# Discrete harmonic analysis seminar - past schedule for the year 2016/2017:

Current page of the seminar is available here.

### Oussi Lahcen (Hassan I University, Maroko)

#### Poisson type limit theorems for a noncommutative independence associated with positive symmetric cones (second part)

We present Poisson type limit theorems for a noncommutative independence (the bm-independence), which is naturally associated with positive symmetric cones in euclidian spaces, including R_+^d, the Lorentz cone in Minkowski spacetime and positive definite (real symmetric or complex hermitian) matrices. The geometry of the cones plays significant role in the study as well as the combinatorics of bm-ordered partitions.

### Bogusław Zegarliński (Imperial College London)

#### Part. I Algebraic and Analytic Questions Concerning Crystallographic Groups for Hoermander Fields

I will introduce a notion of Coxeter group associated to a finite family of fields, (e.g. generators of nilpotent Lie groups), present examples and possibly interesting questions to Algebraists and Analysts. I will also introduce natural Dunkl type extensions of fields and Markov semigroups, and present some (crude) bounds of corresponding heat kernel.

#### Part.II Smoothing and Ergodicity of Dissipative Dynamics for Large Interacting Systems

This would be about Properties of Markov Semigroups with Hormarder type generators as well as some generalisation of Dunkl generators.

### Spectral theory of Fourier-Stieltjes algebras

My talk is devoted to presentation of the most important results from a joint work with Mateusz Wasilewski concerning spectral properties of Fourier-Stieltjes algebras. It is an extensive project including research on elements and Gelfand spaces of the discussed algebras. The most signicant topics are concentrated around the notion of the spectrum of an element in a relation to an image of a function on a group which leads to the notion of the naturality of the spectrum and Wiener-Pitt phenomenon (for non-commutative groups). A lot of other patologies can be transferred from the commutative case but some problems do not have the classical counterparts which is caused by the non-commutativity of group C* -algebras. The subject matter is very broad and the most of the results will only be outlined. Nevertheless I will try to present in an accesible way the dierences between the world of measures on commutative (locally compact) groups and Fourier-Stieltjes algebras (for now only for discrete groups).

### Karol Życzkowski (IF UJ, CFT PAN)

#### Spectral properties of some ensembles of random matrices

We analyze spectral density of ensembles of positive hermitian random matrices related to free convolutions of Marchenko-Pastur distributions. Furthermore, we study asymptotic support of the spectrum and numerical range (field of values) for ensembles of non-hermitian random matrices with independent Gaussian entries.

### Michał Wojciechowski (IMPAN)

#### Anisotropic Ornstein noninequalieties

We investigate the existence of a priori estimates for differential operators in the L1 norm: for anisotropic homogeneous differential operators T1 , . . . , Tl , we study the conditions under which the inequality l\||T_1f\|_1 < \|T_2f\|_1+\|T_3f\|_1+ ......+\|T_lf\|_1 holds true. Properties of homogeneous rank-one convex functions play the major role in the subject. We generalize the notions of quasi- and rank-one convexity to fit the anisotropic situation. We also discuss a similar problem for martingale transforms and provide various conjectures.

### Mateusz Wasilewski (IMPAN)

#### Complete metric approximation property of q-Araki-Woods algebras

We will prove that q-Araki-Woods algebras, which are constructed as a combination of two deformations of the free group factors, have the complete metric approximation property. In the proof we transfer the result from the q-Gaussian algebras (one of the deformations mentioned previously) using a certain central limit theorem and ultraproduct techniques.

### Wend Werner (Universität Münster)

#### Symmetric spaces and K-theory

Roughly a quarter of (Riemannian) symmetric spaces are hermitian and of non-compact type. Each such manifold carries an algebraic structure on its tangent bundle which is similar to (more general than) the algebraic structure of a C*-algebra. We exploit this similarity in order to apply K-theoretical methods to the classification of these manifolds. Whereas this technique reproduces well-known results in finite dimensions, it is still viable for infinite dimensional manifolds and can here be used to e.g. give a K-theoretical classification of inductive limits of bounded symmetric domains.

### Biswarup Das (University of Oulu, Finland)

#### Non-commutative separate continuity: Ellis joint continuity theorem.

Let S be a topological space, which is also a semigroup with identity, such that the multiplication is separately continuous. Such semigroups are called semitopological semigroups. These type of objects occur naturally, if onestudies weakly almost periodic compactification of a topological group. Now if we assume the following: (a) The topology of S is locally compact. (b) Abstract algebraically speaking, S is a group (i.e. every element has an inverse). (c) The multiplication is separately continuous as above (no other assumption. This is the only assumption concerning the interaction of the topology with the group structure). Then it follows that S becomes a topological group i.e. : (a) The multiplication becomes jointly continuous. (b) The inverse is also continuous. This extremely beautiful fact was proven by R. Ellis in 1957 and is known in the literature as Ellis joint continuity theorem. In this talk, we will prove a non-commutative version of this result. Upon briefly reviewing the notion of semitopological semigroup, we will introduce ''compact semitopological quantum semigroup'' which were before introduced by M. Daws in 2014 as a tool to study almost periodicity of Hopf von Neumann algebras. Then we will give a necessary and sufficient condition on these objects, so that they become a compact quantum group. As a corollary, we will give a new proof of the Ellis joint continuity theorem as well. This is the joint work with Colin Mrozinski.

### Rafał Kobuszewski

#### Szacowanie i badanie zbieżności iteracji szeregów

W odczycie przedstawimy metody szacowania i badania zbieżności iteracji szeregu funkcji rozwiniętej w punkcie stałym. Wykażemy pewne związki z partycjami liczb naturalnych i liczbami Catalana.

### Piotr Hajac (IMPAN)

#### Consequences of the torsion noncommutative Borsuk-Ulam Theorem

The goal of this talk is to recall a noncommutative Brouwer fixed-point theorem, and show in detail how it is a special case of the torsion noncommutative Borsuk-Ulam theorem. In particular, the issue of non-contractibility of compact quantum groups shall be explored.

#### Semiclassical asymptotics of GLN(C) tensor products and quantum random matrices

The Littlewood-Richardson process is a discrete random point process which encodes the isotypic decomposition of tensor products of irreducible rational representations of GLN(C). Biane-Perelomov-Popov matrices are a family of quantum random matrices arising as the geometric quantization of random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors. As first observed by Biane, correlation functions of certain global observables of the LR process coincide with correlation functions of linear statistics of sums of classically independent BPP matrices, thereby enabling a random matrix approach to matrix approach to the statistical study of GLN(C) tensor products. In this paper, we prove an optimal result: classically independent BPP matrices become freely independent in any semiclassical/large-dimension limit. This removes all assumptions on the decay rate of the semiclassical parameter present in previous works, and may be viewed as a maximally robust geometric quantization of Voiculescu's theorem on the asymptotic freeness of independent unitarily invariant random Hermitian matrices. In particular, our work proves and generalizes a conjecture of Bufetov and Gorin, and shows that the mean global asymptotics of GLN(C) tensor products are governed by free probability in any and all GLN(C) tensor products are governed by free probability in any and all semiclassical scalings. Our approach extends to global fluctuations, and thus yields a Law of Large Numbers for the LR process valid in all semiclassical scalings.

### Maria Elena Griseta (University of Bari Aldo Moro)

#### On the distributions of position operators on Weakly Monotone Fock Space

In this talk we investigate the distributions for sums of creation and annihilation operators on Weakly Monotone Fock Space. After obtaining a recursive formula for the moments $\mu_{m,n} := \omega_{\Omega}((\sum_{k=1}^{m}(A_{k}+A_{k}^{\dagger}))^{2n})$, where $\Omega$ is the vacuum vector and $\omega_{\Omega}$ is the vacuum expectation, we calculate the Cauchy Transform of the distribution measure.

### Nobuaki Obata (Tohoku University)

#### Distance matrices and quadratic embedding of graphs

General criteria for a graph to admit a quadratic embedding are discussed and, as a quantitative approach, the "QE constant" is introduced. Concrete examples are obtained from well-known graphs with graph operations and the QE constants are determined for all graphs on n vertices, n≤5.

### Oussi Lahcen (Hassan I University, Maroko)

#### Poisson type limit theorems for a noncommutative independence associated with positive symmetric cones

We present Poisson type limit theorems for a noncommutative independence (the bm-independence), which is naturally associated with positive symmetric cones in euclidian spaces, including $\R_+^d$ and the Lorentz cone in Minkowski spacetime. The geometry of the cones plays significant role in the study as well as the combinatorics of bm-ordered partitions.

### Anna Lytova (Uniwersytet Opolski)

#### Linear eigenvalue statistics for sample covariance matrices without independent structure in columns

Here we deal with the sample covariance (SC) matrices of the form M=YY^T, where columns y_i, i=1,...,m of the matrix Y are independent random vectors in R^n. Under the assumption that all entries of Y are independent, the asymptotic spectral analysis of SC matrices has been actively developed since the celebrated work of Marcenko and Pastur (1967). Less is known about large SC matrices with dependent entries in columns of Y.  In papers by Bai and Zhou, Pajor and Pastur, and Yaskov, there were considered limiting spectral distributions  of SC matrices with some general dependence structures of y_i. The next natural step is to study asymptotic fluctuations of linear eigenvalue statistics of the form Tr f(M), where f is a test function.  We show that if y_1,..., y_m  are i.i.d. normalized isotropic random vectors satisfying certain moment conditions, then in the limit when m,n tend to infinity and m/n tends to   c>0, the centered linear eigenvalue statistics converge in distribution to a Gaussian random variable.

### Rachid El Harti (Université Hassan I FST de Settat, Maroko)

#### The amenability from algebraic and analytical perspective

In this talk, we investigate the amenability from the algebraic and analytical point of view and its relationship with the semisimplicity in the case of operator algebras and cross product Banach algebras associeted with a class of C*-dynamical systems.

### Oussi Lahcen (Hassan I University, Maroko)

#### bm-Central Limit Theorem associated with nonsymmetric cone

We formulate and prove the general version of the bm-Central Limit Theorem (CLT) for bm-independent random variables associated with nonsymmetric cones, in particular circular cone $\mathcal{C}_{\theta}^{n}$, such sector $\Omega_{u}^{d}\subset \mathbb{R}^{d}$ and also the Vinberg's cone $\Pi_{v}$ is studied.

### Anna Krystek, Łukasz Wojakowski

#### Direct product of automorphism groups of digraphs

The problem of representability of a permutation group A as the full automorphism group of a digraph G = (V, E) was first studied for regular permutation groups by many authors, the solution of the problem for undirected graphs was first completed by Godsil in 1979. For digraphs, L. Babai in 1980 proved that, except for the groups S_2^2 , S_2^3 , S_2^4 , C_3^2 and the eight element quaternion group Q, each regular permutation group is the automorphism group of a digraph. Later on, the direct product of automorphism groups of graphs was studied by Grech. It was shown that, except for an infinite family of groups S_n × S_n , n ≥ 2, and three other groups D_4 × S_2 , D_4 × D_4 , and S_4 × S_2 × S_2, the direct product of automorphism groups of two graphs is, itself, an automorphism group of a graph. We study the direct product of automorphism groups of digraphs. We show that, except for the infinite family of permutation groups S_n × S_n , n ≥ 2 and four other permutation groups D_4 × S_2 , D_4 × D_4 , S_4 × S_2 × S_2 , and C_3 × C_3 , the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph.

### Ludwik Dąbrowski (SISSA Trieste/IMPAN) and Piotr M. Hajac (IMPAN)

#### Noncommutative Borsuk-Ulam-type conjecture revisited

Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A. Baum, Dąbrowski and Hajac conjectured that there does not exist an equivariant *-homomorphism from A to the equivariant noncommutative join C*-algebra A*H. When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on Z/2Z acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam Theorem. Recently, Chirvasitu and Passer proved the conjecture when H is commutative. In a simple way, we extend this result to a far more general setting assuming only that H admits a character different from the counit. We show how our result implies a noncommutative Brouwer fixed-point theorem and, in particular, the non-contractibility of such compact quantum groups. Moreover, assuming that our compact quantum group is a q-deformation of a compact connected semisimple Lie group, we prove that there exists a finite-dimensional representation of the compact quantum group such that, for any C*-algebra A admitting a character, the finitely generated projective module associated with A*H via this representation is not stably free. Based on joint work with Sergey Neshveyev.

### Rafał Sałapata

#### Kombinatoryka produktów macierzy losowych, cz II

Referat będzie kontynuacją mojego referatu z poprzedniego semestru. Korzystając z metod probabilistyki macierzowo wolnej, wyliczę momenty graniczne iloczynów prostokątnych macierzy losowych. Wyrażę je wsposób kombinatoryczny za pomocą partycji nieprzecinających na odpowiednich słowach oraz podam wzór na S-transformatę miary granicznej,tzn. miary o takich momentach. Ponadto, podam kilka zastosowań w.w.wzorów w kombinatoryce. Wyniki wspólne z R. Lenczewskim.

### Wojciech Tarnowski (Uniwersytet Jagielloński)

#### Druga część twierdzenia Haagerupa-Larsena i funkcja korelacji dla wektorów własnych

Twierdzenia Haagerupa-Larsena opisuje związek między wartościami własnymi a wartościami osobliwymi dla operatorów R-diagonalnych, których realizacją są macierze izotropowe. W trakcie wystąpienia pokażę, że w formaliźmie transformat Cauchy'ego o wartościach w macierzach 2x2, znanym też jako kwaternionizacja lub hermityzacja można otrzymać analogiczną wersję twierdzenia H-L dla funkcji korelacji będącej gęstością spektralną ważoną przez długość lewych i prawych wektorów własnych. Prezentacja opiera się na pracy arXiv:1608.04923 [math-ph].

### Franz Lehner (Technische Universität Graz)

#### Spreadability, cumulants and Hausdorff series

We define spreadability systems as a generalized notion of independence, extending exchangeability systems, to unify various notions of cumulants known in noncommutative probability. To this end we study generalized zeta and M\"obius functions in the context of the incidence algebra of the semilattice of ordered set partitions. The combinatorics are intimately related to the coefficients of the Campbell-Baker-Hausdorff formula and the latter can be seen as a special case of a particular spreadability system.