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Freslon’s Ergodic Theorem for (Finite) Group Algebras

December 17, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

In the hope of gleaning information for the study of aperiodic random walks on (finite) quantum groups, which I am struggling with here, by couching Freslon’s Proposition 3.2, in the language of Fagnola and Pellicer, and in the language of my (failed) attempts (see here, here, and here) to find the necessary and sufficient conditions for a random walk to be aperiodic. It will be necessary to extract the irreducibility and aperiodicity from Freslon’s rather ‘unilateral’ result.

Well… I have an inkling that because dual groups satisfy what I would call the condition of abelianness (under the ‘quantisation’ functor), all (quantum) subgroups are normal… this is probably an obvious thing to write down (although I must search the literature) to ensure it is indeed known (or is untrue?). Edit: Wang had it already, see the last proposition here.

Let $F(G)$ be a the algebra of functions on a finite classical (as opposed to quantum) group $G$ . This has the structure of both an algebra and a coalgebra, with an appropriate relationship between these two structures. By taking the dual, we get the group algebra, $\mathbb{C}G=:F(\widehat{G})$ . The dual of the pointwise-multiplication in $F(G)$ is a coproduct for the algebra of functions on the dual group $\widehat{G}$ … this is all well known stuff.

Recall that the set of probabilities on a finite quantum group is the set of states $M_p(G):=\mathcal{S}(F(G))$ , and this lives in the dual, and the dual of $F(\widehat{G})$ is $F(G)$ , and so probabilities on $\widehat{G}$ are functions on $G$ . To be positive is to be positive definite, and to be normalised to one is to have $u(\delta^e)=1$ .

The ‘simplicity’ of the coproduct,

$\Delta(\delta^g)=\delta^g\otimes\delta^g$ ,

means that for $u\in M_p(\widehat{G})$ ,

$(u\star u)(\delta^g)=(u\otimes u)\Delta(\delta^g)=u(\delta^g)^2$ ,

so that, inductively, $u^{\star k}$ is equal to the (pointwise-multiplication power) $u^k$ .

The Haar state on $\widehat{G}$ is equal to:

$\displaystyle \pi:=\int_{\hat{G}}:=\delta_e$ ,

and therefore necessary and sufficient conditions for the convergence of $u^{\star k}\rightarrow \pi$ is that $u$ is strict. It can be shown that for any $u\in M_p(G)$ that $|u(\delta^g)|\leq u(\delta^e)=1$ . Strictness is that this is a strict inequality for $g\neq e$ , in which case it is obvious that $u^{\star k}\rightarrow \delta_e$ .

Here is a finite version of Freslon’s result which holds for discrete groups.

Freslon’s Ergodic Theorem for (Finite) Group Algebras

Let $u\in M_p(\widehat{G})$ be a probability on the dual of finite group. The random walk generated by $u$ is ergodic if and only if $u$ is not-concentrated on a character on a non-trivial subgroup $H\subset G$ .

Freslon’s proof passes through the following equivalent condition:

The random walk on $\widehat{G}$ driven by $u\in M_p(\widehat{G})$ is not ergodic if $u$ is bimodularwith respect to a non-trivial subgroup $H\subset G$ , in the sense that

$\displaystyle u(\underbrace{\delta^g\delta^h}_{=\delta^{gh}})=u(\delta^g)u(\delta^h)=u\left(\underbrace{\delta^h\delta^g}_{=\delta^{hg}}\right)$ .

Before looking at the proof proper, we might note what happens when $G$ is abelian, in which case $\widehat{G}$ is a classical group, the set of characters on $G$ .

To every positive definite function $u\in M_p(\widehat{G})$ , we can associate a probability $\nu_u\in M_p(\widehat{G})$ such that:

$\displaystyle u(\delta^g)=\sum_{\chi\in\hat{G}} \chi(\delta^g) \nu_u(\chi)$ .

This is Bochner’s Theorem for finite abelian groups. This implies that positive definite functions on finite abelian groups are exactly convex combinations of characters.

Freslon’s condition says that to be not ergodic, $u$ must be a character on a non-trivial subgroup $H\subset G$ . Such characters can be extended in $[G\,:\,H]$ ways.

Therefore, if $u$ is not ergodic, $u_{\left|H\right.}=\eta\in \widehat{H}$ .

For $h\in H$ , we have

$\displaystyle u(h)=\sum_{\chi\in\widehat{G}}\chi(h)\nu_u(\chi)$ ,

dividing both sides by $u(h)=\eta(h)\neq 0$ yields:

$\displaystyle\sum_{\chi \in \widehat{G}} (\eta^{-1}\chi)(h)\nu_u (\chi)=1$ .

As $\nu_u\in M_p(\widehat{G})$ , and $(\eta^{-1}\chi)(h)\in \mathbb{T}$ , this implies that $\nu_u$ is supported on characters such that, for all $h\in H$ :

$\eta^{-1}(h)\chi(h)=1\Rightarrow \chi=\eta\tilde{\chi}$ ,

such that $\tilde{\chi}(H)=\{1\}$ . The set of such $\tilde{\chi}$ is the annihilator of $H$ in $\widehat{G}$ , and it is a subgroup. Therefore $\nu_u$ is concentrated on the coset of a normal subgroup (as all subgroups of an abelian group are normal).

This, via Pontragin duality, is not looking at the ‘support’ of $u$ , but rather of $\nu_u$ . Although we denote $\mathbb{C}G=:F(\widehat{G})$ , and when $G$ is abelian, $\widehat{G}$ is a group (unnaturally, of characters) isomorphic to $G$ . Is it the case though that,

$\Delta(\chi)=\chi\otimes\chi$

gives the same object in as

$\displaystyle\Delta(\chi)=\sum_{g\in G}\chi(\delta^g)\Delta(\delta_g)$

$\displaystyle =\sum_{g\in G}\chi(\delta^g)\sum_{t\in G}\delta_{gt^{-1}}\otimes \delta_t$ ?

Well… of course this is true because $\chi(gh)=\chi(g)\chi(h)$ .

We could proceed to look at Fagnola & Pellicer’s work but first let us prove Freslon’s result, hopefully in the finite case the analysis disappears…

Proof: Assume that $u$ is not strict and let

$\Lambda:=|u|^{-1}(\{1\})$ .

There exists a unitary representation $\Phi:G\rightarrow B(H)$ and a unit vector $\xi$ such that

$u(g)=\langle \Phi(g)\xi,\xi\rangle$

Cauchy-Schwarz implies that

$|u(g)|\leq \|\Phi(g)\xi\|\|\xi\|=\|\xi\|^2$ .

If $h$ is not strict there is an $h$ such that this is an inequality and so $\Phi(h)\xi$ is colinear to $\xi$ , it follows that $\Phi(h)\xi=u(h)\xi$ .

This implies for $h\in \Lambda$ and $g\in G$ :

$|u(gh)|=|\langle \Phi(gh)\xi,\xi\rangle|=|u(h)||\langle \Phi(g)\xi,\xi\rangle|=|u(g)|$ ,

and so $\Lambda$ is closed under multiplication. Also $u(g^{-1})=\overline{u(g)}$ and so $\Lambda$ and so $\Lambda$ is a subgroup. It follows that $u$ is a character on $\Lambda$ , which is not trivial because $u$ is not strict.

I don’t really need to go through the third equivalent condition. If $u$ coincides with a character on a subgroup $\Lambda$ , for $h\in \Lambda$

$|u(h)|^2=u(h)\overline{u(h)}=u(h)u(h^{-1})=u(e)=1$ ,

and so $u$ is not strict $\bullet$

Now let us look at the language of Fagnola and Pellicer. What is a projection in $\mathbb{C}G$ ? First note the involution in $\mathbb{C}G$ is $(\delta^g)^*=\delta^{g^{-1}}$ . The second multiplication is the convolution. This means projections in the algebra are symmetric with respect to the group inverses and they are also idempotents. They are actually equal to Haar states on finite subgroups.

I think periodicity is also wrapped up in Freslon’s result as I think all subgroups of dual groups are normal. Perhaps, oddly, not being concentrated on a subgroup means that the positive function (probability on the dual) is one on that subgroup…

Well… let us start with irreducible. Suppose $u$ fails to be ergodic because it is irreducible. This means there is a projection $p_H=\int_H$ such that that $P_u(p_H)=p_H$ (and support $u$ less than $p_H$ ?)

Let us look at the first condition:

$P_u(p_H)=(u\otimes I)\Delta(p_H)=\cdots=\frac{1}{|H|}\sum_{h\in H}u(h)\delta^h=p_H\Rightarrow u_{\left|H\right.}=1$ .

What now is the support of $u$ ? Well… some work I have done offline shows that the special projections, the group-like projections, correspond to to $\pi_H$ for $H$ a subgroup of $G$ . If $u$ is reducible, it is concentrated on such a quasi-subgroup, and this means that $u$ coincides with a trivial character on $H$ . In terms of Fagnola Pellicer, $P_u(\pi_H)=\pi_H$ .

Now let us tackle aperiodicity. It is going to correspond, I think, with being concentrated on a ‘coset’ of a non-trivial character on $H$ …

Well, we can show that if $u$ is periodic, there is a subset $S\subset G$ such that $u(s)=e^{2\pi i a_s/d}$ for all $s\in S$ . We can use Freslon’s proof to show that $S$ is in a subgroup on which $|u|=1$ .

Now what I want to do is put this in the language of ‘inclusion’ matrices… but the inclusions for cocommutative quantum groups are trivial so no go…

We can reduce Freslon’s conditions down to irreducible and aperiodic: not coinciding with a trivial character, and not coinciding with a character.

Irreducible and Periodic Positive Maps

November 14, 2019 in C*-Algebras & Operator Theory, Probability Theory, Research | 1 comment

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.

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Periodicity of Random Walks on Finite Quantum Groups

September 12, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research Updates | Leave a comment

In the case of a finite classical group $G$ , we can show that if we have i.i.d. random variables $\zeta_i\sim\nu\in M_p(G)$ , that if $\text{supp }\nu\subset Ng$ , for $Ng$ a coset of a proper normal subgroup $N\rhd G$ , that the random walk on $G$ driven by $\nu$ , the random variables:

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

exhibits a periodicity because

$\xi_k\in Ng^{k}$ .

This shows that a necessary condition for ergodicity of a random walk on a finite classical group $G$ driven by $\nu\in M_p(G)$ is that the support of $\nu$ not be concentrated on the coset of a proper normal subgroup.

I had hoped that something similar might hold for the case of random walks on finite quantum groups but alas I think I have found a barrier.

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The Ergodic Theorem for Random Walks: from Finite Groups, to Group Algebras, to Finite Quantum Groups

September 6, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Slides of a talk given at Munster Groups 2019, WIT.

Abstract: It is a folklore theorem that necessary and sufficient conditions for a random walk on a finite group to converge in distribution to the uniform distribution – “to random” – are that the driving probability is not concentrated on a proper subgroup nor the coset of a proper normal subgroup. This is the Ergodic Theorem for Random Walks on Finite Groups. In this talk we will outline the rarely written down proof, and explain why, for example, adjacent transpositions can never mix up a deck of cards. From here we will, in a very leisurely and natural fashion, introduce (and motivate the definition of) finite quantum groups, and random walks on them. We will see how the group algebra of a finite group is the algebra of functions on a finite quantum group. Freslon has very recently proved the Ergodic Theorem in this setting, and we present ongoing work towards an Ergodic Theorem in the more general finite quantum group setting; a result that would generalise both the folklore and group algebra Ergodic Theorems.

Finite Quantum Groups: Pure States

June 25, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Introduction

Every finite quantum group has finite dimensional algebra of functions:

$\displaystyle F(G)=\bigoplus_{j=1}^m M_{n_j}(\mathbb{C})$ .

At least one of the factors must be one-dimensional to account for the counit $\varepsilon:F(G)\rightarrow \mathbb{C}$ , and if this factor is denoted $\mathbb{C}e_1$ , the counit is given by the dual element $e^1$ . There may be more and so reorder the index $j\mapsto i$ so that $n_i=1$ for $i=1,\dots,m_1$ , and $n_i>1$ for $i>m_1$ :

$\displaystyle F(G)=\left(\bigoplus_{i=1}^{m_1} \mathbb{C}e_{i}\right)\oplus \bigoplus_{i=m_1+1}^m M_{n_i}(\mathbb{C})=:A_1\oplus B$ ,

Denote by $M_p(G)$ the states of $F(G)$ . The pure states of $F(G)$ arise as pure states on single factors.

In the case of the Kac-Paljutkin and Sekine quantum groups, the convolution powers of pure states exhibit a periodicity of sorts. Recall for these quantum groups that $B$ consists of a single matrix factor.

In these cases, for pure states of the form $e^i$ , that is supported on $A_1$ (and we can say a little more than is necessary), the convolution remains supported on $A_1$ because

$\Delta(A_1)\subset A_1\otimes A_1+B\otimes B$ .

If we have a pure state $\nu$ supported on $B=M_{\sqrt{\dim B}}(\mathbb{C})$ , then because

$\Delta(B)\subset A_1\otimes B+B\otimes A_1$ ,

then $\nu\star\nu$ must be supported on, because of $\Delta(A_1)\subset A_1\otimes A_1+B\otimes B$ , $A_1$ .

Inductively all of the $\nu^{\star 2k}$ are supported on $A_1$ and the $\nu^{\star 2k+1}$ are supported on $B$ . This means that the convolutions powers of a pure state, in these cases, cannot converge to the Haar state.

The question is, do the results above about the image of $A_1$ and $B$ under the coproduct hold more generally? I believe that the paper of Kac and Paljutkin shows that this is the case whenever $B$ consists of a single factor… but does it hold more generally?

To find out we go back and do some sandboxing with the paper of Kac and Paljutkin. Which is a pleasure because that paper is beautiful. The blue stuff is my own scribbling.

Finite Ring Groups

Let $G$ be a finite quantum group with notation on the algebra of functions as above. Note that $A_1$ is commutative. Let

$p=\sum_{i=1}^{m_1}e_i$ ,

which is a central idempotent.

Lemma 8.1

$S(p)=p$ .

Proof: If $S(\mathbf{1}_G-p)p\neq 0$ , then for some $i>m_1$ , and $f\in M_{n_i}(\mathbb{C})$ , the mapping $f\mapsto S(f)p$ is a non-zero homomorphism from $M_{n_i}(\mathbb{C})$ into commutative $A_1$ which is impossible.

If $S(\mathbf{1}_G-p)p=g\neq 0$ , then one of the $S(I_{n_i})\in A_1\oplus B$ , with ‘something’ in $A_1$ . Using the centrality and projectionality of $p$ , we can show that the given map is indeed a homomorphism.

It follows that $S(p)p=p\Rightarrow S(S(p)p)=S(p)=S(p)p=S(p)$ , and so $p=S(p)$ $\bullet$

Lemma 8.2

$(p\otimes p)\Delta(p)=p\otimes p$

Proof: Suppose that $(p\otimes p)\Delta(f)=b$ for some non-commutative $f\in M_{n_i}(\mathbb{C})$ . This means that there exists an index $k$ such that $f_{(1)_k}\otimes f_{(2)_k}\in A_1\otimes A_1$ . Then for that factor,

$f\mapsto \Delta(f)(p\otimes p)$

is a non-null homomorphism from the non-commutative into the commutative.

We see that $(p\otimes p)\Delta(f)=0$ for all $f\in B$ . Putting $a=\mathbf{1}-p$ we get the result $\bullet$

The following says that $p$ is a group-like projection. We know from previous work that if a state is supported on a group-like projection that it will remain supported on it. In particular, any state supported on $A_1$ will remain there.

Lemma 8.3

$(p\otimes \mathbf{1}_G)\Delta(p)=p\otimes p=(\mathbf{1}_G\otimes p)\Delta(p)$ .

Proof: Since $\Delta$ is a homomorphism, $\Delta(p)$ is an idempotent in $F(G)\otimes F(G)$ . I do not understand nor require the rest of the proof.

Lemma 8.4

$A_1=F(G_1)$ is the algebra of functions on finite group with elements $i=1,\dots,m_1$ , and we write $e_i=\delta_i$ . The coproduct is given by $(p\otimes p)\Delta$ .

We have:

$(p\otimes p)\Delta(e_i)=\sum_{t\in G_1}\delta_{it^{-1}}\otimes \delta_t$ ,

$S(\delta_i)=\delta_{i^{-1}}$ ,

$\varepsilon(e_i)=\delta_{i,1}$ ,

as $e_1=\delta_e$ .

The element $\Delta(\delta_i)$ is a sum of four terms, lying in the subalgebras:

$A_1\otimes A_1,\,B\otimes B,\,A_1\otimes B,\,B\otimes A_1$ .

We already know what is going on with the first summand. Denote the second by $P_i$ . From the group-like-projection property, the last two summands are zero, so that

$\Delta(\delta_i)=\sum_{t\in G_1}\delta_{t}\otimes \delta_{t^{-1}i}+P_i$ .

Since the $\delta_i$ are symmetric ( $\delta_i^*=\delta_i$ ) mutually orthogonal idempotents, $P_i$ has similar properties:

$P_i^*=P_i,\,P_i^2=P_i,\,P_iP_j=0$

for $i\neq j$ .

At this point Kac and Paljutkin restrict to $B=M_{n_{i+1}}(\mathbb{C})$ , that is there is only one summand. Here we try to keep arbitrarily (finitely) many summands in $B$ .

Let the summand $M_{n_i}(\mathbb{C})$ have matrix units $E_{mn}^i$ , where $m,n=1,\dots,n_i$ . Kac and Paljutkin now do something which I think is a little dodgy, but basically that the integral over $G$ is equal on each of the $\delta_i$ , equal on each of the $E_{mm}^i$ , and then zero off the diagonal.

It does follow from above that each $P_i\in B\otimes B$ is a projection.

Now I am stuck!

Ideals & Normal Quantum Subgroups

June 25, 2019 in C*-Algebras & Operator Theory, Quantum Groups, Research | 1 comment

This sandbox is going to take from a variety of sources, mostly Shuzhou Wang.

C*-Ideals

Let $J\subset C(X)$ be a closed (two-sided) ideal in a non-commutative unital $C^*$ -algebra $C(X)$ . Such an ideal is self-adjoint and so a non-commutative $C^*$ -algebra $J=C(S)$ . The quotient map is given by $\pi:C(X)\rightarrow C(X)/C(S)$ , $f\mapsto f+J$ , where $f+J$ is the equivalence class of $f$ under the equivalence relation:

$f\sim_{J} g\Rightarrow g-f\in C(S)$ .

Where we have the product

$(f+J)(g+J)=fg+J$ ,

and the norm is given by:

$\displaystyle\|f+J\|=\sup_{j\in C(S)}\|f+j\|$ ,

the quotient $C(X)/ C(S)$ is a $C^*$ -algebra.

Consider now elements $j_1,\,j_2\in C(S)$ and $f_1,\, f_2\in C(X)$ . Consider

$j_1\otimes f_1+f_2\otimes j_2\in C(S)\otimes C(X)+C(X)\otimes C(S)$ .

The tensor product $\pi\otimes \pi:C(X)\otimes C(X)\rightarrow (C(X)/C(S))\otimes (C(X)/ C(S))$ . Now note that

$(\pi\otimes\pi)(j_1\otimes f_1+f_2\otimes j_2)=(0+J)\otimes(f_1+J)+$

$(f_2+J)\otimes(0+J)=0$ ,

by the nature of the Tensor Product ( $0\otimes a=0$ ). Therefore $C(X)\otimes C(S)+C(S)\otimes C(X)\subset \text{ker}(\pi\otimes\pi)$ .

Definition

A WC*-ideal (W for Woronowicz) is a C*-ideal $J=C(S)$ such that $\Delta(J)\subset \text{ker}(\pi\otimes\pi)$ , where $\pi$ is the quotient map $C(G)\rightarrow C(G)/C(S)$ .

Let $F(G)$ be the algebra of functions on a classical group $G$ . Let $H\subset G$ . Let $J$ be the set of functions which vanish on $H$ : this is a C*-ideal. The kernal of $\pi:F(G)\rightarrow F(G)/J$ is $J$ .

Let $\delta_s\in J$ so that $s\not\in H$ . Note that

$\displaystyle\Delta(\delta_s)=\sum_{t\in G}\delta_{st^{-1}}\otimes\delta_t$

and so

$\displaystyle(\pi\otimes \pi)\Delta(\delta_s)=\sum_{t\in G}\pi(\delta_{st^{-1}})\otimes \pi(\delta_t)$ .

Note that $\pi(\delta_t)=0+J$ if $t\not\in H$ . It is not possible that both $st^{-1}$ and $t$ are in $H$ : if they were $st^{-1}\cdot t\in H$ , but $st^{-1}\cdot t=s$ , which is not in $H$ by assumption. Therefore one of $\pi(\delta_{st^{-1}})$ or $\pi(\delta_t)$ is equal to zero and so:

$(\pi\otimes\pi)\Delta(\delta_s)=0$ ,

and so by linearity, if $f$ vanishes on a subgroup $H$ ,

$\Delta(f)\subset \text{ker}(\pi\otimes\pi)$ .

In this way, WC*-ideals generalise functions which vanish on distinguished subgroups. In fact, without checking all the details, I imagine that first isomorphism theorem can show that $F(G)/ J=F(H)$ . Let $\pi_H:F(G)\rightarrow F(H)$ be the ring homomorphism

$\displaystyle\pi_H\left(\sum_{t\in G}a_t\delta_t\right)=\sum_{t\in H}a_t\delta_t$ .

Then $\text{ker}\,\pi_H=J$ , $\text{im}\,\pi_H=F(H)$ , and so we have the isomorphism of rings, which presumably carries forward to the algebras of functions level…

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Symmetries of Quantum Spaces – Podlés

June 9, 2019 in Quantum Groups, Research | Leave a comment

Just some notes on section 1 of this paper. Flags and notes are added but mistakes are mine alone.

Definition

Let $C(G)$ be the algebra of continuous functions on a compact matrix quantum group. Such an object is given by a matrix $u=\{u_{ij}\}_{i,j=1}^N$ which generates $C(G)$ as a C*-algebra. Furthermore, there exists a C*-algebra homomorphism $\Delta:C(G)\rightarrow C(G)\otimes C(G)$ such that

$\displaystyle \Delta(u_{ij})=\sum_{k=1}^N u_{ik}\otimes u_{kj}$ ,

and both $u$ and $u^T$ are invertible in $M_N(C(G))$ .

Any subgroup $G\subset \text{GL}(N,\mathbb{C})$ is such an object, with the $u_{ij}\in C(G)$ given by $u_{ij}(g)=g_{ij}\in\mathbb{C}$ . Furthermore

$\mathrm{C}_{\text{comm}}\langle u_{ij}\rangle \cong C(G)$ .

We say that $\rho=(\rho_{ij})_{i,j=1}^{d_\rho}\in M_{d_{\rho}}(C(G))$ is a representation if it is invertible and

$\displaystyle \Delta(\rho_{ij})=\sum_{k=1}^{d_\rho}\rho_{ik}\otimes\rho{kj}$ .

The transpose $\rho^T=(\rho_{ji})_{i,j=1}^N\in M_{d_{\rho}}(C(G))$ is also invertible and so we have:

Proposition

The C*algebra generated by the $\rho_{ij}$ is also the algebra of continuous functions on a compact matrix quantum group.

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Periodicity & Aperiodicity for Random Walks on Quantum Groups

June 7, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Background

I am trying to prove an Ergodic Theorem for Random Walks on Finite Quantum Groups. A random walk on a quantum group driven by $\nu\in M_p(G)$ is ergodic if the convolution powers $(\nu^{\star k})_{k\geq 0}$ converge to the Haar state $\int_G$ .

The classical theorem for finite groups:

Ergodic Theorem for Random Walks on Finite Groups

A random walk on a finite group $G$ driven by a probability $\nu\in M_p(G)$ is ergodic if and only if $\nu$ is not concentrated on a proper subgroup nor the coset of a proper normal subgroup.

Not concentrated on a proper subgroup gives irreducibility. A random walk is irreducible if for all $g\in G$ , there exists $k\in\mathbb{N}$ such that $\nu^{\star k}(\{g\})>0$ .

Not concentrated on the coset of a proper normal subgroup gives aperiodicity. Something which should be equivalent to aperiodicity is if

$p:=\gcd\{k>0:\nu^{\star k}(e)>0\}$

is equal to one (perhaps via invariance $\mathbb{P}[\xi_{i+1}=t|\xi_{i}=s]=\mathbb{P}[\xi_{i+1}=th|\xi_{i}=sh]$ ).

If $\nu$ is concentrated on the coset a proper normal subgroup $N\rhd G$ , specifically on $Ng\neq Ne$ , then we have periodicity ( $Ng\rightarrow Ng^2\rightarrow \cdots\rightarrow Ng^{o(g)}=Ne\rightarrow Ng\rightarrow \cdots$ ), and $p=o(g)$ , the order of $g$ .

In Markov chain theory, ergodicity is equivalent to irreduciblity and aperiodicity.

The theorem in the quantum case should look like:

Ergodic Theorem for Random Walks on Finite Quantum Groups

A random walk on a finite quantum group $G$ driven by a state $\nu\in M_p(G)$ is ergodic if and only if “X”.

Irreducibility

At the moment I have some part of X;the irreducibility bit. As is well known since Pal (1996), it is possible to have a probability not concentrated on a quantum subgroup be reducible. This led Franz & Skalski to generalise quantum subgroups to group-like-projections, which I will say correspond to quasi-subgroups following Kasprzak & Sołtan.

I have shown that if $\nu$ is concentrated on a proper quasi-subgroup $S$ , in the sense that $\nu(P_S)=1$ for a group-like-projection $P_S$ , that so are the $\nu^{\star k}$ . The analogue of irreducible is that for all $q$ projections in $F(G)$ , there exists $k\in\mathbb{N}$ such that $\nu^{\star k}(q)>0$ . If $\nu$ is concentrated on a quasi-subgroup $S$ , then for all $k$ , $\nu^{\star k}(Q_S)=0$ , where $Q_S=\mathbf{1}_G -P_S$ .

I have also shown on the other hand that if the random walk is reducible that it must be concentrated on a proper quasi-subgroup. Franz & Skalski show that group-like-projections also have a correspondence with idempotent states. The Césaro Means

$\displaystyle \nu_n:=\frac{1}{n}\sum_{k=1}^n\nu^{\star k}$ ,

converge to an idempotent state $\nu_\infty$ . If $\nu^{\star k}(q)=0$ for all $k$ then the $\nu_{\infty}(q)=0$ also, so that $\nu_\infty\neq \int_G$ (as the Haar state is faithful). I was able to prove that $\nu$ is supported on the quasi-subgroup given by the idempotent $\nu_\infty$ .

I believe this result — irreducible if and only if not concentrated on a quasi-subgroup — holds more generally than just in finite quantum groups.

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Lattice of Idempotent States on a Locally Compact Quantum Group — Kasprzak & Sołtan

May 30, 2019 in C*-Algebras & Operator Theory, Quantum Groups | Leave a comment

Just some notes on the pre-print. I am looking at this paper to better understand this pre-print. In particular I am hoping to learn more about the support of a probability on a quantum group. Flags and notes are added but mistakes are mine alone.

Abstract

From this paper I will look at:

lattice operations on $\mathcal{I}(G)$ , for $G$ a LCQG (analogues of intersection and generation)

1. Introduction

Idempotent states on quantum groups correspond with “subgroup-like” objects. In this work, on LCQG, the correspondence is with quasi-subgroups (the work of Franz & Skalski the correspondence was with pre-subgroups and group-like projections).

Let us show the kind of thing I am trying to understand better.

Let $F(G)$ be the algebra of function on a finite quantum group. Let $\nu,\,\mu\in M_p(G)$ be concentrated on a pre-subgroup $S$ . We can associate to $S$ a group like projection $p_S$ .

Let, and this is another thing I am trying to understand better, this support, the support of $\nu$ be ‘the smallest’ (?) projection $p\in F(G)$ such that $\nu(p)=1$ . Denote this projection by $p_\nu$ . Define $p_\mu$ similarly. That $\mu,\,\nu$ are concentrated on $S$ is to say that $p_\nu\leq p_S$ and $p_\mu\leq p_S$ .

Define a map $T_\nu:F(G)\rightarrow F(G)$ by

$a\mapsto p_\nu a$ (or should this be $ap_\nu$ or $p_\nu a p_\nu$ ?)

We can decompose, in the finite case, $F(G)\cong \text{Im}(T_\nu)\oplus \ker(T_\nu)$ .

Claim: If $\nu$ is concentrated on $S$ , $\nu(ap_S)=\nu(a)$ … I don’t have a proof but it should fall out of something like $p_\nu\leq p_S\Rightarrow \ker p_\nu\subseteq \ker p_S$ together with the decomposition of $F(G)$ above. It may also require that $\int_G$ is a trace, I don’t know. Something very similar in the preprint.

From here we can do the following. That $p_S$ is a group-like projection means that:

$\Delta (p_s)(\mathbf{1}_G\otimes p_S)=p_S\otimes p_S$

$\Rightarrow \sum p_{S(1)}\otimes (p_{S(2)}p_S)=p_S\otimes p_S$

Hit both sides with $\nu\times \mu$ to get:

$\sum \nu(p_{S(1)})\mu(p_{S(2)}p_S)=\nu(p_S)\mu(p_S)$ .

By the fact that $\nu,\,\mu$ are supported on $S$ , the right-hand side equals one, and by the as-yet-unproven claim, we have

$\sum \nu(p_{S(1)})\mu(p_{S(2)})=1$ .

However this is the same as

$(\nu\otimes\mu)\Delta(p_S)=1\Rightarrow (\nu\star \mu)(p_S)=1$ ,

in other words $p_{\nu\star \mu}\leq p_S$ , that is $\nu\star \mu$ remains supported on $S$ . As a corollary, a random walk driven by a probability concentrated on a pre-subgroup $S\subset G$ remains concentrated on $S$ .

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Some Unresolved and Unexplored Aspects of Random Walks on Quantum Groups

May 14, 2019 in Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

Slides of a talk given to the Functional Analysis seminar in Besancon.

Some of these problems have since been solved.

“e in support” implies convergence

Consider a $\nu\in M_p(G)$ on a finite quantum group such that where

$M_p(G)\subset \mathbb{C}\varepsilon \oplus (\ker \varepsilon)^*$ ,

$\nu=\nu(e)\varepsilon+\psi$ with $\nu(e)>0$ . This has a positive density of trace one (with respect to the Haar state $\int_G\in M_p(G)$ ), say

$\displaystyle a_\nu=\nu(e)\eta+b_\psi\in \mathbb{C}\eta\oplus \ker \varepsilon$ ,

where $\eta$ is the Haar element.

An element in a direct sum is positive if and only if both elements are positive. The Haar element is positive and so $b_\psi\geq 0$ . Assume that $b_\psi\neq 0$ (if $b_\psi=0$ , then $\psi=0\Rightarrow \nu=\varepsilon\Rightarrow \nu^{\star k}=\varepsilon$ for all $k$ and we have trivial convergence)

Therefore let

$\displaystyle a_{\tilde{\psi}}:=\frac{b_\psi}{\int_G b_\psi}$

be the density of $\tilde{\psi}\in M_p(G)$ .

Now we can explicitly write

$\displaystyle \nu=\nu(e)\varepsilon+(1-\nu(e))\tilde{\psi}$ .

This has stochastic operator

$P_\nu=\nu(e)I_{F(G)}+(1-\nu(e))P_{\tilde{\psi}}$ .

Let $\lambda$ be an eigenvalue of $P_\nu$ of eigenvector $a$ . This yields

$\nu(e)a+(1-\nu(e))P_{\tilde{\psi}}(a)=\lambda a$

and thus

$\displaystyle P_{\tilde{\psi}}a=\frac{\lambda-\nu(e)}{1-\nu(e)}a$ .

Therefore, as $a$ is also an eigenvector for $P_{\tilde{\psi}}$ , and $P_{\tilde{\psi}}$ is a stochastic operator (if $a$ is an eigenvector of eigenvalue $|\lambda|>1$ , then $\|P_\nu a\|_1=|\lambda|\|a\|_1\leq \|a\|_1$ , contradiction), we have

$\displaystyle \left|\frac{\lambda-\nu(e)}{1-\nu(e)}\right|\leq 1$

$\Rightarrow |\lambda-\nu(e)|\leq 1-\nu(e)$ .

This means that the eigenvalues of $P_\nu$ lie in the ball $B_{1-\nu(e)}(\nu(e))$ and thus the only eigenvalue of magnitude one is $\lambda=1$ , which has (left)-eigenvector the stationary distribution of $P_\nu$ , say $\nu_\infty$ .

If $\nu$ is symmetric/reversible in the sense that $\nu=\nu\circ S$ , then $P_\nu$ is self-adjoint and has a basis of (left)-eigenvectors $\{\nu_\infty=:u_1,u_2,\dots,u_{|G|}\}\subset \mathbb{C}G$ and we have, if we write $\nu=\sum_{t=1}^{|G|}a_tu_t$ ,

$\displaystyle \nu^{\star k}=\sum_{t=1}^{|G|}a_t\lambda_t^ku_t$ ,

which converges to $a_1\nu_\infty$ (so that $a_1=1$ ).

If $\nu$ is not reversible, it is a standard argument to show that when put in Jordan normal form, that the powers $P_{\nu}^k$ converge and thus so do the $\nu^{\star k}$ $\bullet$

Total Variation Decrasing

Uses Simeng Wang’s $\|a\star_Ab\|_1\leq \|a\|_1\|b\|_1$ . Result holds for compact Kac if the state has a density.

Periodic $e^2$ is concentrated on a coset of a proper normal subgroup of $\mathfrak{G}_0$

$e_2+e_4$ is a minimal projection (coset) in the quotient space of the normal subgroup (to be double checked) given by $\langle e_1,e_3\rangle$

Supported on Subgroup implies Reducible

I have a proof that reducible is equivalent to supported on a pre-subgroup.

	Freslon’s Ergo… on Ideals & Normal Quantum…
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	J.P. McCarthy on MATH6040: Winter 2019, Week…
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Category Archive

Freslon’s Ergodic Theorem for (Finite) Group Algebras

Freslon’s Ergodic Theorem for (Finite) Group Algebras

Irreducible and Periodic Positive Maps

Introduction

Stochastic Operators and Operator Algebras

Periodicity of Random Walks on Finite Quantum Groups

The Ergodic Theorem for Random Walks: from Finite Groups, to Group Algebras, to Finite Quantum Groups

Finite Quantum Groups: Pure States

Introduction

Finite Ring Groups

Lemma 8.1

Lemma 8.2

Lemma 8.3

Lemma 8.4

Ideals & Normal Quantum Subgroups

C*-Ideals

Definition

Symmetries of Quantum Spaces – Podlés

Definition

Proposition

Periodicity & Aperiodicity for Random Walks on Quantum Groups

Background

Ergodic Theorem for Random Walks on Finite Groups

Ergodic Theorem for Random Walks on Finite Quantum Groups

Irreducibility

Lattice of Idempotent States on a Locally Compact Quantum Group — Kasprzak & Sołtan

Abstract

1. Introduction

Some Unresolved and Unexplored Aspects of Random Walks on Quantum Groups

“e in support” implies convergence

Total Variation Decrasing

Periodic $e^2$ is concentrated on a coset of a proper normal subgroup of $\mathfrak{G}_0$

Supported on Subgroup implies Reducible

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Periodic $e^2$ is concentrated on a coset of a proper normal subgroup of $\mathfrak{G}_0$