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The Frucht property in the quantum group setting

November 19, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

Abstract

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

with Teo Banica, Glasgow Math J., to appear. Arxiv link here.

A state-space approach to quantum permutations

April 6, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

Abstract

An exposition of quantum permutation groups where an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and posits that states on the algebra of continuous functions on a quantum permutation group can be interpreted as quantum permutations. This interpretation allows talk of an element of a compact quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated with the Kac-Paljutkin quantum group, the duals of finite groups, as well as by other finite quantum group phenomena.

Arxiv link here.

Talk: Quantum Group Seminar

August 21, 2020 in C*-Algebras & Operator Theory, Quantum Groups, Research | Leave a comment

Giving a talk 17:00, September 1 2020:

See here for more.

Slides.

The Ergodic Theorem for Random Walks on Finite Quantum Groups

April 2, 2020 in C*-Algebras & Operator Theory, Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research, Research Updates | 1 comment

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.

Link to journal here.

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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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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Projection Distance Equal to Total Variation Distance?

November 9, 2017 in C*-Algebras & Operator Theory, Probability Theory, Quantum Groups, Random Walks on Finite Quantum Groups, Research | 2 comments

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).

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Direct Limits and Tensor Products: Direct Limits of C*-algebras

January 7, 2012 in C*-Algebras & Operator Theory, Research | Leave a comment

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 $m\geq n$ .

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Representations of C*-Algebras: Irreducible Representations and Pure States

October 5, 2011 in C*-Algebras & Operator Theory, Research | 1 comment

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 $a\in A$ .

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Direct Limits and Tensor Products: Tensor Products of C*-algebras

September 9, 2011 in C*-Algebras & Operator Theory, Research | 1 comment

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.

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Category Archive

The Frucht property in the quantum group setting

Abstract

A state-space approach to quantum permutations

Abstract

Talk: Quantum Group Seminar

The Ergodic Theorem for Random Walks on Finite Quantum Groups

Abstract

Irreducible and Periodic Positive Maps

Introduction

Stochastic Operators and Operator Algebras

Ideals & Normal Quantum Subgroups

C*-Ideals

Definition

Projection Distance Equal to Total Variation Distance?

Distances between Probability Measures

Quantum Projection Distance $\neq$ Quantum Variation Distance?

Direct Limits and Tensor Products: Direct Limits of C*-algebras

Representations of C*-Algebras: Irreducible Representations and Pure States

Theorem 5.1.1

Direct Limits and Tensor Products: Tensor Products of C*-algebras

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Introduction

Stochastic Operators and Operator Algebras

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Definition

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Distances between Probability Measures

Quantum Projection Distance Quantum Variation Distance?

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Theorem 5.1.1

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Quantum Projection Distance $\neq$ Quantum Variation Distance?