Discrete Harmonic Analysis and Noncommutative Probability

The main organizer: prof. Marek Bożejko

Time and place: every Thursday, 10.00 - 12.00 Institute of Mathematics, Wrocław University, room 604.

Topics: commutative and non-commutative harmonic analysis, quantum groups, combinatorics, quantum probability, free probability, Young diagrams, random matrices, convolutions of measures, ...

People you may meet here: Marek Bożejko (IMPAN), Biswarup Das, Wiktor Ejsmont, Anna Krystek (Politechnika Wrocławska), Romuald Lenczewski (Politechnika Wrocławska), Wojciech Młotkowski, Lahcen Oussi, Piotr Śniady, Anna Wysoczańska-Kula, Janusz Wysoczański

# Current schedule:

## Entropy for general quantum systems

We revisit the notion of relative entropy for both classical and quantum systems, and provide some new descriptions of this notion, respectively based on the theories of the Connes cocycle derivative, and noncommutative $L^p$-spaces. We then introduce the notion of entropy for a single state of a general quantum system, and show that this notion agrees with von Neumann entropy in the case of semifinite von Neumann algebras. In closing we investigate the relationship between this notion of entropy and relative entropy, and identify an Orlicz space which forms the home for all states with good'' entropy. This is a joint work with Adam Majewski.

# Previous talks:

#### Thursday, January 31, 2019, 10:15-12:00, room 604 Ewa Damek (Uniwersytet Wrocławski) Solutions to smoothing equations - existence of density.

We study the stochastic equation $$Y=_{law}\sum _{j=1}^N T_jY_j, \quad \quad \qquad (1)$$ where $=_{law}$ denotes equality in law, $(T_1,T_2,....)$ is a given sequence of complex variables i.e. random variables with values in $\mathbb{C}$ and $Y_1,Y_2,...$ are independent copies of the complex random variable $Y$ and independent of $(T_1,T_2,....)$. Let $N$ be the (possibly random) number of $T_j$ that are non zero. As long as $\mathbb{E}N>1$, the law of $\sum _{j= 1}^N T_jY_j$ behaves much better with respect to local regularity than the law of $Y$ which justifies the name smoothing''. For the real valued case and $T_j\geq 0$ the first result about absolute continuity of $Y$ was obtained by Liu. Recently, under suitable assumptions on $(T_1,T_2,....)$, a complete description of the set of solutions to (1) has been provided by Meiners and Mentemeier (2017). Using that we (Damek and Mentemeier) have proved absolute continuity of $Y$ in the complex case. Limit theorems for a broad class of random recursive structures and algorithms are derived by a so called'' contraction method. It was introduced by R\"osler in 1991 for the distributional analysis of complexity of Quicksort and later on has been extended to a variety of random recursive structures like recursive algorithms, data structures, Polya urn models and random tree models. Limit distributions derived by the contraction method are given implicitly as solutions to (1). Often there is no explicit formula for the distribution of $Y$ but still many properties of these limits may be derived from the equation. One of them is the absolute continuity, possibly differentiability which is important for the sake of possible applications.

#### Thursday, January 23, 2019, 10:15-12:00, room 604 Xumin Wang (Universite Bourgogne Franche Comte) Invariant Markov semigroups on the quantum homogeneous spaces

We obtained some one-to-one correspondences of invariant quantum Markov semigroups acting on expected coideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The classical sphere $S^{N-1}$, the free sphere $S^{N-1}_+$, and the half-liberated sphere $S^{N-1}_*$ are considered as examples. On these spheres, we classifying the generators of the invariant quantum Markov semigroups by calculating their eigenvalues and eigenvector spaces. The generators obtained in this way can naturally be viewed as Laplace operators on these spaces.
The talk is a part of the Polonium programme, co-financed by the Polish National Agency for Academic Exchange (NAWA).

#### Thursday, January 3, 2019, 10:15-12:00, room 604 Jakub Gismatullin (IM PAN & UWr)On counting multifurcating trees and metric ultraproducts of groups

In the first part I will present work in progress with A. Żywot on counting rooted phylogenetic multifurcating trees. We study recurrence $T_{n,m}=(n+m-2)T_{n-1,m-1} + m T_{n-1,m}$, $n>m>1$, due to J. Felsenstein, where $T_{n,m}$ is the number of all such trees trees with $n$ leaves and $m$ internal leaves. After certain transformation this recurrence gives apparently new family of polynomials of binomial type. Generating functions and cumulants of it is closely related to Maclaurin expansion of $\frac{x^2}{\exp(x)-x-1}$, and maybe also to some generalized Bernoulli numbers. In the second part I will speak about groups with conjugacy-invariant norms and metric ultraproducts, focusing on stronger (uniform) versions of simplicity and amenability of metric ultraproducts of groups, for example uniform metric amenability, which is strictly weaker that uniform amenability and strictly stronger than amenability. I will also present a Foelner condition for amenability of topological groups (due to Thom and Schneider) in this context. This part is a joint work with M. Ziegler.

#### Thursday, January 10, 2019, 10:15-12:00, room 604 Ignacio Vergara (IM PAN)Radial Schur multipliers

A Schur multiplier on a set $X$ is a function $\varphi:X\times X\to\mathbb{C}$ defining a map $M_\varphi$ on the algebra of bounded operators on $\ell_2(X)$ by $\langle M_\varphi(T)\delta_y,\delta_x\rangle=\varphi(x,y)\langle T\delta_y,\delta_x\rangle$, for all $T\in\mathcal{B}(\ell_2(X))$. In this talk, I will focus on the particular case when $X$ is (the set of vertices of) a graph and the function $\varphi$ depends only on the distance between each pair of vertices. Such functions are said to be radial. For homogeneous trees, radial Schur multipliers were characterised by Haagerup, Steenstrup and Szwarc by the fact that a certain Hankel matrix belongs to the trace class. I will present some extensions of this result to larger classes of graphs: products of trees, CAT(0) cube complexes and products of hyperbolic graphs. The study of such graphs is motivated by some results on weak amenability for groups acting on them.

#### Thursday, January 17, 2019, 10:15-12:00, room 604 Marek Bożejko Central Limit Theorem, Generalized Gaussian processes and realizations on some Fock spaces

In my talk we will consider the following subjects: 1. Abstract CLT and connections with generalized Hermite polynomials like: (a) q-continuus (b) q-discrete (c) q-Meixner -Pollaczek (d) Free Meixner (Kesten) (e) arbitrary central orthogonal polynomial can be considered as generalized Hermite polynomial. 2. Connections with positive definite functions on permutation (hyperoctahedral) groups and Thoma characters on that class of groups.

#### Thursday, December 17, 2018, 14:00-15:30, room 604 Natasha Blitvic (Lancaster University)Noncommutative Central Limit Theorems

We will discuss noncommutative versions of the Central Limit Theorem. We will describe the general ideas behind such constructions, along with several recent examples and some current work in progress. Parts of this talk will be based on recent joint work with W. Ejsmont.

#### Thursday, December 13, 2018, 10:15-12:00, room 604 Simon Schmidt (Universität des Saarlandes)Quantum symmetries of finite graphs

The symmetry of a graph is captured by its automorphism group. This talk will concern a generalization of this concept in the framework of Woronowicz's compact matrix quantum groups, the so-called quantum automorphism group. A natural question is: When does a graph have no quantum symmetry, i.e. when does the quantum automorphism group coincide with the classical automorphism group? We will see that the Petersen graph has no quantum symmetry. Furthermore, we show that if the automorphism group of a graph contains a certain pair of automorphisms, this graph has quantum symmetry.

#### Thursday, December 6, 2018, 10:15-12:00, room 604 Biswarup Das (Uniwersytet Wrocławski)Admissibility conjecture for quantum group representations

A long-standing open problem in the representation theory of quantum groups is to decide whether the following statement is true or false: Every finite dimensional, unitary representation of (locally) compact quantum group ''factors'' in a suitable sense, through a representation of some matrix quantum group. This conjecture came up first in a work of P. Sołtan, and later on it kept featuring in many subsequent works on representation theory of (locally) compact quantum groups. We will give a partial solution to this conjecture, through proving that for a large class of (locally) compact quantum groups, this is true. Based on a joint work with P. Salmi and M. Daws.

#### Thursday, November 29, 2018, 10:15-12:00, room 604 Adrian Dacko (Politechnika Wrocławska)V-monotone independence. Part III

Kontynuacja referatu z dnia 22 listopada.

#### Thursday, November 22, 2018, 10:15-12:00, room 604 Adrian Dacko (Politechnika Wrocławska)V-monotone independence. Part II

Kontynuacja referatu z dnia 11 października.

#### Thursday, 8 November, 2018, 10:00-12:00, room 604 Adam Paszkiewicz (Uniwersytet Łódzki) Fenomeny związane ze składaniem kontrakcji

Omówię paradoksy związane z trajektoriami $x_n := T_n \ldots T_1 x_0$ dla $n \in \mathbb{N}$, gdy $(T_1,T_2,\ldots)$ przebiega ciągi o wyrazach w skończonym zbiorze kontrakcji $\{Q_1,\ldots ,Q_k \}$, w ustalonej przestrzeni Hilberta \textit {H}. Silne metody dają twierdzenia o dylatacjach. Gdy $Q_1, \ldots ,Q_k$ muszą należeć do algebry von Neumanna $\mathcal{M}$, decydującą rolę odgrywa istnienie śladu skończonego $\tau$ na $\mathcal{M}$.

#### Thursday, 8 November, 2018, 10:00-12:00, room 604 Mitsuru Wilson (IM PAN) Quantum symmetry of the toric noncommutative manifolds

In his framework, Rieffel showed that compact Lie groups of rank at least 2 admit nontrivial $\theta$-deformations as compact quantum groups. In my recent work, I showed that an action of such a Lie group $G$ on a manifold $M$ with a toric action can be extended to an action in the deformed setting. Of course, an action cannot be extended to an action of the deformed algebras for arbitrary $\theta$-parameters. First, I will explain what these deformations mean and I will explain exactly when an action in the classical setting extends to the noncommutative setting. I will also explain how the noncommutative 7-sphere S^7_Θ can be viewed as a quantum homogeneous space.

#### Thursday, October 25, 2018, 10:15-12:00, room 604 Patryk Pagacz (Uniwersytet Jagielloński)Rozkłady typu Wolda dla stacjonarnych pól losowych oraz dla komutujących izometrii na przestrzeni Hilberta

Punktem wyjścia referatu jest twierdzenie Wolda, zarówno w pierwotnej wersji dotyczącej procesów stochastycznych jak i w abstrakcyjnej teorio-operatorowej wersji. Podczas referatu przyjrzymy się późniejszym analogicznym wynikom dotyczącym dwuwymiarowych pól losowych jak i niezależnie uzyskiwanym rezultatom dotyczącym rozkładów par izometrii na przestrzeni Hilberta. Przedstawimy znaczenie wyników, uzyskanych w ramach Teorii Operatorów, dla Procesów Stochastycznych i vice versa.

#### Thursday, October 11, 2018, 10:15-12:00, room 604 Adrian Dacko (Politechnika Wrocławska)V-monotone independence

We introduce and study a new notion of noncommutative independence, called V-monotone independence, which generalizes the monotone independence of Muraki. We investigate the combinatorics of mixed moments of V-monotone random variables and prove the central limit theorem. We obtain a combinatorial formula for the limit moments and we find the solution of the differential equation for the moment generating function in the implicit form.

#### Thursday, October 4, 2018, 10:15-12:00, room 604 Alexander Bendikov (Uniwersytet Wrocławski)On the spectrum of the hierarchical Schrödinger operator: the case of fast decreasing potential

This is the spectral analysis of the Schrödinger operator $H=L-V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=D^{\alpha}$ by a potential $V$. Assuming that $V$ belonges to a class of fast decreasing potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V$, and low energy part which lies in the spectrum of certain bounded Schrödinger operator acting on the Dyson hierarchical lattice. The spectral asymptotics strictly depend on the transience versus recurrence properties of the underlying hierarchical random walk. In the transient case we will prove results in spirit of CLR theory, for the recurrent case we will provide Bargmann's type asymptotics.