## 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). Pour le moment, les expos és sont virtuels (pour le lien, contactez la responsable).

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 8 septembre, 13:30 (en ligne): Jacek Krajczok, IMPAN, Warsaw
Type I locally compact quantum groups: coamenability and applications

We say that a locally compact quantum group is type I if its universal C* algebra (which is equal to $C^u_0(\hat{G})$) is type I. This class of quantum groups can be though of as an intermediate step between compact and general locally compact quantum groups; they are significantly more general than compact ones, but still have tractable representation theory. Similarly to the compact case, one can define "character-like" operators associated with suitable representations. I will discuss a result which states that coamenability of G is equivalent to a certain condition on spectra of these operators. If time permits, I will also discuss how one can use theory of type I locally compact quantum groups to show that the quantum space underlying the Toeplitz algebra does not admit a quantum group structure (joint work with Piotr Sołtan).

-Mardi 22 septembre, 13:45 (en ligne): Biswarup Das, Wroclaw University
Towards quantizing separate continuity: A quantum version of 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.

### Aout

-Mardi 25 aout, 13:30 (en ligne): Martijn Caspers, TU Delft, Analysis Group
Weak type estimates for multiple operator integrals and generalized absolute value functions

This talk is concerned with the following question. Let f be an n times differentiable function on the reals with bounded n-th derivative. Let $f_n$ be its n-th order divided difference function. For instance $f_1(s,t) = (f(s) - f(t))/(s-t)$. Is it true that the multiple operator integral $T_{f_n}$ maps $S_{p_1} \times \dots \times S_{p_n}$ to $S_{1, \infty}$ boundedly? Here $S_p$ is the Schatten non-commutative Lp-space and $S_{1, \infty}$ is the non-commutative weak L_1 space. In case n=1 the question boils down on whether the Schur multiplier with symbol $(f_1(s,t))_{s,t}$ is bounded from $S_1$ to $S_{1, \infty}$. We give a positive answer to a class of functions involving the function $a(t) = sign(t) t^n$. If n =1 we find a complete solution and the asnwer is affirmative. We give further details and definitions in the talk, including the theory of multiple operator integrals. This is joint work with Fedor Sukochev, Dima Zanin as well as Denis Potapov.

### Juin

-Mardi 2 juin, 13:30 (en ligne): Mario Klisse, TU Delft, Analysis Group
Graph product Khintchine inequalities and their applications to Hecke C*-algebras

A graph product of groups is a group theoretic construction that generalizes both free products and Cartesian products. It admits an operator algebraic conterpart, which interpolates between free products and tensor products. Both constructions preserve many properties of the underlying groups/algebras. Important examples of operator algebras that can be realized in terms of graph products are right-angled Hecke algebras, special cases of mixed q-Gaussian algebras as well as several group C*-algebras. In this talk we will discuss so called Khintchine inequalities for general C*-algebraic graph products. These are inequalities which estimate the operator norm of a reduced operator of a given length with the norm of certain Haagerup tensor products of column and row Hilbert spaces. Inequalities of this kind turn out to be very useful. We will demonstrate this in the case of (right-angled) Hecke C*-algebras, by investigating their ideal structures.

### Mai

-Mardi 5 mai, 14:00 (en ligne): Adam Skalski, IMPAN, Varsovie
Furstenberg boundary of a discrete quantum group (based on joint work with Mehrdad Kalantar, Paweł Kasprzak and Roland Vergnioux).

The notion of topological boundary actions has found recently striking applications in the study of operator algebras associated to discrete groups. We will discuss the analogue concept for discrete quantum groups, show that also here there always exists a maximal boundary action - the so-called Furstenberg boundary - and explain how its properties can be used to deduce certain features of the associated C*-algebras. We will focus on difficulties and differences arising from the quantum context and present an example related to free orthogonal groups.

-Mardi 19 mai, 13:30 (en ligne): Orr Shalit, Isreal Institure of Technology, Haifa
Assumed knowledge: Completely positive maps and C*-algebras

We introduce a framework for studying dilations of semigroups of completely positive maps on von Neumann algebras. The heart of our method is the systematic use of families of Hilbert C*-correspondences that behave nicely with respect to tensor products: these are product systems, subproduct systems and superproduct systems. Although we developed our tools with the goal of understanding the multi-parameter case, they also lead to new results even in the well studied one parameter case. In my talk I will give a broad outline and a taste of the dividends our work.
The talk is based on a recent joint work with Michael Skeide.

### Avril

-Mardi 21 avril, 14:00: Jorge Castillejos, IMPAN, Varsovie.
En ligne sur https://webconf.math.cnrs.fr/b/fra-wng-3ap

The Toms-Winter conjecture

The classification programme of C*-algebras seeks to classify all separable simple unital nuclear C*-algebras using K-theory and traces. After enjoying tremendous success during several years, some exotic examples were found and certain regularity properties emerged as necessary conditions in the classification programme. The Toms-Winter conjecture asserts that these regularity properties are all equivalent. In this talk, I will discuss the current state of the Toms-Winter conjecture and the classification program.

### Mars

-Mardi 3 mars: Alexandros Eskenazis, Institut de Mathématiques de Jussieu-Sorbonne Université et Trinity College Cambridge.

Discrete Littlewood-Paley-Stein theory and Pisier's inequality for superreflexive targets

In modern terminology, Enflo's conjecture (1978) asserts that a Banach space $X$ has Rademacher type $p$ if and only if $X$ satisfies a metric property called Enflo type $p$. Loosely speaking, the conjecture suggests that all $X$-valued functions on the Hamming cube satisfy a dimension independent $L_p$ Poincaré inequality if and only if the same inequality holds merely for linear functions. In his 1986 work, Pisier showed that Banach spaces of Rademacher type $p$ have Enflo type $q$ for every $q\lt p$ and proved the endpoint Enflo type $p$ inequality with an additional logarithmic factor in the dimension of the Hamming cube. In this talk, I shall present joint work in progress with A. Naor, in which we improve Pisier's bound for Banach spaces which admit an equivalent uniformly convex norm. The proof relies on (either new or recently proven) vector valued Littlewood-Paley-Stein theory on the Hamming cube.

### Janvier

-Mardi 14 janvier: Gilles Lancien, Besançon.
Théorie descriptive et espaces de Banach grossièrement universels

Il est connu depuis peu que la classe des espaces réflexifs et asymptotiquement $c_0$ est stable par plongements grossiers ou uniformes. On pourrait penser que c'est parce que cette classe est très restreinte. Ce n'est pas faux, car le premier exemple d'un tel espace a été construit en 1974 par Boris Tsirelson. Nous expliquerons cependant qu'un espace de Banach qui contient grossièrement (ou uniformément) tous les espaces de Banach séparables réflexifs et asymptotiquement $c_0$, contient grossièrement tous les espaces métriques séparables. Une partie de l'argument reposera sur l'étude de la complexité, au sens de la théorie descriptive, de cette classe.Travail en commun avec F. Baudier, P. Motakis et Th. Schlumprecht.

-Mardi 21 janvier: Luc Deléaval, Marne-la-Vallée.
Autour du théorème maximal de Hardy-Littlewood

### Décembre

-Mardi 10 décembre: Tony Prochazka, Besançon.
Plongements des espaces Lipschitz libres dans $\ell _1$

We show that, for a separable and complete metric space M, the Lipschitz-free space F(M) embeds linearly and almost-isometrically into $\ell_1$ if and only if M is a subset of an R-tree with length measure 0. Moreover, it embeds isometrically if and only if the length measure of the closure of the set of branching points of M (taken in any minimal R-tree that contains M) is negligible. We also prove that, for any subset M of an R-tree, every extreme point of the unit ball of F(M) is an element of the form (δ(x)−δ(y))/d(x,y) for x≠y∈M. Joint work with R. Aliaga and C. Petitjean.

-Jeudi 6 décembre à 13h45: Panu Lahti, University of Augsburg.
A new Federer-type characterization of sets of finite perimeter

A new Federer-type characterization of sets of finite perimeter}{Federer’s characterization, which is a key result in the theory of functions of bounded variation (BV functions), states that a set is of finite perimeter (i.e. the set's indicator function is a BV function) if and only if the n−1-dimensional Hausdorff measure of the set's measure-theoretic boundary is finite. The measure-theoretic boundary consists of those points where both the set and its complement have positive upper density. I discuss recent work in which I show that the characterization remains true if the measure-theoretic boundary is replaced by a smaller boundary consisting of those points where the lower densities of both the set and its complement are at least a given positive constant.

-Mardi 3 décembre: Cristina Benea, Nantes.
Des inégalités de Brascamp-Lieb pour des intégrales singulières multilinéaires

Des inégalités de Brascamp-Lieb pour des intégrales singulières multilinéaires}{Les inégalités du type Brascamp-Lieb ou Loomis-Whitney permettent d’estimer en $\mathbb{R}^d$ le produit de $d$ fonctions, chacune dependant que de $d-1$ variables. Un résultat similaire existe pour des intégrales multilinéaires singulières, dont le symbol en fréquence est singulier le long d’un espace de dimension supérieure. Les techniques sont basées sur des estimations dans des espaces $L^p$ mixtes. Travail en commun avec C. Muscalu.

### Novembre

-Mardi 5 novembre à 13h30: Séance du groupe de travail

-18-22 novembre: Ecole d'hiver: Multipliers in non-commutative analysis.
Page web de l'école

-25-26 novembre: Journées de jeunes analystes non-commutatifs
Programme

### Octobre

-Jeudi 3 octobre à 15h00: Pierre Fima, Paris.

-Mardi 8 octobre: Relâche (Journées du GDR AFHA)

-Mardi 15 octobre: Loris Arnold, LMB.
Calcul fonctionnel des opérateurs half-plane type et $\gamma$-bornitude

-Mercredi 23 octobre à 09h30: Gilles Godefroy, Paris.

-Mardi 29 octobre: Relâche (Vacances)

### Septembre

-Mardi 10 septembre: Malte Gerhold, Greifswald.
Dilations and matrix ranges of free operators

## Agenda

• ### Mardi 22 septembre 13:45-15:00 - Biswarup Das - Wroclaw University

Towards quantizing separate continuity : A quantum version of Ellis joint continuity theorem

Résumé : 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.

• ### Mercredi 19 mai 2021 13:45-15:00 - Victor Nistor - Metz

Séminaire d’Analyse Fonctionnelle