## Séminaire d’Analyse Fonctionnelle

publié le , mis à jour le

Le séminaire a lieu le mardi à 13h45, en salle 316Bbis du bâtiment de
Métrologie (plan d’accès).

Vous trouverez ci-dessous le planning du séminaire d’Analyse
Fonctionnelle pour l’année universitaire en cours.
L’historique des séminaires des années précédentes se trouve
ici.

Pour contacter la responsable (Yulia Kuznetsova) : yulia.kuznetsova@univ-fcomte.fr.
Pour s’abonner au séminaire : ACM.

### Exposés à venir

-Mardi 16 janvier: Un Cig Ji, Chungbuk National University, Korea.
Inequalities for Positive Module Operators on von Neumann Algebras

We establish the Cauchy-Schwarz and Golden-Thompson inequalities for module operators, a generalization of a (noncommutative) conditional expectation, on a von Neumann algebra. We apply these inequalities to the Bennett inequality and a uncertainty relation, a generalization of the Schrödinger uncertainty relation, for conditional expectations. This is a joint work with B. J. Choi and Y. Lim.

### Décembre

-Mardi 12 décembre: Philipp Varso, Greifswald.
Central Limit Theorem for General Universal Products

To model independence in quantum probability theory one uses so-called universal products, which are in general described by tensor categories of algebraic quantum probability spaces $(A,\varphi )$. In [1] Muraki has shown how to classify such products and in particular obtained that only -five universal products exist. But there are examples which do not fit into Muraki's framework, for instance if one wants to deal with a tuple of linear functionals $\varphi ^(i)$ on the algebra $A$ like it has been done in the case of $c$-freeness by Bozejko and Speicher in [2].The case of bi-freeness by Voiculescu [3], where in particular the underlying algebra $A$ is isomorphic to the free product of two algebras $A^(1)$ and $A^(2)$, is also not covered by Muraki's classi-cation. In [4] Schürmann and Manzel present a unified approach to cumulants, which includes the above concepts of independence. This is achieved by considering a certain category of algebraic non-commutative probability spaces, denoted by $algP_d,m$, which consist of an $m$-tuple of subalgebras and a $d$-tuple of linear functionals and therefore allows to investigate $(d,m)$-independence induced by a so-called u.a.u.-product.The first noncommutative version of a central limit theorem dates back to von Waldenfels [5]. In this talk we want to present a noncommutative version of a central limit theorem for a u.a.u.-product in $algP_d,m$, where we make use of the so-called Lachs functor [6], which operates between certain tensor categories. References: [1] Muraki, Naofumi : The five independences as natural products. In-n. Dimens. Anal. Quantum Probab. Relat. Top. 6.3 (2003), 337-371. [2] Bozejko, Marek ; Speicher, Roland : $\psi$ -independent and symmetrized white noises. In : Quantum probability \& related topics. QP-PQ, VI, 219-236. World Sci. Publ., River Edge, NJ, 1991. [3] Voiculescu, Dan-Virgil : Free probability for pairs of faces I. Comm. Math. Phys. 332.3 (2014), 955-980. [4] Manzel, Sarah ; Schürmann, Michael : Non-Commutative Stochastic Independence and Cumulants. Preprint arXiv:1601.06779 (2016), 42 pages. To appear in IDAQP 20.2 (2017). [5] von Waldenfels, W. : An algebraic central limit theorem in the anti-commuting case. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42.2 (1978), 135-140. [6] Lachs, Stephanie : A New Family of Universal Products and Aspects of a Non-Positive Quantum Probability Theory. PhD thesis. Ernst-Moritz-Arndt-Universität Greifswald, 2015.

-Mardi 19 décembre: Ignacio Vergara, ENS Lyon.

Les multiplicateurs de Schur sont des fonctions à deux variables sur un ensemble $X$ qui définissent des opérateurs bornés sur $B(\ell _2(X))$ par multiplication des coefficients matriciels. Lorsque l'ensemble $X$ est un graphe, on peut étudier le cas particulier des multiplicateurs radiaux, c'est-à-dire, des fonctions que ne dépendent que de la distance entre chaque paire de sommets. Les multiplicateurs radiaux sur un arbre homogène ont été caractérisés par Haagerup, Steenstrup et Szwarc en termes de certaines matrices de Hankel. Dans cet exposé, je présenterai des extensions de ce résultat à des produits d'arbres, des produits de graphes hyperboliques et aux complexes cubiques CAT(0) de dimension finie.

### Novembre

-Mardi 21 novembre: Safoura Zadeh, IMPAN, Warsaw.
Isometric algebra isomorphisms between weighted $L^p$-algebras

In Chapter 11 of his renowned book "Théorie des opérations linéaires" S. Banach gave a description of linear norm preserving operators on the spaces $L^p$ and $l^p$, $1\leq p<\infty,\ p\not =2$. The proofs are not stated completely and the theorems are not given in their full generality. This was fulfilled by J. Lamperti who provided new proofs for a more general theorem ; besides being set in arbitrary ($\sigma$-finite) measure spaces, Lamperti's result accepts values $p<1$. Later, Parrot and Strichartz independently extended Lamperti's result to convolution $L^p$-algebras. They showed that if $G$ and $H$ are compact topological groups and if $T:L^p(G)\to L^p(H)$, $1\leq p<\infty,\ p\not =2$, is an isometric algebra isomorphism then there is an isomorphisms of topological groups $\phi :G\to H$, a continuous character $\gamma :G\to (0,+\infty )$ and a constant $c$ such that $$Tf(y)=c\gamma \circ \phi ^-1(y) f\circ \phi ^-1(y)\ \ \ (y\in H).$$ In this talk I give a description of isometric algebra isomorphisms between weighted $L^p$-algebras on locally compact groups. This is based on a join work with Yulia Kuznetsova.

-Mardi 28 novembre: Adam Skalski, IMPAN, Warsaw.
Translation invariant Dirichlet forms in the context of locally compact quantum groups

Since the work of Cipriani on one hand and Goldstein and Lindsay on the other in the 1990s it is known that certain natural class of symmetric Markov semigroups on a von Neumann algebra M equipped with a faithful normal state admitsextensions to associated Haagerup $L^p$-spaces and is characterised via a Dirichlet property of the generating quadratic form on the$L^2$-space.Recently Cipriani, Franz and Kula studied a special class of such semigroups associated to compact quantum groups. In this talk I will discuss how their results extend to the framework of locally compact quantum groups, where two new important technical features appear : there is no natural `algebraic' domain for the generator and one needs to work with weights, as opposed to finite states (using the appropriate Dirichlet form result provided by Goldstein and Lindsay). I will also present some applications of Dirichlet forms to the study of geometric properties of quantum groups.This is based on the joint work with A.Viselter.

### Octobre

-Mardi 10 octobre à 14h00: Runlian Xia, LMB.
Les espaces de Hardy locaux à valeur opératorielle et les applications sur les opérateurs pseudo-différentiels

La soutenance de thèse

-Mardi 17 octobre: Takahiro Hasebe, Hokkaido University.
Séminaire d'Analyse Fonctionnelle

-Jeudi 19 octobre à 11h00: Ping Zhong, University of Waterloo and Wuhan University.
Some noncommutative probability aspects of meandric systems

The talk will consider a family of diagrammatic objects (well-known to combinatorialists and mathematical physicists) which go under the names of "meandric systems" or "semi-meandric systems". I will review some connections which these objects are knownto have with free probability, and I will show in particular how the so-called "semi-meandric polynomials" can be retrieved from a natural consideration of operators on the $q$-Fock space. This is joint work with Alexandru Nica.

-Mardi 24 janvier: Srinivasan Raman, .
$E_0-$semigroups on factors

I will review the current progress on endomorphism semigroups on factors, particularly on non-type-I factors. This is mostly my joint work with Oliver T Margetts.

### Septembre

- Mardi 26 septembre 2017: Colin Petitjean, UFC.
The linear structure of some dual Lipschitz free spaces

Consider a metric space $M$ with a distinguished point $0_M$. Let $Lip_0(M)$ be the Banach space of Lipschitz functions from $M$ to $\mathbb R$ satisfying $f(0_M) = 0$ (the canonical norm being the best Lipschitz constant). The Lipschitz-free space $\mathcal F(M)$ over $M$ is defined as the closed linear span in $Lip_0(M)^*$ of $\delta(M)$ where $\delta (x)$ denotes the Dirac measure defined by $\langle \delta (x) , f \rangle = f(x)$. The Lipschitz free space $\mathcal F(M)$ is a Banach space such that every Lipschitz function on $M$ admits a canonical linear extension defined on $\mathcal F(M)$. It follows easily from this fundamental linearisation property that the dual of $\mathcal F (M)$ is in fact $Lip_0(M)$. A considerable effort to study the linear structure and geometry of these spaces has been undergone by many researchers in the last two or three decades. In this talk, we first focus on some classes of metric spaces $M$ for which $\mathcal F(M)$ is isometrically isomorphic to a dual Banach space. After a quick overview of the already known results in this line, we define and study the notion of "natural predual". A natural predual is a Banach space $X$ such that $X^* = \mathcal F(M)$ isometrically and $\delta(M)$ is $\sigma(\mathcal F(M),X)$-closed. As we shall see, $\delta(M)$ is always $\sigma(\mathcal F(M),Lip_0(M))$-closed but it may happened that it is not $\sigma(\mathcal F(M),X)$-closed for some predual $X$. We characterise the existence of a natural predual in some particular classes of metric spaces. Notably, we concentrate on the class of uniformly discrete and bounded (shortened u.d.b.) metric spaces, for which it is well known that $\mathcal F(M)$ is isomorphic to $\ell_1$. In particular, we exhibit an example of a u.d.b. metric space $M$ for which $\mathcal F(M)$ is a dual isometrically but which does not have any natural predual. We also provide a u.d.b. metric space $M$ such that $\mathcal F(M)$ is not a dual isometrically. We finish with the study of the extremal structure of Lipschitz free spaces admitting a natural predual. This is part of a joint work with L. García-Lirola, A. Procházka and A. Rueda Zoca.

### Mai

-Mardi 2 mai: Evguenii Abakoumouv, Marne-la-Vallée.

-Mercredi 3 mai à 11h00: Gilles Pisier, IMJ.

-Mardi 30 mai: Jean Roydor, Bordeaux.

### Avril

-Mardi 4 avril: Kangwei Li, Basque Center for Applied Mathematics.
Sawyer's conjecture : history and beyond

In this talk, I will introduce the Sawyer's conjecture, which was proposed by Sawyer in 1980s. Then in 2005, Cruz-Uribe, Martell and Pérez finally solved this conjecture, they also proposed a new conjecture, which can be viewed as the most singular case of several extensions of Sawyer's conjecture. I will give a positive answer to the new conjecture. Several quantitative estimates are also obtained. This work is jointly with Sheldy Ombrosi and Carlos Pérez.

-Mardi 11 avril: Marat Aukhadiev, Münster.
C*-algebras of left cancellative semigroups. Inverse semigroup approach.

We will discuss the new approach to the definition of the C*-algebra of a left cancellative semigroup. We will see how it resolves the problems of the old construction. Some benefits of this new approach, such as amenability vs. nuclearity, crossed products, connections with partial crossed products, will also be decribed.

### Mars

-Mardi 7 mars: Runlian Xia, Besancon.
TBA

-Mardi 14 mars: Robert Yuncken, Clermont-Ferrand.
TBA

-Mardi 21 mars: Malte Gerhold, Greifswald.
New examples of generalized Brownian motions

-Mardi 28 mars: Leonard Cadilhac, Caen.
TBA

### Février

-Mardi 14 février: Gilles Lancien, LMB.
Normes équivalentes avec la propriété $(\beta )$ de Rolewicz et applications

-Mardi 21 février: Relâche, .
Vacances d'hiver

-Mardi 28 février: Elizabeth Strouse, Bordeaux.
Truncated Toeplitz operators

-Mardi 7 février: Un Cig Ji, .
Anticipating Quantum Stochastic Integrals

### Janvier

-Mardi 10 janvier: Colin Petitjean, .
Schur properties over some Lipschitz-free spaces.

It has been known that the free spaces of countable compacts enjoy the Schur property. In this talk we will show that these (and other) spaces in fact satisfy a stronger property, the so called 1-Schur property. Also, we are going to show that $F(\ell _p)$ enjoys the (usual) Schur property whenever $p<1$. This establishes the first example of $F(M)$ with Schur property when $M$ is neither countable, nor snowflaking of another metric space.

-Mardi 17 janvier: Relâche, Trop d'absents.

-Mardi 24 janvier: Quanhua Xu, UFC.
La théorie vectorielle de Littlewood-Paley-Stein revisitée.

-Mardi 31 janvier: Relâche, École de Barboux.

### Décembre

-Mardi 6 décembre: Pawel Józiak, IMPAN, Warsowie.
On quantum increasing sequences.

Quantum increasing sequences were introduced by S. Curran to characterize free conditional independence by means of comparing joint distributions of initial segments of a sequence of random variables to joint distributions of initial segments of a subsequence of that sequence of random variables, à la Ryll-Nardzewski. This is a de Finetti type theorem, but with weakened assumptions. I will explain the rôle of increasing sequences in free probability and discuss some results of mine in the theory of compact quantum groups, that grew out of the study of the connection of quantum increasing sequences and quantum permutations.

-Mardi 13 décembre: Aris Daniilidis, Université du Chili.
De la dynamique du gradient au processus de rafle

Le processus de rafle a été introduit par Jean-Jacques Moreau dans les années 70 pour modéliser certaines problèmes de la mécanique non-régulière. On établit une variante de la technique de desingularisation de Kurdyka pour desingulariser les co-dérivées du processus de rafle dans le cas définissable, et garantir ainsi la finitude de longueur de ses orbites. Ce résultat, dans le cas particulier où le processus de rafle correspond aux sous-niveaux d'une fonction (non nécessairement régulière), généralise les résultats connus pour les orbites des systèmes dynamiques de type sous-gradient.

### Novembre

-Jeudi 3 novembre, 15h: Martin Lindsay, Lancaster University.
Multiple quantum Wiener integrals

-Mardi 8 novembre: Ignacio Vergara, ENS Lyon.
La propriété $p$-AP pour le groupe SL(3,R)

La $p$-AP est une propriété d’approximation pour les groupes localement compacts. On peut la voir comme une “version $L^p$” de la AP de Haagerup et Kraus. Je commencerai par définir cette propriété et en suite j’expliquerai comment on peut montrer que le groupe SL(3,R) ne satisfait pas $p$-AP pour aucun 1< p <$\infty$.

-Jeudi 17 novembre, 13h45: Serguei Kisliakov, Steklov Mathematical Institute, Saint-Pétersbourg.
Certains nouvelles estimations dans le théorème de la couronne.

-Mardi 22 novembre 13h30: Sebastien Schleissinger, Université de Wuerzburg.
The Loewner Equation and Monotone Probability Theory

The Loewner differential equation is an important tool in geometric function theory. It has been introduced by C. Loewner in 1923 in order to attack the Bieberbach conjecture (proven by L. de Branges in 1985). In 2000, O. Schramm considered a stochastic version of this equation, which turned out to have striking applications, in particular in statistical physics and conformal field theory. Schramm’s equation has become a field which is now called Schramm-Loewner Evolution (SLE). In this talk we consider a simple relation of Loewner theory to monotone probability theory. Certain Loewner equations can be interpreted as evolution equations for quantum processes.

-Mardi 22 novembre, 14h45: Hun Hee Lee, Seoul National University.
Similarity degree of Fourier algebras

In this talk we will focus on the Dixmier type of similarity question for Fourier algebras and their similarity degrees by Pisier. We will explain that for a locally compact group $G$, amongst a class which contains amenable and small invariant neighbourhood groups (especially discrete groups), that its Fourier algebra $A(G)$ satisfies a completely bounded version of Pisier's similarity property with similarity degree at most 2. Specifically, any completely bounded homomorphism $\pi:A(G)\to B(H)$ admits an invertible $S$ in $B(H)$ for which $\|S\|\|S^{-1}\|\leq \|\pi\|^2_{cb}$ and $S^{-1}\pi(\cdot)S$ extends to a $*$-representation of the C*-algebra $C_0(G)$.

-Mardi 29 novembre: Gilles Godefroy, Université Paris 6.
The complexity of the isomorphism class of some Banach spaces

### Octobre

-Mardi 4 octobre: François Netillard, UFC.
Plongements grossièrement Lipschitz entre espaces de James

Il est connu que $\ell_q$ ne se plonge pas grossièrement Lipschitz dans $\ell_p$ pour $q\neq p$ ($p, q \geq 1$). On essaie d’adapter les méthodes utilisées alors au cas des espaces de James.

-Mardi 11 octobre: relâche, Journées GDR, Toulouse.

-Mardi 18 octobre: Sergey Tikhonov , CRM, Barcelona.
Measures of smoothness and Fourier transforms

In this talk we discuss some recent results related to the quantitative Riemann-Lebesgue lemma on relationship between behavior of the Fourier transform at infinity and smoothness of a function.

-Mardi 25 octobre: Hubert Klaja, École Centrale de Lille.
Image numérique et calcul fonctionnel

Si $T$ est un opérateur linéaire borné, alors pour tout polynôme $p$, le spectre de $p(T)$ verifie $\sigma(p(T)) = p(\sigma(T))$. Ce n'est plus vrai si l'on remplace le spectre par l'image numérique. Dans cet exposé on discutera d'une nouvelle preuve d'un résultat de Drury qui permet de localiser l'image numérique de $p(T)$. C'est un travail en collaboration avec Javad Mashreghi et Thomas Ransford.

### Septembre

-Mardi 13 septembre: Marek Cúth, Université Charles, Prague.
Embedding of ℓ1 into Lipschitz-free Banach spaces and ℓ∞ into their duals

Given a metric space M, it is possible to construct a Banach space ℱ(M) in such a way that the Lipschitz structure of M corresponds to the linear structure of ℱ(M). This space ℱ(M) is sometimes called the "Lipschitz-free space over M". The study of Lipschitz-free Banach spaces became an active field of study. I will present our recent result with M. Johanis that ℓ∞ embeds isometrically into the dual of every infinite-dimensional Lipschitz-free Banach space and that it is often the case that a Lipschitz-free Banach space contains a 1-complemented subspace isometric to ℓ1. We do not know whether the later is true for every infinite-dimensional Lipschitz-free Banach space.

-Mardi 20 septembre: Safoura Jafar-Zadeh , UFC.
Isometric isomorphisms of the annihilator of $C_0(G)$ in $LUC(G)^*$

For a locally compact group $G$, let $C_b(G)$ be the space of all complex-valued, continuous and bounded functions on $G$ equipped with the sup-norm, and $LUC(G)$ be the subspace of $C_b(G)$ consisting of all functions $f$ such that the map $G\to C_b(G);x\mapsto l_xf$ is continuous, where $l_xf$ is the function defined by $l_xf(y)=f(xy)$, for each $y\in G.$ In this talk, I will show that if $G$ is a locally compact group and $H$ is a discrete group then whenever there exists a weak-star continuous isometric isomorphism between $C_0(G)^\perp$ (the annihilator of $C_0(G)$ in $LUC(G)^*$) and $C_0(H)^\perp$, then $G$ is isomorphic to $H$ as a topological group. Several related results will also be discussed.

-Mardi 27 septembre: Uwe Franz, UFC.
HUNT FORMULA FOR SUq(n) AND Uq(n)

Joint work with Anna Kula, Martin Lindsay, and Michael Skeide. We provide a Hunt type formula for the infinitesimal generators of Lévy process on the compact quantum groups SUq(N )and Uq(N ). In particular, we obtain a decomposition of such generators into a gaussian part and a "jump" type part, similar to the classical Hunt formula.

## Agenda

• ### Mardi 13 février 13:45-15:00 - Kangwei Li

Séminaire d’Analyse Fonctionnelle

• ### Mardi 13 mars 13:45-15:00 - Anna Skripka - University of New Mexico

Séminaire d’Analyse Fonctionnelle

• ### Mardi 27 mars 10:00-11:00 -

Séminaire d’Analyse Fonctionnelle