# Discrete harmonic analysis seminar - past schedule for the year 2008/2009:

Current page of the seminar is available here.

#### Jacek Wesołowski (Warsaw University of Technology)

##### Quadratic harnesses, q-commutation relations and Askey-Wilson polynomials

Kwadratowe harnessy to procesy stochastyczne mające liniową warunkową wartość oczekiwaną i kwadratową warunkową wariancję przy warunkowaniu przez przeszłość i przyszłość łącznie. Przykłady kwadratowych harnessów to: proces Wienera, proces Poissona, proces Dirichleta, wolny proces Wienera, procesy q-gaussowskie, proces bi-poissonowski. Problem, którym się zajmujemy, to próba kompletnego opisu tej klasy procesów stochastycznych. Okazuje się, że są to zawsze procesy markowskie, co więcej są one wyznaczone jednoznacznie przez 5 stałych rzeczywistych (jedną z nich jest q). Stałe te są parametrami równania q-komutacyjnego wiążącego dwie części macierzy Jacobiego ortogonalnych wielomianów martyngałowych rozważanego procesu. Macierz Jacobiego jest funkcją afiniczną czasu i wspomniane dwie części to odpowiednie współczynniki macierzowe tej właśnie funkcji. W taki schemat po odpowiednim przeparametryzowaniu zanurzyć można szeroką klasę wielomianów Askey-Wilsona. Miara ortogonalizująca te wielomiany jest rozkładem jednowymiarowym odpowiedniego kwadratowego harnessa. Wyniki, o których będę mówił, zostały otrzymane wspólnie z W. Brycem i częściowo z W. Matysiakiem.

#### Romuald Lenczewski (Wrocław University of Technology)

##### Asymptotic properties of random matrices and pseudomatrices

I will discuss asymptotic properties of sums of matricially free random variables. The concept of matricial freeness can be viewed as a generalization of both freeness of Voiculescu and monotone independence of Muraki. The main idea is that we consider arrays of random variables. In particular, the rows' of square arrays lead to free random variables whereas the rows' of lower- (upper-) triangular arrays lead to monotone (anti-monotone) independent random variables. At the same time, the sums of all random variables in the given array, called a random pseudomatrix, have the same asymptotics as a Gaussian random matrix if we assume (in both cases) that the variance matrix is symmetric and has block-identical variances. The same feature is exhibited by the blocks of both objects.

#### Benoit Collins (Université d'Ottawa)

##### Random matrix models arising from quantum information theory and applications

We will focus this talk on the study of a random matrix model describing the output of a specific input state (the Bell state) under well chosen random quantum channels. This matrix model is quite new from the point of view of random matrix theory and has many interesting properties. In particular, under appropriate scalings we can compute its asymptotic eigenvalue distribution and provide new examples and counterexamples for various additivity conjectures in quantum information theory.

#### Karol Życzkowski (Jagiellonian University, Cracow, Poland and Center for Theoretical Physics, Polish Academy of Science)

##### Random density matrices and random quantum maps

A link between properties of quantized chaotic systems and random matrices will be reviewed. Statistical properties of periodically driven quantum chaotic systems described in a finite dimensional Hilbert space H_N can be described by circular ensembles of random unitary matrices. To describe the effect of a possible interaction of the system in question with an environment one needs to work with density operators, which are Hermitian, positive and normalized. Discrete time evolution of a density matrix can be represented by so-called quantum operation (completely positive, trace preserving map). We introduces several ensembles of random operations, discuss algorithms to generate them and investigate spectral properties of the corresponding superoperators with spectrum consisting of N^2 eigenvalues inside the unit disk. A quantum analogue of the Frobenius-Perron theorem concerning the spectrum of stochastic matrices is formulated. Obtained predictions for random operations are compared with spectral properties of quantized chaotic systems, interacting with an environment.

#### Ilona Królak

##### Complex fermion hypercontractivity

I will present a new proof of the hypercontractivity inequalities for holomorphic algebras generated by elements fulfilling Canonial Anicommutation Relations (CAR).

#### Tim Steger (Universita degli Studi di Sassari)

##### Free group representations from vector-valued multiplicative functions

Let \Gamma denote a noncommutative free group, and let \Omega stand for its boundary. We construct a large class of unitary representations of \Gamma. This class contains many previously studied representations, and is closed under several natural operations. Each of the constructed representations is in fact a representation of \Gamma\ltimes_\lambda C(\Omega). As representations of \Gamma\ltimes_\lambda C(\Omega), they are all irreducible. As representations of \Gamma, each of them is either irreducible, or the direct sum of exactly two irreducible, inequivalent \Gamma-representations.

#### Troels Steenstrup Jensen (University of Southern Denmark)

##### Herz-Schur multipliers and spherical functions on the non–abelian free groups

There will be a short introduction to the basic concepts of the talk, namely spherical functions and Herz-Schur multipliers. The main focus of the talk is a criterion for determining when a radial function on a non-abelian free group is a Herz-Schur multiplier, together with an explicit formula for the Herz-Schur norm. This result, obtained by Haagerup-Szwarc, has remained unpublished but is to appear as part of a joint paper. As an application of this result one can find closed expressions for the Herz-Schur norm of the spherical functions on the non-abelian free groups, which extends previous work by Pytlik-Szwarc. Haagerup used these formulas to show that there are Herz-Schur multipliers on the non-abelian free groups which are not coefficients of uniformly bounded representations. This has remained unpublished, but an alternate proof has since been published by Pisier. Actually, a generalization can be found for countable discrete groups, using that such groups are amenable if and only if the Herz-Schur multipliers coincide with the Fourier-Stieltjes algebra (due to Bozejko) and that each Herz-Schur multiplier can be realized as the coefficient of a (not necessarily uniformly bounded) representation (due to Bozejko-Fendler). Finally, we will discuss various extensions to (some) Lie groups and general locally compact groups.

#### Maciej Burnecki (Wrocław University of Technology)

##### Pewna operatorowa charakteryzacja przestrzeni L^{p}

Rozważamy zanurzenie grupy transformacji odwracalnych odcinka [0,1] w algebrę operatorów ograniczonych na przestrzeni Orlicza. Dowodzimy, że włożenie to zachowuje działanie grupowe wtedy i tylko wtedy, gdy przestrzeń Orlicza jest przestrzenią L^p dla pewnego 1\le p<\infty.

#### Melanie Hinz

##### Multiplicative square of the free Poisson measure

We compute moments and free cumulants of the measure \rho_t:=\pi_t\boxtimes\pi_t, where \pi_t denotes the free Poisson law with parameter t>0. We also compute free cumulants of the symmetrization of \rho_t. Finally we introduce free symmetrization of a probability measure on \mathbb{R} and provide some examples.

#### Paweł J. Szabłowski (Warsaw University of Technology)

##### On conditional q-normal distributions

We expand Chebishev polynomials and some of its linear combination in linear combinations of q-Hermite and Al Salam-Chihara polynomials. We use these expansions to expand q-Normal and related densities into infinite series of Chebishev polynomials and thus study probabilistic properties of these distributions including efficient simulation. The analysis includs also case q>1. Then distributions are discrete. In the case of conditional normal distribution the support consists of zeros of some Al-Salam-Chihara polynomilas. We find those zeros. We use this result to prove the existence of stationary random fields with linear regressions and thus close an open question posed by W. Bryc et al.. We prove this result by describing a discrete 1 dimensional conditional distribution. As a by product we generalize to q-series case, a well known formula (x+y)^{n}=\sum_{i=0}^{n} \binom{n}{k}i^{k}H_{n-k}\left( x\right) H_{k}\left( -iy\right) , where H_{k}\left( x\right) denotes k-th Hermite polynomial.

#### Ana-Maria Stan (Universite de Franche Comte, Besancon)

##### The completely bounded multiplier norm of some Leinert sets in the free group

In this talk, we present a way to compute the completely bounded multiplier norm of the characteristic function of the set of generators in the free group

#### Nizar Demni (Universit\'e Pierre et Marie Curie, Paris)

##### Ultraspherical type generating functions, generalized Stieltjes transforms and Markov transforms

I give the classification of families of orthogonal polynomials with ultraspherical type generating functions. This gives identities where a generalized Stieltjes transform of some probability distribution is expressed as the geometric mean of two Stieltjes transforms on the one side and as a powered Stieltjes transform of a probability distribution on the other side. Using Markov transforms, we get a family of symmetric probability distributions interpolating between the arcsine and the Wigner distributions. As a matter of fact, interesting, not yet solved, problems emerge: finding a convolution operation that interpolates between the r-convolution defined by M. Bozejko for r=1/2 and r=1, find a parallel to Young diagrams for which this probability distribution is the rescaled limiting shape.

#### Marcin Marciniak (University of Gdańsk)

##### Some problems concerning extremal positive maps acting on type I factors

We will consider extremal elements of the cone of all positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. We will show that every positive map with the property that rank \phi(P)\leq 1 for any one-dimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2-positive turns out to be automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank \phi(P)\leq 1 for some one-dimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson's problem in some special cases.

#### Stefan Neuwirth (Université de Franche-Comté, Besancon)

##### Transfer between Fourier and Schur multipliers

We will review well-known techniques and a new theorem that permit to associate a Toeplitz Schur multiplier to a Fourier multiplier, and vice versa, and provide some applications.

#### Aleksander Pełczyński (Polish Academy of Sciences)

##### Banach spaces of functions and measures orthogonal to a Sidon set

G-compact abelian group, \Gamma-its dual, S\subset\Gamma- an infinite Sidon set; \lambda-the normalized Haar measure on G, L_1(G)=L_1(G,\lambda), M(G)-the space of all regular Borel measures on G. We consider Banach spaces: L_1^{S^\perp}(G)=\{f\in L_1(G)\vert \int_G \gamma^{-1}(g)\lambda(dg)=0 \foral \gamma\in S\}, M^{S^\perp}(G)=\{\mu\in M(G) \vert \int_G\gamma^{-1}(g)\mu(dg) \forall \gamma\in S\}. Clearly L_1^{S^\perp}(G) is naturally isometric to a subspace of M^{S^\perp}(G). We investigate Banach space properties of these spaces. Answering a question of Bourgain [B] we show that L_1^{S^\perp}(G) and M^{S^\perp}(G) are not L_1-spaces, although they share some properties of the classical spaces. \begin{theorem} (i) L_1^{S^\perp}(G) (resp. M^{S^\perp}(G)} is not isomorphic to any Banach lattice. (ii) There are bounded non absolutely summing operators from L_1^{S^\perp}(G) (resp M^{S^\perp}(G) ) to a Hilbert space. (iii) If E is a separable subspace of M^{S^\perp}(G) which contains L_1^{S^\perp}(G) then E is uncomplemented in M^{S^\perp}(G). \end{theorem} (iii) answers in negative a question of Plichko and Yost who asked whether so called separable complementation property (=SCP) is hereditary. Note that M^{S^\perp}(G) is a subspace of M(G) which has SCP, i.e. every separable subspace of M(G) is contained in a separable complemented subspace of M(G). \begin{theorem} (j) If G is metrizable then L_1^{S^\perp}(G) has a basis. (jj) L_1^{S^\perp}(G) has the Dunford-Pettis property, i.e. every weakly compact operator from L_1^{S^\perp}(G) to an arbitrary Banach space takes weakly null sequences into norm null sequences. (jjj) L_1^{S^\perp}(G) and M^{S^\perp}(G) have the uniform approximation property =(UAP). \end{theorem} Recall that a Banach space X has UAP if \exists c\ge 1 and a function \Phi:N \to N such that for every finite-dimensional subspace F\subset X there is a finite rank operator T:X\to X such that ||T||\le c, T(f)=f for f\in F \dim T(X)\le \Phi(\dim F).

#### Romuald Lenczewski (Wrocław University of Technology)

##### Matricially free random variables, part 2

I will discuss a new concept of noncommutative independence called matricial freeness, which lies somewhere in between freeness and random matrices. My talk will be similar to that in Bedlewo a few months ago, but the presentation may contain some more details.

#### Romuald Lenczewski (Wrocław University of Technology)

##### Matricially free random variables

I will discuss a new concept of noncommutative independence called matricial freeness, which lies somewhere in between freeness and random matrices. My talk will be similar to that in Bedlewo a few months ago, but the presentation may contain some more details.

#### Wednesday, 19 November, 2008, 10:15-12:00, room 603,

##### Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.

#### Wednesday, 29 October, 2008, 10:15-12:00, room 603,

##### Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R_2,R_3,... of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.

#### Marek Bożejko

##### New examples of Generalized Brownian Motions and positive definite functions on Coxeter groups

We present some new classes of Generalized Brownian Motions for which the vacuum is trace state. The elementary model using random matrix was obtained by Bryc, Dembo and Jiang (Annals of Probab.) and next by another method was done by Artur Buchholz. The corresponding measure is the free product of classical Gaussian and the Wigner law or q-Gaussian and Wigner law. Connections with new classes of positive definite functions on Coxeter groups also will be done.

#### Wojciech Młotkowski

##### Liczby Fussa-Catalana w nieprzemiennej probabilistyce

Używając metod z nieprzemiennej probabilistyki pokazujemy ze ciąg Fussa-Catalana \left({mp+r\atop m}\right)\frac{r}{mp+r} jest dodatnio określony o ile p\ge1, 0\le r\le p lub p\le0, p-1\le r\le0. Odpowiednią miarę probabilistyczną na \mathbb{R} oznaczamy przez \mu(p,r). Pokazujemy różne własności miar \mu(p,r), na przykład zależności \begin{eqnarray*} \mu(1+p,1)^{\boxtimes t}&=&\mu(1+tp,1),\\ \mu(p_1,r)\boxtimes\mu(1+p_2,1)&=&\mu(p_1+rp_2,r),\\ \mu(a,b)\vartriangleright\mu(a+r,r)&=&\mu(a+r,b+r), \end{eqnarray*} wyliczamy też momenty potegi boolowskiej \mu(p,1)^{\uplus t}. Pokazujemy też że jeśli 0\le 2r\le p, r+1\le p lub p\le 2r+1, p\le r\le 0 to \mu(p,r) jest \boxplus-nieskończenie podzielna.