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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 $M_n(\mathbb{C})$ 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 $\alpha$ — see Section 5.7).

I also want to understand how sharp (or otherwise) the order $n^n$ convergence for the random walk on the dual of $S_n$ is — $n^n$ 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 $\sim$ subsets via $G \supset S\leftrightarrow \mathbf{1}_S$. If I can show that for a positive linear functional $\rho$ that $|\rho(a)|\leq \rho(|a|)$ then using these ideas I can. More on this soon hopefully. No, this approach won’t work.

The next thing I might like to do is look at a random walk on the Sekine quantum groups with an $n$-dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for $k\ll t_n$, 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 $\nu\in M_p(\mathbb{G})$ 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 $\{G_n\}_{n=1}^\infty$ is a sequence of groups, and if for each $n$ we have a homomorphism $\varphi_n:G_n\rightarrow G_{n+1}$, then we call $\{G_n\}_{n\geq1}$ a direct sequence of groups. Given such a sequence and positive integers $n\leq m$, we set $\varphi_{nn}=I_{G_n}$ and we define $\varphi_{nm}:G_n\rightarrow G_m$ inductively on $m$ by setting

$\varphi_{n,m+1}=\varphi_m\varphi_{nm}$.

If $n\leq m\leq k$, we have $\varphi_{nk}=\varphi_{mk}\varphi_{nm}$.

If $G'$ is a group and we have homomorphisms $\theta^n:G^n\rightarrow G'$ such that the diagram

commutes for each $n$, that is $\theta^n=\theta^{n+1}\varphi_n$, then $\theta^n=\theta^m\varphi_{nm}$ for all $m\geq n$.

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 $(H,\varphi)$ is a representation of a C*-algebra $A$, we say $x\in H$ is a cyclic vector for $(H,\varphi)$ if $x$ is cyclic for the C*-algebra $\varphi(A)$ (This means that cyclic vector is a vector $x\in H$ such that the closure of the linear span of $\{\varphi(a)x\,:\,a\in A\}$ equals $H$). If $(H,\varphi)$ 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 $A$ be a C*-algebra and $\rho\in S(A)$. Then there is a unique vector $x_\rho\in H_\rho\in H$ such that

$\rho(a)=\langle a+N_\rho,x_\rho\rangle$, for $a\in A$.

Moreover, $x_\rho$ is a unit cyclic vector for $(H_\rho,\varphi_\rho)$ and

$\varphi_\rho(a)x_\rho=a+N_\rho$, for $a\in A$.

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

If $H$ and $K$ are vector spaces, we denote by $H\otimes K$ their algebraic tensor product. This is linearly spanned by the elements $x\otimes y$ ($x\in H$$y\in K$).

One reason why tensor products are useful is that they turn bilinear maps (a bilinear map $\varphi$ has $\lambda\varphi(x,y)=\varphi(\lambda x,y)=\varphi(x,\lambda y)$) into linear maps ($\lambda\varphi(x,y)=\varphi(\lambda x,\lambda y)$). More precisely, if $\varphi:H\times K\rightarrow L$ is a bilinear map, where $H,\,K$ and $L$ are vector spaces, then there is a unique linear map $\varphi_1:H\otimes K\rightarrow L$ such that $\varphi_1(x\otimes y)=\varphi(x,y)$ for all $x\in H$ and $y\in K$.

If $\rho,\,\tau$ are linear functionals on the vector spaces $H,\,K$ respectively, then there is a unique linear functional $\rho\otimes\tau$ on $H\otimes K$ such that

$(\rho\otimes\tau)(x\otimes y)=\rho(x)\tau(y)$

since the function

$H\times K\rightarrow\mathbb{C}$$(x,y)\mapsto \rho(x)\tau(y)$,

is bilinear.

Suppose that the finite sum $\sum_jx_j\otimes y_j=0$, where $x_j\in H$ and $y_j\in K$. If $y_1,\dots,y_n$ are linearly independent, then $x_1=\cdot=x_n=0$. For, in this case, there exist linear functionals $\rho_j:K\rightarrow \mathbb{C}$ such that $\rho_j(y_i)=\delta_{ij}$. If $\rho:H\rightarrow\mathbb{C}$ is linear, we have

$0=(\rho\otimes \rho_j)(\sum_i x_j\otimes y_j)=\sum_i\rho(x_i)\rho_j(y_i)=\rho(x_j)$.

Thus $\rho(x_j)=0$ for arbitrary $\rho$ and this shows that all the $x_j=0$.

Similarly if the finite sum $\sum_jx_j\otimes y_j=0$ with the $x_j$ linearly independent, implies that all the $y_j$ 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 $H$ is a Hilbert space, the involution $T\mapsto T^*$ is strongly continuous when restricted to the set of normal operators of $B(H)$.

### Proof

Let $x\in H$ and suppose that $T,S$ are normal operators in $B(H)$. Then

$\|(S^*-T^*)(x)\|^2=\langle S^*x-T^*x,S^*x-T^*x\rangle$

$=\|Sx\|^2-\|Tx\|^2+\langle TT^*x,x\rangle-\langle ST^*x,x\rangle$

$+\langle TT^*x,x\rangle-\langle TS^*x,x\rangle$

$=\|Sx\|^2-\|Tx\|^2+\langle (T-S)T^*x,x\rangle+\langle x,(T-S)T^*x\rangle$

$\leq \|Sx\|^2-\|Tx\|^2+2\|(T-S)T^*x\|\|x\|.$

If $\{T_\lambda\}_{\lambda\in\Lambda}$ is a net of normal operators strongly convergent to a normal operator $T$, then the net $\|T_\lambda x\|^2$ is convergent to $\|Tx\|^2$ and the net $\{(T-T_\lambda)T^*x\}$ is convergent to $0$, so $\{T_\lambda^*x-T^*x\}$ is convergent to $0$. Therefore, $\{T_\lambda^*\}$ is strongly convergent to $T^*$ $\bullet$

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

Preparatory to our introduction of the weak and ultraweak topologiesm we show now that $L^1(H)$ is the dual of $K(H)$, and $B(H)$ is the dual of $L^1(H)$.

Let $H$ be a Hilbert space, and suppose that $T\in L^1(H)$. 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

$\text{tr}(T\cdot):K(H)\rightarrow\mathbb{C}$$S\mapsto \text{tr}(TS)$,

is linear and bounded, and $\|\text{tr}(T\cdot)\|\leq \|T\|$. We therefore have a map

$L^1(H)\rightarrow K(H)^\star$$T\mapsto \text{tr}(T\cdot)$,

which is clearly linear and norm-decreasing. We call this map the canonical map from $L^1(H)$ to $K(H)^\star$.

## Theorem 4.2.1

If $H$ is a Hilbert space, then the canonical map from $L^1(H)$ to $K(H)^\star$ 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 $L^\infty(\Omega,\mu)$, where $(\Omega,\mu)$ 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.

# Question 1

Let $a,b$ be normal elements of a C*-algebra $A$, and $c$ an element of $A$ such that $ac=cb$. Show that $a^*c=cb^*$, using Fuglede’s theorem and the fact that the element

$d=\left(\begin{array}{cc}a &0\\ 0&b\end{array}\right)$

is normal in $M_2(A)$ and commutes with

$d'=\left(\begin{array}{cc} 0&c\\ 0&0\end{array}\right)$.

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

## Solution

Fuglede’s theorem states that if $a$ is a normal element commuting with some $b\in A$, then $b^*$ also commutes with $a$. Now we can show that $d^*d=d^*d$ using the normality of $a$ and $b$. We can also show that $d$ and $d'$ commute. Hence by the theorem $d$ and $d^*$ commute. This yields:

$bc^*=c^*a$.

Taking conjugates:

$cb^*=a^*c$,

as required $\bullet$

# 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 $B(H)$ for some Hilbert space $H$. 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 $A$ is a pair $(H,\varphi)$ where $H$ is a Hilbert space and $\varphi:A\rightarrow B(H)$ is a *-homomorphism. We say $(H,\varphi)$ is faithful if $\varphi$ is injective.

For abelian C*-algebras we were able to completely determine the structure of the algebra in terms of the character space, that is, in terms of the one-dimensional representations. For the non-abelian case this is quite inadequate, and we have to look at representations of arbitrary dimension. There is a deep inter-relationship between the representations and the positive linear functionals of  a C*-algebra. Representations will be defined and some aspects of this inter-relationship investigated in the next section. In this section we establish the basis properties of positive linear functionals.