## Séminaire d’analyse fonctionnelle 2021-2022

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• 21/09/2021 Yulia Kuznetsova, LmB, Multiplicateurs engendrés par le laplacien sur les groupes ax+b’
Soit L l’opérateur de Laplace et f une fonction bornée. La question centrale de cet exposé est l’estimation des normes de l’opérateur $f(L)$ de $L^p$ dans $L^q$, une question qui dans le cas classique se traduit en langage des multiplicateurs de Fourier. Le problème a attiré beaucoup d’attention, d’abord sur l’espace $\mathbb R^n$, puis sur des groupes de Lie ou sur des variétés riemanniennes. Je vais parler du contexte de ce sujet et de différences qu’on observe sur différents groupes. Dans le travail présenté, nous nous avons occupés le plus de fonctions "oscillantes" de type $f(x)\exp(itx)$, avec $f \in C_0(R)$. D. Müller et C. Thiele ont montré en 2007 que sur les groupes ax+b’ de dimension n, la norme $L^1 \to L^1$ de ces opérateurs est bornée par $C(1+t)$ tandis que la norme $L^1 \to L^\infty$ et majorée par une constante. Cet ordre est essentiellement plus grand que dans $\mathbb R^n$. Nos résultats récents montrent que ces estimations sont optimales.
C’est un travail commun avec Rauan Akylzhanov, Michael Ruzhansky et Haonan Zhang.
• 5/10/2021 Yujia ZHAI, Université de Nantes, Study of biparameter BMO and application to directional Hilbert transforms
We will study the properties of functions in the biparameter BMO space and illustrate the difference from its classical one-parameter variant. We will then focus on the behavior of the bi-parameter BMO space under the action of a rotation and show that this BMO space is not preserved by a rotation. By imposing additional regularity, we will prove interpolation inequalities which imply a boundedness property of directional Hilbert transforms. This is joint work with Frédéric Bernicot.
• 12/10/2021 Guillaume Grelier, Universidad de Murcia, Extremal structure in ultrapowers of Banach spaces
Given a bounded convex subset $C$ of a Banach space $X$ and a free ultrafilter $\mathcal U$, we study which points $(x_i)_\mathcal U$ are extreme points of the ultrapower $C_\mathcal U$ in $X_\mathcal U$. In general, we obtain that when $(x_i)$ is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then $(x_i)_\mathcal U$ is an extreme point (respectively denting point, strongly exposed point) of $C_\mathcal U$. We also show that extreme points and strongly extreme points of $C_\mathcal U$ coincide provided $\mathcal U$ is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of $C_\mathcal U$ in the case that $C$ is a super weakly compact or uniformly convex set. Joint work with L. C. García-Lirola and A. Rueda Zoca.
• 19/10/2021 Eric Ricard, Université de Caen, Inégalités pour des anticommutateurs
Je vais présenter plusieurs inégalités qui permettent de terminer l’étude du module de continuité des applications de Mazur entre Lp-non commutatifs pour des indices plus petits que 1. Elles sont intimement liées aux inégalités de Khintchine non commutatives et permettent d’en donner une preuve simple.
• 19/10/2021 Janusz Wysoczański, Uniwersytet Wrocławski, Joint monotone and boolean numerical and spectral radii of d-tuples of operators
• 20/10/2021 Anna Wysoczańska-Kula, Uniwersytet Wrocławski, Does Levy-Khinchine decomposition exists in the noncommutative framework ?
Known since 1930ies, the Lévy-Khintchine formula provides a classification of Lévy processes on $\mathbb R^n$ in terms of their generators. It shows how the generators of Lévy processes are combinations of continuous (or Gaussian) parts and jump parts. In my talk I will discuss the problem of the existence of an analogous decomposition for Lévy processes’ generators on $*$-bialgebras and compact quantum groups, and comment on recent developments in this area.
• 26/10/2021 Matias Raja, Universidad de Murcia, Uniformly convex functions and applications
The notion of uniform convexity for functions was introduced by Levitin and Polyak in the 60′s, that localises in a certain way some good properties of functions defined on uniformly convex spaces. Since then, the properties of uniformly convex functions have been studied by several authors, notably Vladimirov, Nesterov, Chekanov, Zaˇlinescu, Borwein, Vanderwerff, Guirao and Hájek.
Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key for further constructions.
We will provide two applications of our results and techniques in this talk. Firstly, we will show how Enflo’s uniformly convex renorming of super-reflexive spaces can be retrieved in a quite natural way.
The second application is related to the notion of super weakly compactness. After a brief overview of the state of the art on this topic, we will discuss several fashions to quantify the non-super weakly compactness of a subset of a Banach space and then we will prove that they all are equivalent.
This is a joint work with G. Grelier recently published in JMAA.
• 9/11/2021 Louis Labuschagne, North-West University, Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras
We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a noncommutative Hoffman-Rossi theorem’ giving the existence of weak* continuous noncommutative representing measures’ for so-called $D$-characters. These results may also be viewed as module’ Hahn-Banach extension theorems for weak* continuous characters’ into possibly noninjective von Neumann algebras. In closing we introduce the notion of `noncommutative Jensen measures’, and show that as in the classical case representing measures of logmodular algebras are Jensen measures.
• 23/11/2021 Bernhard Haak, Université de Bordeaux, Observabilité exacte pour des groupes continues et discrets
Dans cet exposé nous nous intéressons à l’observabilité exacte de groupes bornés sur un espace de Hilbert ou de Banach, soit dans un cadre discret, soit dans un cadre continu. L’analyse commence avec un analogue discret du "critère de Hautus" qui caractérise l’observabilité d’un groupe unitaire sur un espace de Hilbert. Il s’avère que l’analyse de Fourier utilisée ne sert qu’à "engendrer des carrés" dans un sens que l’on précisera ; vu que cela peut se faire également avec des sommes aléatoires (par exemple Gaussiennes ou Rademacher), la porte s’ouvre à une généralisation Banachique qui nous donne une caractérisation complète de l’observabilité pour des groupes bornées (discrets et continues).
• 29/11/2021 Yemon Choi, Lancaster University, Operator space tensor products, and cocycles on Fourier algebras
When studying Fourier algebras of locally compact groups, it is commonly accepted that we need to use the projective tensor product of operator spaces. On the other hand, the study of derivations on Fourier algebras has revealed a potentially rich area for investigation, but such derivations can never be completely bounded,Retour ligne automatique
and hence there would seem to be no reason why they should interact well with operator space tensor products.
In this talk I will explain more about these two themes, and why they presented an obstacle until recently when trying to construct non-trivial 2-cocycles on Fourier algebras. I will then outline how the obstacle can be overcome by making use of extra structure for certain derivations, together with a "twisted inclusion" result for operator space tensor products.
• 30/11/2021 Trond A. Abrahamsen, University of Agder, Almost square Banach spaces and relatives
• 7/12/2021 Thomas Schlumprecht, Texas A&M, Stochastic embeddings of Metric Spaces into trees and applications to embeddings of Lamplighter and Wasserstein spaces into $L_1$
Understanding how a group or a graph, viewed as a geometric object, can be faithfully embedded into certain Banach spaces is a fundamental topic with applications to geometric group theory and theoretical computer science.
In this joint work with Florent Baudier, Pavlos Motakis and Andras Zsak we observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the $L_1$-distortion of lamplighter metrics Wasserstein spaces follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on a $n$-point metric space embeds bi-Lipschitzly into $L_1$ with distortion $O(\log n)$.
In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved.