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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.

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

**Abstract** *A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.*

After a long time I have finally completed my PhD studies when I handed in my hardbound thesis (a copy of which you can see here).

It was a very long road but thankfully now the pressure is lifted and I can enjoy my study of quantum groups and random walks thereon for many years to come.

I have finally finished the first draft of my PhD thesis. My advisor Dr Stephen Wills is presently reading through it and will get back to me with his comments in the next few weeks. The project was successful in that I managed to prove the Diaconis-Shahshahani Upper Bound Lemma for finite quantum groups… how successful my application of the Lemma to concrete examples is probably open to debate. First draft of abstract and introduction — without references — below the fold.

Let be a finite quantum group described by with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) . Under the assumption of finiteness, there is a unique Haar state, on .

# Representation Theory

A *representation* of is a linear map that satisfies

The dimension of is given by . If has basis then we can define the *matrix elements* of by

One property of these that we will use it that .

Two representations and are said to be *equivalent*, , if there is an invertible *intertwiner* between them. An intertwiner between and is a map such that

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if is a maximal family of pairwise inequivalent irreducible representation that is a basis of . When we refer to “the matrix elements” we always refer to such a family. We define the span of as , the *space of matrix elements of* .

Given a representation , we define its *conjugate*, , where is the conjugate vector space of , by

so that the matrix elements of are .

Timmermann shows that the matrix elements have the following orthogonality relations:

- If and are inequivalent then for all and .
- If is such that the conjugate, , is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

This second relation is more complicated without the assumption and refers to the entries and trace of an intertwiner from to the coreprepresention with matrix elements . If , then this intertwiner is simply the identity on and so the the entries and the trace is .

Denote by the set of unitary equivalence classes of irreducible unitary representations of . For each , let be a representative of the class where is the finite dimensional vector space on which acts.

# Diaconis-Van Daele Fourier Theory

The following runs a thread through what I’ve looked at over the past year: Progression Report.

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I managed to get through two sections last week: *Compact Hilbert Space Operators *and *The Spectral Theorem*. I also have 9 of 12 chapter 2 exercises completed. I have been writing my study up here and this is proving fruitful on three counts:

- I can put questions in red for my supervisor to see
- I am not happy putting up something on this page that I haven’t justified to myself. This means I have to fill in some extra steps (in blue)
- I should have a nice set of notes to peruse should I need them

Unfortunately this week will be mostly concerned with preparing lectures for two modules that I will be lecturing in CIT:

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

Before the Christmas break I finished off the chapter 1 exercises.

*Chapter 2: C*-Algebras and Hilbert Space Operators.*

**2.1 C*-Algebras**

Initially we defined a C*-algebra, , as a complete normed algebra, together with a conjugate-linear *involution ** that satisfies the *C*-equation:*

,

*Self-adjoint *or *Hermitian *elements are defined to have the property . As a consequence of this, and the C*-equation, the spectral radius of a self-adjoint element, , is equal to its norm, . As a corollary of this, of all the norms that can be put on the *-algebra, only one makes it into a C*-algebra – i.e. satisfying the C*-equation.

In the previous chapter we have seen that an algebra, , can be unitised to form a new algebra, , which contains as a subspace. In general, the norm got by extending the norm on to a norm on does not make into a C*-algebra. However Theorem 2.1.6 shows that there does exist a (unique) norm on making it a C*-algebra. In many examples we may now assume that a general C*-algebra is unital – replacing it with the unique unitisation, , if necessary.

One such result which depends on this fact is that the the spectrum of a self-adjoint element is real.

A central result in this chapter is that all abelian C*-algebras are , for some locally compact Hausdorff space, . In fact is the character space (as with Belton, this is via the Gelfand transformation). This identification allows the development of the powerful *functional calculus. *Briefly, if is a normal element of a C*-algebra , (), and is the inclusion map from , then there exists a unique *-homomorphism such that . This unique *-homorphism is called *the functional calculus at *. This particular section ended with the Belton result that if is a compact Hausdorff space, (via ).

**2.2 Positive Elements of C*-Algebras**

This section introduces a partial order on (the set of self-adjoint elements of ). Namely, an element is *positive *if . The partial order is defined in the obvious way.

As a consequence of the Gelfand transformation and the functional calculus, we can show that positive elements of a C*-algebra possess unique positive square roots. Another prominent result is that for an arbitrary element , is positive.

**2.3 Operators and Sesquilinear Forms**

As a first move, we prove that bounded operators on Hilbert spaces have adjoints. Next projections are examined and partial isometries are examined. This leads onto the polar decomposition theorem. Namely, if is a continuous linear operator on a Hilbert space , there exists a unique partial isometry such that ; where . The rest of the section focusses on the connection between operators and sesquilinear forms.

**2.4 Compact Hilbert Space Operators**

At first this chapter looks at some of the basic properties of these objects – e.g. if is compact so are and . Thus is self-adjoint and thus a C*-algebra (it is a closed ideal in ). We see that normal compact operators are diagonalisable.

We look at the finite rank operators, and see that they are dense in . Next the operator is examined:

These are rank-one, and the are rank-one projections if is a unit vector. This leads on to the fact that is linearly spanned by these rank-one projections.

**This is a synopsis of what I covered up until recently (up to p.56). As an experiment I am attempting to do my study of Murphy by way of fully presenting the details on this webpage. I am unsure of whether or not this is too time consuming. Presently I am on page 63 and I will have to cover the rest of the chapter material (10 pages) in one day or similar if I am going to consider this tactic feasible.**

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I have finished off section 1.4 including Atkinson’s Theorem and a first look at the unilateral shift. I have done exercises 1-7. In terms of progress, I am on p.31 of 265, with 13 exercises left in this section. Following discussions with my supervisor, I may be able to leave out sections 3.2, 3.5, 4.4, 5.2-6 and the whole of chapter 7.

I have finished off my revision of sections 1.2 (The Spectrum and the Spectral Radius) & 1.3 (The Gelfand Representation). Section 1.4 is a new topic for me – Compact and Fredholm Operators. A linear map between Banach spaces is *compact* if is totally bounded. As a corollary, all linear maps on finite dimensional spaces are compact. The transpose has been introduced by Murphy is this chapter, and I have seen that if is compact, then so is . A linear map is *Fredholm* if the and are finite dimensional. In terms of progress, I am on p.25 of 265.

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