Vous trouverez ci-dessous le planning du séminaire d’Analyse
Fonctionnelle pour l’année 2020-2021. Le programme de l’année en cours se trouve ici.

Juin

• Mardi 8 juin : Haonan Zhang, IST Austria
  (en ligne)
  Curvature-dimension conditions for symmetric quantum Markov semigroups
  

The curvature-dimension condition consists of the lower Ricci curvature bound and upper dimension bound of the Riemannian manifold, which has a number of geometric consequences and is very helplful in proving many functional inequalities. In this talk I will speak about two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras and depolarizing semigroups. Joint work with Melchior Wirth (IST Austria).

Mai

• Mardi 11 mai : Fabio Cipriani, Politecnico di Milano
  (en ligne)
  On a noncommutative Sierpiński gasket

We illustrate the construction of a C*-algebra A that can be genuinely interpreed as a quantization of the classical Sierpiński gasket, the most studied instance of a self-similar fractal space. We further describe the discrete and continuous spectrum of A, the structure of the traces on A as well as the construction of a Dirichlet form E and of a spectral triple (A,D,H).

• Mardi 25 mai : David Kyed, University of Southern Denmark, Odense
  (en ligne)
  The Podleś spheres converge to the sphere

The Podleś spheres, which are q-deformed analogues of the 2-sphere, are by now among the most classical objects in non-commutative geometry, but only recently their structure as non-commutative Riemannian manifolds has begun to unravel. In my talk, I will first provide an introduction to Rieffel’s notion of compact quantum metric spaces and his non-commutative counterpart to the Gromov-Hausdorff distance, and then present some recent progress within this field which shows that the quantised 2-spheres actually converge (in the quantum Gromov-Hausdorff distance) to the classical round 2-sphere as the deformation parameter q tends to 1. The talk is based on joint works with Konrad Aguilar and Jens Kaad.

Avril

• Mardi 27 avril : Loris Arnold, LMB
  

Mars

• Mardi 9 mars : Ryosuke Sato, Nagoya University
  (en ligne)
  Markov dynamics on unitary duals of compact quantum groups

In this talk, we will discuss Markov semigroups on unitary duals (i.e., the set of all irreducible representations) of compact quantum groups. First, we will construct quantum Markov semigroups on the group von Neumann algebra of compact quantum group based on its Hopf-algebra structure and characters of the compact quantum group. Then we will show the dynamics preserve the center of the group von Neumann algebra, and it gives the dynamics on the unitary dual. Moreover, the dynamics have generators, and we can describe it explicitly by the representation theory. In particular, we will deal with the case of quantum unitary groups.

• Mardi 16 mars : Gilles Lancien, LMB
  Plongements non linéaires dans les duaux séparables

Le théorème d’Aharoni (1974) assure que tout espace métrique séparable se plonge de façon bi-Lipschitz dans $c_0$. C’est une question ouverte importante de savoir si tout Banach contenant une copie bi-Lipschitz de $c_0$ contient un sous-espace linéairement isomorphe à $c_0$. Dans cet exposé nous considérerons des questions similaires en relation avec la notion plus faible de plongement grossier. Dans un papier publié en 2007, un grand pas a été fait par N. Kalton, qui a prouvé qu’un Banach contenant grossièrement $c_0$ ne peut pas être réflexif. Cependant, on ignore encore si un tel espace peut être un dual separable. Dans cet exposé, nous discuterons de certains aspects de cette question. L’argument de Kalton est basé sur l’utilisation de graphes métriques particuliers, dits ``entrelacés’’. Nous donnerons des résultats sur les duaux contenant de façon équi-Lipschitz ou équi-grossière ces graphes, en relation avec leur indice de Szlenk et nous prouverons leur optimalité.
Travail en commun avec B. de Mendonça Braga, C. Petitjean and A. Procházka.

• Mardi 30 mars : Hua Wang, LMB
  Exemples de biproduits croisés ayant propriété (RD)
  

Je vais d’abord parler rapidement la propriété (RD) pour les groupes
quantiques discrets de type Kac et dire quelques mots comme motivation. Puis
je vais présenter un critère pour voir si les biproduits croisés possèdent
cette propriété. Enfin, je vais introduire une procédure pour construire
explicitement exemples de biproduits croisés ayant ou sans cette propriété,
et parler la limitation de cette procédure.

Février

• Mardi 23 février à 16h00 : Michael Brannan, Texas A&M University
  (en ligne)
  Complete logarithmic Sobolev inequalities and non-commutative Ricci curvature

I will give a brief introduction to the study of log-Sobolev type inequalities (LSI’s) for quantum Markov semigroups and some of their applications. In the context of classical heat semigroups on compact Riemannian manifolds, the famous Bakry-Emery theorem provides a beautiful connection between the geometry of the underlying manifold and the LSI, showing that a positive lower bound on the Ricci curvature implies an LSI for the heat semigroup. I will discuss an information-theoretic approach to obtain modified log-Sobolev inequalities based on non-positive non-commutative Ricci curvature lower bounds previously developed by Carlen and Maas. Using these tools, we are able to find new examples of quantum Markov semigroups satisfying a completely bounded version of the modified LSI, including heat semigroups on free quantum groups. This talk is based on joint work with Li Gao (TUM) and Marius Junge (UIUC).

Janvier

• Mardi 12 janvier à 13h45 : Marek Bożejko, University of Wrocław
  Remarks on Generalized Gaussian processes and positive
  definite functions on some Coxeter groups

In my talk I will present the following topics :
1. Strong connections between generalized Gaussian processes and some class of positive definite functions on permutations group.
2. Type B Fock spaces and new Gaussian processes of type B , relations with q-Meixner-Pollaczek polynomials and Meixner probability measures like 1/cosh.
3. Thoma repsentation of central positive definite functions on Coxeter groups of type A and B and new classes of generalized Gaussian processes.
4. Open problems.

Novembre

• Mardi 3 novembre à 13h30 : Tony Prochazka, LMB
  Compact reduction in Lipschitz-free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are : ℱ(X) is weakly sequentially complete for every superreflexive Banach space X, and ℱ(M) has the Schur property and the approximation property for every scattered complete metric space M. This is a joint work with R. Aliaga, C. Noûs and C. Petitjean.

Octobre

• Mardi 6 octobre, 13:45 : Jean-Christophe Bourin, LMB
  Une décomposition pour les matrices partitionnées

On établit "un théorème de Pythagore" pour les matrices partitionnées qui entraîne de nombreuses inégalités.

• Mardi 20 octobre, 13:45 (en ligne) : Jonas Wahl, Hausdorff Center for Mathematics, Bonn
  Markov dynamics on branching graphs of diagram algebras

Thoma’s famous theorem on the classification of characters on
the infinite symmetric group has been very influential in different
areas of mathematics such as combinatorics and probability theory. In
this talk, we explain versions of Thoma’s theorem for different diagram
algebras arising out of subfactor theory and Banica and Speicher’s
theory of easy quantum groups. As Thoma’s classical theorem, these
results can be formulated in a probabilistic language and we find
interesting new connections to random lattice paths and random walks on
trees.

Septembre

• 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 $C^u_0(\hatG)$) 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.

• Mardi 29 septembre, 13:45, salle 316B : Uwe Franz, LMB
  Quelques applications de la cohomologie de Hochschild aux probabilités non commutatives

Les semi-groupes de convolution d’états sur un espace de probabilités non commutatif sont en général caractérisés par leur dérivée en t=0, qu’en appelle leur fonctionnelle génératrice. Une telle fonctionnelle $\psi$ peut être complétée en un triplet $(\rho,\eta,\psi)$, dont les composantes sont caractérisées par des relations cohomologiques. Dans mon expose je vais rappeler la cohomologie de Hochschild ainsi que cette construction. Ensuite nous allons regarder quelques applications du premier et deuxième groupes de cohomologie à l’étude des fonctionnelles génératrices.