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In May 2017 I wrote down some problems that I hoped to look at in my study of random walks on quantum groups. Following work of Amaury Freslon, a number of these questions have been answered. In exchange for solving these problems, Amaury has very kindly suggested some other problems that I can work on. The below hopes to categorise some of these problems and their status.

## Solved!

• Show that the total variation distance is equal to the projection distance. Amaury has an a third proof. Amaury suggests that this should be true in more generality than the case of $\nu$ being absolutely continuous (of the form $\nu(x)=\int_G xa_{\nu}$ for all $x\in C(G)$ and a unique $a_{\nu}\in C(G)$). If the Haar state is no longer tracial Amaury’s proof breaks down (and I imagine so do the two others in the link above). Amaury believes this is true in more generality and says perhaps the Jordan decomposition of states will be useful here.
• Prove the Upper Bound Lemma for compact quantum groups of Kac type. Achieved by Amaury.
• Attack random walks with conjugate invariant driving probabilitys: achieved by Amaury.
• Look at quantum generalisations of ‘natural’ random walks and shuffles. Solved is probably too strong a word, but Amaury has started this study by looking at a generalisation of the random transposition shuffle. As I suggested in Seoul, Amaury says: “One important problem in my opinion is to say something about analogues of classical random walks on $S_n$ (for instance the random transpositions or riffle shuffle)”. Amaury notes that “we are blocked by the counit problem. We must therefore seek bounds for other distances. As I suggest in my paper, we may look at the norm of the difference of the transition operators. The $\mathcal{L}^2$-estimate that I give is somehow the simplest thing one can do and should be thought of as a “spectral gap” estimate. Better norms would be the norms as operators on $\mathcal{L}^\infty$ or even better, the completely bounded norm. However, I have not the least idea of how to estimate this.”

## Results to be Improved

• I have recently received an email from Isabelle Baraquin, a student of Uwe Franz, pointing out a small error in the thesis (a basis-error with the Kac-Paljutkin quantum groups).
• Recent calculations suggest that the lower bound for the random walk on the dual of $S_n$ is effective at $k\sim (n-1)!$ while the upper bound shows the walk is random at time order $n!$.  This is still a very large gap but at least the lower bound shows that this walk does converge very slowly.
• Get a much sharper lower bound for the random walk on the Sekine family of quantum groups studied in Section 5.7. Projection onto the ‘middle’ of the $M_n(\mathbb{C})$ factor may 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).

## More Questions on Random Walks

• Irreducibility is harder than the classical case (where ‘not concentrated’ on a subgroup is enough). Can anything be said about aperiodicity in the quantum case? (U. Franz).
• Prove an Ergodic Theorem (Theorem 1.3.2) for Finite Quantum Groups. Extend to Compact Quantum Groups. 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 and cosets.
• Look at a random walk on the Sekine quantum groups with an $n$-dependent driving probability and see if the cut-off phenomenon (Chapter 4) can be detected. This will need good lower bounds for $k\ll t_n$, some cut-off time.
• Convolutions of the Random Distribution: such a study may prove fruitful in trying to find the Ergodic Theorem. See Section 6.5.
• Amaury mentions the problem of considering random walks associated to non-central states (in the compact case). “The difficulty is first to build non-central states (I do not have explicit examples at hand but Uwe Franz said he had some) and second to be able to compute their Fourier transform. Then, the computations will certainly be hard but may still be doable.”
• A study of the Cesaro means: see Section 6.6.
• Spectral Analysis: it should be possible to derive crude bounds using the spectrum of the stochastic operator. More in Section 6.2.

## Future Work (for which I do not yet have the tools to attack)

• Amaury/Franz Something perhaps more accessible is to investigate quantum homogeneous spaces. The free sphere is a noncommutative analogue of the usual sphere and a quantum homogeneous space for the free orthogonal quantum group. We can therefore define random walks on it and the whole machinery of Gelfand pairs might be available. In particular, Caspers gave a Plancherel theorem for Gelfand pairs of locally compact quantum groups which should apply here yielding an Upper Bound Lemma and then the problem boils down to something which should be close to my computations. There are probably works around this involving Adam Skalski and coauthors.
• Amaury: If one can prove a more general total variation distance equal to half one norm result, then Amaury suggests one can consider random walks on compact quantum groups which are not of Kac type. The Upper Bound Lemma will then involve matrices $Q$ measuring the modular theory of the Haar state and some (but not all) dimensions in the formulas must be replaced by quantum dimensions. The main problem here is to define explicit central states since there is no Haar-state preserving conditional expectation onto the central algebra. However, there are tools from monoidal equivalence to do this.

## Distances between Probability Measures

Let $G$ be a finite quantum group and $M_p(G)$ be the set of states on the $\mathrm{C}^\ast$-algebra $F(G)$.

The algebra $F(G)$ has an invariant state $\int_G\in\mathbb{C}G=F(G)^\ast$, the dual space of $F(G)$.

Define a (bijective) map $\mathcal{F}:F(G)\rightarrow \mathbb{C}G$, by

$\displaystyle \mathcal{F}(a)b=\int_G ba$,

for $a,b\in F(G)$.

Then, where $\|\cdot\|_1^{F(G)}=\int_G|\cdot|$ and $\|\cdot\|_\infty^{F(G)}=\|\cdot\|_{\text{op}}$, define the total variation distance between states $\nu,\mu\in M_p(G)$ by

$\displaystyle \|\nu-\mu\|=\frac12 \|\mathcal{F}^{-1}(\nu-\mu)\|_1^{F(G)}$.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative $\mathcal{L}^p$ machinary shows that:

$\displaystyle \|\nu-\mu\|=\sup_{\phi\in F(G):\|\phi\|_\infty^{F(G)}\leq 1}\frac12|\nu(\phi)-\mu(\phi)|$.

(supremum presentation)

In the classical case, using the test function $\phi=2\mathbf{1}_S-\mathbf{1}_G$, where $S=\{\nu\geq \mu\}$, we have the probabilists’ preferred definition of total variation distance:

$\displaystyle \|\nu-\mu\|_{\text{TV}}=\sup_{S\subset G}|\nu(\mathbf{1}_S)-\mu(\mathbf{1}_S)|=\sup_{S\subset G}|\nu(S)-\mu(S)|$.

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

$\displaystyle \|\nu-\mu\|_P=\sup_{p\text{ a projection}}|\nu(p)-\mu(p)|$.

(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 $\neq$ 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 $M_n(\mathbb{C})$ factor of $F(G)$ 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 $M_n(\mathbb{C})$ 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. (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 $n$-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 $\bullet$