*) is a C*-ideal such that , where is the quotient map .

Let be the algebra of functions on a classical group . Let . Let be the set of functions which vanish on : this is a C*-ideal. The kernal of is .

Let so that . Note that

and so

.

Note that if . It is not possible that both and are in : if they were , but , which is not in by assumption. Therefore one of or is equal to zero and so:

,

and so by linearity, if vanishes on a subgroup ,

.

In this way, WC*-ideals generalise functions which vanish on distinguished subgroups. In fact, without checking all the details, I imagine that first isomorphism theorem can show that . Let be the ring homomorphism

.

Then , , and so we have the isomorphism of rings, which presumably carries forward to the algebras of functions level…

*Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.*

#### Abstract

From this paper I will look at:

- lattice operations on , for a LCQG (analogues of intersection and generation)

## 1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with *quasi-subgroups *(the work of Franz & Skalski the correspondence was with *pre-subgroups *and *group-like projections*).

Let us show the kind of thing I am trying to understand better.

Let be the algebra of function on a finite quantum group. Let be concentrated on a pre-subgroup . We can associate to a group like projection .

Let, and this is another thing I am trying to understand better, this support, the support of be ‘the smallest’ (?) projection such that . Denote this projection by . Define similarly. That are concentrated on is to say that and .

Define a map by

(or should this be or ?)

We can decompose, in the finite case, .

**Claim: **If is concentrated on , … I don’t have a proof but it should fall out of something like together with the decomposition of above. It may also require that is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That is a group-like projection means that:

Hit both sides with to get:

.

By the fact that are supported on , the right-hand side equals one, and by the as-yet-unproven claim, we have

.

However this is the same as

,

in other words , that is remains supported on . As a corollary, a random walk driven by a probability concentrated on a pre-subgroup remains concentrated on .

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