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## Distances between Probability Measures

Let be a finite quantum group and be the set of states on the -algebra .

The algebra has an invariant state , the dual space of .

Define a (bijective) map , by

,

for .

Then, where and , define the total variation distance between states by

.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative machinary shows that:

.

(supremum presentation)

In the classical case, using the test function , where , we have the probabilists’ preferred definition of total variation distance:

.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in , and therefore the classical total variation distance is equal to:

.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

## Quantum Projection Distance Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a factor of might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

Just back from a great workshop at Seoul National University, I am just going to use this piece to outline in a relaxed manner my key goals for my work on random walks on quantum groups for the near future.

In the very short term I want to try and get a much sharper lower bound for my random walk on the Sekine family of quantum groups. I believe the projection onto the ‘middle’ of the might provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller — see Section 5.7).

I also want to understand how sharp (or otherwise) the order convergence for the random walk on the dual of is — sounds awfully high. Furthermore it should be possible to get a better lower bound that what I have.

It should also be possible to redefine the quantum total variation distance as a supremum over projections subsets via . If I can show that for a positive linear functional that then using these ideas I can. More on this soon hopefully. No, this approach won’t work. (I have since completed this objective with some help: see here).

The next thing I might like to do is look at a random walk on the Sekine quantum groups with an -dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for , some cut-off time.

Going back to the start, the classical problem began around 1904 with the question of Markov:

Which card shuffles mix up a deck of cards and cause it to ‘go random’?

For example, the perfect riffle shuffle does not mix up the cards at all while a riffle shuffle done by an amateur will.

In the context of random walks on classical groups this question is answered by the Ergodic Theorem 1.3.2: when the driving probability is not concentrated on a subgroup (irreducibility) nor the coset of a normal subgroup (aperiodicity).

Necessary and sufficient conditions on the driving probability for the random walk on a *quantum *group to converge to random are required. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).

This will involve a study of subgroups of quantum groups as well as *normal *quantum subgroups.

It should be straightforward to extend the Upper Bound Lemma (Lemma 5.3.8) to the case of compact Kac algebras. Once that is done I will want to look at quantum generalisations of ‘natural’ random walks and shuffles.

I intend also to put the PhD thesis on the Arxiv. After this I have a number of options as regard to publishing what I have or maybe waiting a little while until I solve the above problems — this will all depend on how my further study progresses.

*Taken from C*-Algebras and Operator Theory by Gerald J. Murphy.*

Although the principal aim of this section is to construct direct limits of C*-algebras, we begin with direct limits of groups.

If is a sequence of groups, and if for each we have a homomorphism , then we call a *direct sequence of groups. *Given such a sequence and positive integers , we set and we define inductively on by setting

.

If , we have .

If is a group and we have homomorphisms such that the diagram

commutes for each , that is , then for all .

*Taken from C*-Algebras and Operator Theory by Gerald Murphy.*

This section is concerned with positive linear functionals and representations. Pure states are introduced and shown to be the extreme points of a certain convex set, and their existence is deduced from the Krein-Milman theorem. From this the existence of irreducible representations is proved by establishing a correspondence between them and the pure states.

If is a representation of a C*-algebra , we say is a *cyclic vector for *if is cyclic for the C*-algebra (This means that cyclic vector is a vector such that the closure of the linear span of equals ). If admits a cyclic vector, then we say that it is a *cyclic representation.*

We now return to the GNS construction associated to a state to show that the representations involved are cyclic.

## Theorem 5.1.1

*Let be a C*-algebra and . Then there is a unique vector such that*

*, for* .

*Moreover, is a unit cyclic vector for and*

,* for *.

*Taken from C*-algebras and Operator Theory by Gerald Murphy.*

If and are vector spaces, we denote by their algebraic tensor product. This is linearly spanned by the elements (, ).

One reason why tensor products are useful is that they turn bilinear maps (a bilinear map has ) into linear maps (). More precisely, if is a bilinear map, where and are vector spaces, then there is a unique linear map such that for all and .

If are linear functionals on the vector spaces respectively, then there is a unique linear functional on such that

since the function

, ,

is bilinear.

Suppose that the finite sum , where and . If are linearly independent, then . For, in this case, there exist linear functionals such that . If is linear, we have

.

Thus for arbitrary and this shows that all the .

Similarly if the finite sum with the linearly independent, implies that all the are zero.

*Taken from C*-algebras and Operator Theory by Gerald Murphy.*

We prepare the way for the density theorem with some useful results on strong convergence.

## Theorem 4.3.1

*If is a Hilbert space, the involution is strongly continuous when restricted to the set of normal operators of .*

### Proof

Let and suppose that are normal operators in . Then

If is a net of normal operators strongly convergent to a normal operator , then the net is convergent to and the net is convergent to , so is convergent to . Therefore, is strongly convergent to

*Taken from C*-algebras and Operator Theory by Gerald Murphy.*

Preparatory to our introduction of the weak and ultraweak topologiesm we show now that is the dual of , and is the dual of .

Let be a Hilbert space, and suppose that . It follows from Theorem 2.4.16 (https://jpmccarthymaths.wordpress.com/2011/01/18/c-algebras-and-operator-theory-2-4-compact-hilbert-space-operators/) that the function

, ,

is linear and bounded, and . We therefore have a map

, ,

which is clearly linear and norm-decreasing. We call this map the *canonical map from to .*

## Theorem 4.2.1

*If is a Hilbert space, then the canonical map from to is an isometric linear isomorphism.*

*Taken from C*-algebras and Operator Theory by Gerald Murphy.*

A useful way of thinking of the theory of C*-algebras is as “non-commutative topology”. This is justified by the correspondence between abelian C*-algebras and locally compact Hausdorff spaces given by the Gelfand representation. The algebras studied in this chapter, von Neumann algebras, are a class of C*-algebras whose study can be thought of as “non-commutative measure theory”. The reason for the analogy in this case is that the abelian von Neumann algebras are (up to isomorphism) of the form , where is a measure space.

The theory of von Neumann algebras is a vast and very well-developed area of the theory of operator algebras. We shall be able only to cover some of the basics. The main results of this chapter are the von Neumann double commutant theorem and the Kaplansky density theorem.

## Read the rest of this entry »

# Question 1

*Let be normal elements of a C*-algebra **, and ** an element of ** such that **. Show that **, using Fuglede’s theorem and the fact that the element*

*is normal in ** and commutes with*

*.*

*This more general result is called the Putnam-Fuglede theorem.*

## Solution

Fuglede’s theorem states that if is a normal element commuting with some , then also commutes with . Now we can show that using the normality of and . We can also show that and commute. Hence by the theorem and commute. This yields:

.

Taking conjugates:

,

as required

* Read the rest of this entry »*

In this section we introduce the important GNS construction and prove that every C*-algebra can be regarded as a C*-subalgebra of for some Hilbert space . It is partly due to this concrete realisation of the C*-algebras that their theory is so accessible in comparison with more general Banach spaces.

A *representation *of a C*-algebra is a pair where is a Hilbert space and is a *-homomorphism. We say is *faithful *if is injective.

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