*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 .

Back to the paper, the “subgroup-like” objects are called *quasi-subgroups. *They are related to group-like projections.

I don’t fully have the functional analysis background (to… read this paper?!) to fully understand these objects, but we are denoting by the reduced -algebra of the locally compact quantum group . Let be the universal -algebra, and be the von Neumann algebra.

Denote the right Haar measure on by and the GNS Hilbert space of .

A von Neumann subalgebra is a left *coideal *if .

## 2. Idempotent States, Coideals, and Group-Like Projections

Let be a state on . The dual of this universal C*-algebra is Banach together with the convolution. Denote the set of idempotent states on the universal C*-algebra by .

Where is the half-lifted Kac-Takesaki Operator (??), the formula

,

for , defines a normal conditional expectation on and denote by its range:

,

which is a left-coideal in . Let

.

us a group-like projection in and is the orthogonal projection onto .

There is a bijective correspondence between:

- idempotent states on the universal C*-algebra
- left-coideals (+integrable + scaling group invariant)
- group like projections (+ invariant under the scaling group)

.

Let . We say that *dominates * if (I would usually be doing something similar with the support, perhaps… perhaps not) and we write . This is equivalent to (which could be something to use in the finite/compact case). The convention is the opposite to that of Franz & Skalski so beware!

#### Lemma 2.1

*Let . TFAE*

## 3. The Lattice of Compact Quasi-Subgroups

Let be a compact subgroup. This gives a surjective *-homomorphism from the universal C*-algebra of down to the universal C*-algebra of . We then have

.

This is a *Haar type *idempotent. Classically, all idempotents arise like this but not in the quantum case. In the quantum case let an idempotent correspond to a *quasi-subgroup *of . Under this line of thinking, the coideal corresponds to the von Neumann algebra of essentially bounded functions on the set of cosets of the quasi-subgroup.

### 3.1 Intersection of quasi-subgroups

Let . The von Neumann algebra generated by and , is an (integrable, -invariant) coideal, and thus of the form for some . Denote , and the corresponding quasi-subgroup as the *intersection *of the quasi-subgroups given by and . This operation is commutative and associative.

### 3.2 Quasi-subgroup generated by two quasi-subgroups

#### Theorem 3.1

- is weak*-convergent to
- iff iff is the idempotent state corresponding to the (-invariant, integrable, left) coideal

At this point I have no intuition for the following as I don’t understand what is… if we denote the above by , this vee is an associative and commutative map from pairs of idempotents to … hang on…

We call the quasi-subgroup corresponding to the quasi-subgroup generated by and . If , we say the quasi-subgroup is non-compact.

In particular, if is compact, then is non-zero, because is unital, and hence the set of states is closed in the weak* topology.

#### Remark 3.2

For , is the *smallest *(left) coideal containing both and . In particular, . Also is the *largest *idempotent dominated by both :

,

with a similar result for the quasi-group generated by .

#### Theorem 3.3

*Let such that*

*(weird, no?)*

*Then .
*

Note TFAE

## Open Compact Quasi-Subgroups

Consider the *reducing morphism*

,

*we embed the predual into (???).* The predual is a closed ideal of the Banach algebra given by . Consider idempotents that lie in this ideal, called normal idempotents:

.

Such states correspond to *open compact quasi-subgroups. *The following generalises the fact that if a subgroup of a topological group contains an open subgroup, then it is itself open.

#### Proposition 4.1

*For any , , such that , .*

*Proof*: Recall that the predual of the von Neumann algebra is an ideal in the convolution algebra. Therefore is in the predual also, and is thus an open compact quasi-subgroup

#### Corollary 4.2

*If is a discrete quantum group, then all quasi-subgroups are open compact.*

*Proof*: Presumably is in the predual (something to do with (co)amenability?), and of course any has , and thus corresponds to an open compact quasi-subgroup

For each open compact quasi-subgroup, consider its *left kernel*

.

This left-kernel is a -weakly closed left ideal in the algebra of measurable functions, and hence of the form (just what I need?!) for a unique projection . A measurable is in this left kernel if and only if . Note as a projection, itself satisfies this and thus , and by Schwarz for any measurable (yes, this is very useful). Moreover putting , for all measurable we have:

.

#### Remark 4.3

If is a positive, measurable, and , then (yelp!). Indeed for some , so that for some measurable . Thus .

Now I think I understand what is — in fairness the paper spoke about it. Think classically, indeed finitely. Let be the uniform distribution on a subgroup . The big scary definition for above… I am fairly sure is nothing as bad in this easier category than

.

Let . Note

.

Let . Then

.

Therefore this conditional expectation takes functions defined on the group to functions defined on the left cosets of .

We want to say that is the set of functions constant on left cosets. This will feed into the following lemma.

#### Lemma 4.4

*Let be an open compact quasi-subgroup. Then*

*iff .*

Let us illustrate the first (one direction) in the finite, classical case. Let be constant on left cosets of . That is

.

We have that is the projection while . We have

,

and

.

We can show that

.

Hitting both with shows that they are equal. Indeed both are equal to zero if . If then both are equal to .

*Proof: *Assume that . Hit both sides with . What we get (!) is the right-hand side gives

.

Now the left-hand side gives:

.

That is , that is is constant on left cosets.

Now assume that is constant on left cosets. Then so will (I actually can’t see this… or the next part)

…

#### Theorem 4.5

*Let be a projection, satisfying*

,

*, and . Then is group-like.*

Recall that is a projection such that for functions such that , .

#### Corollary 4.6

*Let be an open compact quasi-subgroup. Then is a group-like projection.*

Call the *support projection *of .

#### Theorem 4.8

*The support projection of any open compact quasi-subgroup is constant on conjugacy-classes (like one and zero). Moreover is minimal **(?) **and central among projections constant on conjugacy classes.*

#### Theorem 4.11

*For any open compact quasi-subgroup, for *

.

One more result of interest (to me).

#### Corollary 4.15

*Let give open compact quasi-subgroups. Then iff .*

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