Propriétés de relèvement pour les $C^*$-algèbres : du local au global ?

• Gilles PISIER (Sorbonne Université - Texas A&M University)

Room: Centre de conférences Marilyn et James Simons The main problem we will consider is whether the local lifting property (LLP) of a $C^*$-algebra implies the (global) lifting property (LP). Kirchberg showed that this holds if the Connes embedding problem has a positive solution, but it might hold even if its solution is negative. We will present several new characterizations of the lifting property for a $C^*$-algebra $A$ in terms of the maximal tensor product of A with the (full) $C^*$-algebra of the free group ${\mathbb F}_{\infty}$. We will recall our recent construction of a non-exact $C^*$-algebra with both LLP and WEP. This prompted us to try to prove that LLP implies LP for a WEP $C^*$-algebra. While our investigation is not conclusive we obtain a fairly simple condition in terms of tensor products that is equivalent to the validity of the latter implication.

Non-commutative Pointwise Ergodic Theorem for Actions of Amenable Groups

• Léonard CADILHAC (Sorbonne Université)

Room: Centre de conférences Marilyn et James Simons Birkhoff's famous theorem asserts the pointwise convergence of ergodic averages associated with a measure preserving transformation of a measure space. In this talk, I will discuss generalizations of this theorem in two directions: the transformation will be replaced by the action of an amenable group, and the measure space by a von Neumann algebra equipped with a trace. A central role will be played by the notion of non-commutative maximal function, which extends for our purposes the notion of supremum to families of operators. The talk is based on joint work with Simeng Wang.

Tracial and G-invariant States on Quantum Groups

• Benjamin ANDERSON-SACKANEY (Université de Caen-Normandie)

Room: Centre de conférences Marilyn et James Simons For a discrete group G, the tracial states on its reduced group $C^*$-algebra $C^∗_r (G)$ are exactly the conjugation invariant states. This makes the traces on $C^∗_r (G)$ amenable to group dynamical techniques. In the setting of a discrete quantum group ${\mathbb G}$, there is a quantum analog of the conjugation action of $G$ on $C^∗_r (G)$. Recent work of Kalantar, Kasprzak, Skalski, and Vergnioux shows that ${\mathbb G}$-invariant states on the quantum group reduced $C^*$-algebra $C_r( \widehat{\mathbb G})$ are in one-to-one correspondence with certain KMS-states, exhibiting a disparity between tracial states and ${\mathbb G}$-invariant states unless ${\mathbb G}$ is unimodular. We will show there is still enough of a connection between traceability and G-invariance to say interesting things about the tracial states of $C_r( \widehat{\mathbb G})$.

From the Littlewood-Paley-Stein Inequality to the Burkholder-Gundy Inequality

• Xu ZHENDONG (Université de Bourgogne-Franche-Comté)

Room: Centre de conférences Marilyn et James Simons We solve a question asked by Xu about the order of optimal constants in the Littlewood-Paley-Stein inequality. This relies on a construction of a special diffusion semi-group associated with a martingale which relates the Littlewood G-function with the martingale square function pointwise. This can also be done in vector-valued and noncommutative cases.

Schoenberg Correspondence and Semigroup of k-(super)positive Operators

• Purbayan CHAKRABORTY (Université de Bourgogne-Franche-Comté)

Room: Centre de conférences Marilyn et James Simons The famous Lindblad, Kossakowski, Gorini, and Sudarshan's (LKGS) theorem characterizes the generator of a semigroup of completely positive maps. Motivated by this result we study the characterization of the generators of other positive maps e.g. k-positive and k-super positive maps. We prove a Schoenberg-type correspondence for a general non-unital semigroup of operators and apply this result to different cones of positive operators in $L(M_n, M_n)$ which are interesting for quantum information. As a corollary of our result, we re-establish the LKGS's theorem.

Free Wreath Products as Fundamental Graph C*-algebras

• Arthur TROUPEL (Université Paris-Cité)

Room: Centre de conférences Marilyn et James Simons The free wreath product of a compact quantum group by the quantum permutation group S+N has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such groups is well-known, but some results about their operator algebras were still open, for example, the Haagerup property, K-amenability, or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental $C^*$-algebras of certain graphs of $C^*$-algebras, and we deduce these properties from these constructions

Fonctions sphéroïdales et triplets spectraux

• Alain CONNES (IHES)

Room: Centre de conférences Marilyn et James Simons J'expliquerai la construction à partir de l'opérateur différentiel W du second ordre qui apparait par séparation des variables dans le Laplacien d'un ellipsoide, et des fonctions propres de W, de triplets spectraux reproduisant les comportement infrarouge et ultraviolets des zeros de zeta. Ce sont des résultats récents en collaboration avec Katia Consani et Henri Moscovici.

Le problème du bicentralisateur de Connes

• Amine MARRAKCHI (ENS Lyon)

Room: Centre de conférences Marilyn et James Simons À la fin des années 1970, Connes formula une conjecture portant sur les facteurs de type III1 connue sous le nom de "problème du bicentralisateur" et montra qu'une solution positive à ce problème permettrait de prouver l'unicité du facteur moyennable de type III1. Cette conjecture de Connes fut résolue dans le cas des facteurs moyennables par Haagerup en 1984. Mais elle reste encore largement ouverte aujourd'hui dans le cas non moyennable. Dans cet exposé, j'expliquerai le problème du bicentralisateur, son histoire, ses motivations et je présenterai quelques résultats nouveaux obtenus ces dernières années.

The Godbillon-Vey Invariant in $KK$-theory with Real Coefficients

• Sarah AZZALI (Università di Bari)

Room: Centre de conférences Marilyn et James Simons The Godbillon-Vey invariant is a 3-degree cohomology class associated with a foliation of codimension 1 of a closed manifold M. This classical invariant has been shown to be closely related to measure theory and dynamics of the foliation. It also plays a crucial role in index theory, as proved by Alain Connes. We construct a natural class in bivariant $KK$-theory with real coefficients representing the Godbillon-Vey invariant. We shall explain these construction, see how the Godbillon-Vey invariant deals with a (densely defined) infinite trace, and the relation to the index theorem for measured foliations. This is joint work with Paolo Antonini (Unisalento) and Georges Skandalis (Université Paris Cité).

Quantum Automorphism Groups of Some Classes of Graphs

• Paul MEUNIER (KU Leuven)

Room: Centre de conférences Marilyn et James Simons Simple combinatorial objects like finite graphs can reveal hidden endemically quantum behaviors. In the same way that the symmetries of a graph are encoded in its automorphism group, its quantum symmetries are encoded in its quantum automorphism group. Surprisingly, the latter can be very different from the former, and a graph can have much more symmetries in the quantum world than it has in the classical world. In this talk, after introducing the topic, I will present some of these examples as well as recent computations of quantum automorphism groups for some classes of graphs.

Schatten Properties of Commutators

• Kai ZENG (Université de Bourgogne-Franche-Comté)

Room: Centre de conférences Marilyn et James Simons Given a quantum tori $\mathbb{T}_{\theta}^d$, we can define the Riesz transforms $\mathfrak{R}_j$ on the quantum tori and the commutator $đx_i$ := [$\mathfrak{R}_i,M_x$], where $M_x$ is the operator on $L^2(\mathbb{T}_{\theta}^d)$ of pointwise multiplication by $x \in L^\infty (\mathbb{T}_{\theta}^d)$. In this talk, we will characterize the Schatten properties of the commutator [$\mathfrak{R}_i,M_x$] by showing that $x \in B_{p,q}^{\alpha} ({\mathbb T}_{\theta}^d)$, where $B_{p,q}^{\alpha} ({\mathbb T}_{\theta}^d)$ is the Besov space on quantum tori. Futhermore, we will extend this characterisation to the more general case where $\mathfrak{R}_j$ replaced by an arbitrary Calderon-Zygmund operator. To date, these new results treat the quantum differentiability in the strictly noncommutative setting.

Paires de Hecke et K-théorie

• Clément DELL’AIERA (ENS Lyon)

Room: Centre de conférences Marilyn et James Simons Introduites par Shimura en théorie des nombres dans les années 50, les paires de Hecke sont des inclusions de sous-groupes qui sont presque normales : leurs conjugués sont tous commensurables. À une paire de Hecke est associée un groupe localement compact totalement discontinu : sa complétion de Schlichting. Dans cet exposé, nous relions l’existence de sous-groupes presque normaux à la géométrie à grande échelle des complétions de Schlichting. Cela permet de prouver divers résultats de stabilité pour les conjectures de Baum-Connes et de Novikov, et de les valider sur de nouveaux exemples. Si le temps le permet, nous présenterons un travail en cours sur l’application de ces techniques au calcul de K-théorie de C*-algèbres de Hecke.

Schatten Properties for Noncommutative Martingale Paraproduct

• Zhenguo WEI (Université de Bourgogne-Franche-Comté)

Room: Centre de conférences Marilyn et James Simons As is well-known, the martingale paraproducts are Hankel-type operators. In this talk, I will present some Schatten class memberships of the d-adic martingale paraproducts in the semi-commutative setting. Then I will use the transference method to give a characterization of the Sp-norms of the martingale paraproducts for some particular noncommutative martingales