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### Abstract

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

In my pursuit of an Ergodic Theorem for Random Walks on (probably finite) Quantum Groups, I have been looking at analogues of Irreducible and Periodic. I have, more or less, got a handle on irreducibility, but I am better at periodicity than aperiodicity.

The question of how to generalise these notions from the (finite) classical to noncommutative world has already been considered in a paper (whose title is the title of this post) of Fagnola and Pellicer. I can use their definition of periodic, and show that the definition of irreducible that I use is equivalent. This post is based largely on that paper.

## Introduction

Consider a random walk on a finite group $G$ driven by $\nu\in M_p(G)$. The state of the random walk after $k$ steps is given by $\nu^{\star k}$, defined inductively (on the algebra of functions level) by the associative

$\nu\star \nu=(\nu\otimes\nu)\circ \Delta$.

The convolution is also implemented by right multiplication by the stochastic operator:

$\nu\star \nu=\nu P$,

where $P\in L(F(G))$ has entries, with respect to a basis $(\delta_{g_i})_{i\geq 1}$ $P_{ij}=\nu(g_jg_{i^{-1}})$. Furthermore, therefore

$\nu^{\star k}=\varepsilon P^k$,

and so the stochastic operator $P$ describes the random walk just as well as the driving probabilty $\nu$.

The random walk driven by $\nu$ is said to be irreducible if for all $g_\ell\in G$, there exists $k\in\mathbb{N}$ such that (if $g_1=e$) $[P^k]_{1\ell}>0$.

The period of the random walk is defined by:

$\displaystyle \gcd\left(d\in\mathbb{N}:[P^d]_{11}>0\right)$.

The random walk is said to be aperiodic if the period of the random walk is one.

These statements have counterparts on the set level.

If $P$ is not irreducible, there exists a proper subset of $G$, say $S\subsetneq G$, such that the set of functions supported on $S$ are $P$-invariant.  It turns out that $S$ is a proper subgroup of $G$.

Moreover, when $P$ is irreducible, the period is the greatest common divisor of all the natural numbers $d$ such that there exists a partition $S_0, S_1, \dots, S_{d-1}$ of $G$ such that the subalgebras $A_k$ of functions supported in $S_k$ satisfy:

$P(A_k)=A_{k-1}$ and $P(A_{0})=A_{d-1}$ (slight typo in the paper here).

In fact, in this case it is necessarily the case that $\nu$ is concentrated on a coset of a proper normal subgroup $N\rhd G$, say $gN$. Then $S_k=g^kN$.

Suppose that $f$ is supported on $g^kN$We want to show that for $Pf\in A_{k-1}$Recall that

$\nu^{\star k-1}P(f)=\nu^{\star k}(f)$.

This shows how the stochastic operator reduces the index $P(A_k)=A_{k-1}$.

A central component of Fagnola and Pellicer’s paper are results about how the decomposition of a stochastic operator:

$P(f)=\sum_{\ell}L_\ell^*fL_{\ell}$,

specifically the maps $L_\ell$ can speak to the irreducibility and periodicity of the random walk given by $P$. I am not convinced that I need these results (even though I show how they are applicable).

## Stochastic Operators and Operator Algebras

Let $F(X)$ be a $\mathrm{C}^*$-algebra (so that $X$ is in general a  virtual object). A $\mathrm{C}^*$-subalgebra $F(Y)$ is hereditary if whenever $f\in F(X)^+$ and $h\in F(Y)^+$, and $f\leq h$, then $f\in F(Y)^+$.

It can be shown that if $F(Y)$ is a hereditary subalgebra of $F(X)$ that there exists a projection $\mathbf{1}_Y\in F(X)$ such that:

$F(Y)=\mathbf{1}_YF(X)\mathbf{1}_Y$.

All hereditary subalgebras are of this form.

In a recent preprint, Haonan Zhang shows that if $\nu\in M_p(Y_n)$ (where $Y_n$ is a Sekine Finite Quantum Group), then the convolution powers, $\nu^{\star k}$, converges if

$\nu(e_{(0,0)})>0$.

The algebra of functions $F(Y_n)$ is a multimatrix algebra:

$F(Y_n)=\left(\bigoplus_{i,j\in\mathbb{Z}_n}\mathbb{C}e_{(i,j)}\right)\oplus M_n(\mathbb{C})$.

As it happens, where $a=\sum_{i,j\in\mathbb{Z}_n}x_{(i,j)}e_{(i,j)}\oplus A$, the counit on $F(Y_n)$ is given by $\varepsilon(a)=x_{(0,0)}$, that is $\varepsilon=e^{(0,0)}$, dual to $e_{(0,0)}$.

To help with intuition, making the incorrect assumption that $Y_n$ is a classical group (so that $F(Y_n)$ is commutative — it’s not), because $\varepsilon=e^{(0,0)}$, the statement $\nu(e_{(0,0)})>0$, implies that for a real coefficient $x^{(0,0)}>0$,

$\nu=x^{(0,0)}\varepsilon+\cdots= x^{(0,0)}\delta^e+\cdots$,

as for classical groups $\varepsilon=\delta^e$.

That is the condition $\nu(e_{(0,0)})>0$ is a quantum analogue of $e\in\text{supp}(\nu)$.

Consider a random walk on a classical (the algebra of functions on $G$ is commutative) finite group $G$ driven by a $\nu\in M_p(G)$.

The following is a very non-algebra-of-functions-y proof that $e\in \text{supp}(\nu)$ implies that the convolution powers of $\nu$ converge.

Proof: Let $H$ be the smallest subgroup of $G$ on which $\nu$ is supported:

$\displaystyle H=\bigcap_{\underset{\nu(S_i)=1}{S_i\subset G}}S_i$.

We claim that the random walk on $H$ driven by $\nu$ is ergordic (see Theorem 1.3.2).

The driving probability $\nu\in M_p(G)$ is not supported on any proper subgroup of $H$, by the definition of $H$.

If $\nu$ is supported on a coset of proper normal subgroup $N$, say $Nx$, then because $e\in \text{supp}(\nu)$, this coset must be $Ne\cong N$, but this also contradicts the definition of $H$.

Therefore, $\nu^{\star k}$ converges to the uniform distribution on $H$ $\bullet$

Apart from the big reason — that this proof talks about points galore — this kind of proof is not available in the quantum case because there exist $\nu\in M_p(G)$ that converge, but not to the Haar state on any quantum subgroup. A quick look at the paper of Zhang shows that some such states have the quantum analogue of $e\in\text{supp}(\nu)$.

So we have some questions:

• Is there a proof of the classical result (above) in the language of the algebra of functions on $G$, that necessarily bypasses talk of points and of subgroups?
• And can this proof be adapted to the quantum case?
• Is the claim perhaps true for all finite quantum groups but not all compact quantum groups?

## Quantum Subgroups

Let $C(G)$ be a the algebra of functions on a finite or perhaps compact quantum group (with comultiplication $\Delta$) and $\nu\in M_p(G)$ a state on $C(G)$. We say that a quantum group $H$ with algebra of function $C(H)$ (with comultiplication $\Delta_H$) is a quantum subgroup of $G$ if there exists a surjective unital *-homomorphism $\pi:C(G)\rightarrow C(H)$ such that:

$\displaystyle \Delta_H\circ \pi=(\pi\otimes \pi)\circ \Delta$.

## The Classical Case

In the classical case, where the algebras of functions on $G$ and $H$ are commutative,

$\displaystyle \pi(\delta_g)=\left\{\begin{array}{cc}\delta_g & \text{ if }g\in H \\ 0 & \text{ otherwise}\end{array}\right..$

There is a natural embedding, in the classical case, if $H$ is open (always true for $G$ finite) (thanks UwF) of $\imath: C(H) \xrightarrow\, C(G)$,

$\displaystyle \sum_{h\in H}a_h \delta_h \mapsto \sum_{g\in G} a_g \delta_g$,

with $a_g=a_h$ for $h\in G$, and $a_g=0$ otherwise.

Furthermore, $\pi$ is has the property that

$\pi\circ\imath\circ \pi=\pi$,

which resembles $\pi^2=\pi$.

In the case where $\nu$ is a probability on a classical group $G$, supported on a subgroup $H$, it is very easy to see that convolutions $\nu^{\star k}$ remain supported on $H$. Indeed, $\nu^{\star k}$ is the distribution of the random variable

$\xi_k=\zeta_k\cdots \zeta_2\cdot \zeta_1$,

where the i.i.d. $\zeta_i\sim \nu$. Clearly $\xi_k\in H$ and so $\nu^{\star k}$ is supported on $H$.

We can also prove this using the language of the commutative algebra of functions on $G$, $C(G)$. The state $\nu\in M_p(G)$ being supported on $H$ implies that

$\nu\circ\imath\circ \pi=\nu\imath\pi=\nu$.

Consider now two probabilities on $G$ but supported on $H$, say $\mu,\,\nu$. As they are supported on $H$ we have

$\mu=\mu\imath\pi$ and $\nu=\nu\imath\pi$.

Consider

$(\mu\star \nu)\imath\pi=(\mu\otimes \nu)\circ \Delta\circ \imath\pi$

$=((\mu\imath\pi)\otimes(\nu\imath\pi))\circ \Delta\circ\imath\pi =(\mu\imath\otimes \nu\imath)(\pi\circ \pi)\Delta\circ\imath\pi$

$=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi\circ \imath\circ \pi)=(\mu\imath\otimes\nu\imath)(\Delta_H\circ \pi)$

$=(\mu\imath\otimes \nu\imath)\circ (\pi\circ \pi)\circ\Delta=(\mu\imath\pi\otimes \nu\imath\pi)\circ\Delta$

$=(\mu\otimes\nu)\circ\Delta=\mu\star \nu$,

that is $\mu\star \nu$ is also supported on $H$ and inductively $\nu^{\star k}$.

## Some Questions

Back to quantum groups with non-commutative algebras of functions.

• Can we embed $C(H)$ in $C(G)$ with a map $\imath$ and do we have $\pi\circ \imath\circ \pi=\pi$, giving the projection-like quality to $\pi$?
• Is $\nu\circ\imath\circ \pi=\nu$ a suitable definition for $\nu$ being supported on the subgroup $H$.

If this is the case, the above proof carries through to the quantum case.

• If there is no such embedding, what is the appropriate definition of a $\nu\in M_p(G)$ being supported on a quantum subgroup $H$?
• If $\pi$ does not have the property of $\pi\circ \imath\circ \pi=\pi$, in this or another definition, is it still true that $\nu$ being supported on $H$ implies that $\nu^{\star k}$ is too?

## Edit

UwF has recommended that I look at this paper to improve my understanding of the concepts involved.

Slides of a talk given at the Irish Mathematical Society 2018 Meeting at University College Dublin, August 2018.

Abstract Four generalisations are used to illustrate the topic. The generalisation from finite “classical” groups to finite quantum groups is motivated using the language of functors (“classical” in this context meaning that the algebra of functions on the group is commutative). The generalisation from random walks on finite “classical” groups to random walks on finite quantum groups is given, as is the generalisation of total variation distance to the quantum case. Finally, a central tool in the study of random walks on finite “classical” groups is the Upper Bound Lemma of Diaconis & Shahshahani, and a generalisation of this machinery is used to find convergence rates of random walks on finite quantum groups.

## 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).

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.

Taken from Condition Expectation in Quantum Probabilty by Denes Petz.

In quantum probability there are a number of fundamental questions that ask how faithfully can one quantise classical probability. Suppose that $(\Omega,\mathcal{S},P)$ is a (classical) probability space and $\mathcal{G}\subset\mathcal{A}$ a sub-$\sigma$-algebra. The conditional expectation of some integrable function $f$ (with respect to some $L$-space) relative to $\mathcal{G}$ is the orthogonal projection onto the closed subspace $L(\mathcal{G})$:

$\mathbb{E}^{\mathcal{G}}:L(\mathcal{A})\rightarrow L(\mathcal{G})$$f\mapsto \mathbb{E}(f|\mathcal{G})$.

Suppose now that $(A,\rho)$ is a quantum probability space and that $B$ is some C*-subalgebra of $A$. Can we always define a conditional expectation with respect to $B$? The answer turns out to be not always, although this paper gives sufficient conditions for the existence of such a projection. Briefly, things work the other way around. Distinguished states give rise to quantum conditional expectations — and these conditional expectations define a subalgebra. We can’t necessarily start with a subalgebra and find the state which gives rise to it — Theorem 2 gives necessary conditions in which this approach does work.

Taken from Real Analysis and Probability by R.M. Dudley.

For a sequence of $n$ repeated, independent trials of an experiment, some probability distributions and variables converge as $n$ tends to infinity. In proving such limit theorems, it is useful to be able to construct a probability space on which a sequence of independent random variables is defined in a natural way; specifically, as coordinates for a countable Cartesian product.

The Cartesian product of finitely many $\sigma$-finite measure spaces gives a $\sigma$-finite measure space. For example, Cartesian products of Lesbesgue measure on the line give Lesbesgue measure on finite-dimensional Euclidean spaces. But suppose we take a measure space $\{0,1\}$ with two points each having measure $1$$\mu(\{0\})=1=\mu(\{1\})$, and form a countable Cartesian product of copies of this space, so that the measure of any countable product of sets equals the product of their measures. Then we would get an uncountable space in which all singletons have measure $1$, giving the measure usually called counting measure. An uncountable set with counting measure is not a $\sigma$-finite space, although in this example it was a countable product of finite measure spaces. By contrast, the the countable product of probability measures will again be a probability space. Here are some definitions.

For each $n=1,2,\dots$ let $(\Omega_n,S_n,P_n)$ be a probability space. Let $\Omega$ be the Cartesian product $\displaystyle \prod_{n\geq 1}\Omega_n$, that is, the set of all sequences $\{\omega_n\}_{n\geq 1}$ with $\omega_n\in\Omega_n$ for all $n$. Let $\pi_n$ be the natural projection of $\Omega$ onto $\Omega_n$ for each $n$$\pi_n\left(\{\omega_m\}_{m\geq 1}\right)=\omega_n$ for all $n$. Let $S$ be the smallest $\sigma$-algebra of subsets of $\Omega$ such that for all $m$$\pi_m$ is measurable from $(\Omega,S)$ to $(\Omega_m,S_m)$. In other words, $S$ is the smallest $\sigma$-algebra containing all sets $\pi_n^{-1}(A)$ for all $n$ and all $A\in S_n$.

Let $\mathcal{R}$ be the collection of all sets $\displaystyle \prod_{n\geq 1}A_n\subset\Omega$ where $A_n\in \mathcal{S}_n$ for all $n$ and $A_m=\Omega_m$ except for at most finitely many values of $n$. Elements of $\mathcal{R}$ will be called rectangles. Now recall the notion of semiring. $\mathcal{R}$ has this property.