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Abstract: Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-empty weak*-compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

Link to arxiv.

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BCRI Mini-Symposium: Noncommutative Probability & Quantum Information

Monday, 10th October 2022 from 12:00 to 15:00

Organizers: Claus Koestler (UCC), Stephen Wills (UCC)

SPEAKER: J.P. McCarthy (Munster Technological University)
TITLE: The Kawada-Itô theorem for finite quantum groups.
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 compact quantum groups, where a state on the algebra of functions plays the role of the driving probability. A random walk on a compact quantum group can fail to be irreducible without being concentrated on a proper quantum subgroup. In this talk we will explore this phenomenon. Time allowing, we will talk about periodicity, and as a conclusion, I give necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group in terms of the support projection of the driving state.

In the end the talk (below) didn’t quite match the abstract.

Quantum Group Seminar, Monday 24 January, 2022.

Abstract: A classical theorem of Frucht states that every finite group is the automorphism group of a finite graph. Is every quantum permutation group the quantum automorphism group of a finite graph? In this talk we will answer this question with the help of orbits and orbitals.

This talk is based on joint work with Teo Banica.

This article has been accepted to Expositiones Mathematicae.

Abstract

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.

Anyone clicking here before March 04, 2022 will be taken directly to the latest version of the article on ScienceDirect, which they are welcome to read or download. No sign up, registration or fees are required.

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.

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.

Giving a talk 17:00, September 1 2020:

See here for more.

Slides.

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.

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^kNWe 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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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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