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# Programme de la rencontre

Le but de cette rencontre est de rassembler les membres du projet ainsi que quelques invités extérieurs. De longs moments sont prévus pour faciliter les échanges.

Le premier exposé commencera à 15h15 le lundi 007 novembre et la rencontre se terminera à midi le mercredi 09 novembre juin. au sweet-home à Cabourg

Programme détaillé :

Mardi 8

09:45-10h35
M. de la Salle

11h-11h30
S. Roy

14h-14h50
R. Xia

15h-15h30
C. Duquet

16h30-17h20
L. Cadilhac

- Benjamin Anderson-Sackaney Relative Amenability, Amenability, and Coamenability of Coideals Amenability is a deeply studied property of groups, with many interesting reformulations and connections to the operator algebraic aspects of groups. For example, the reduced $C^*$-algebra $C^*_r(G)$ of a discrete group has a unique tracial state if and only if there are no non-trivial amenable normal subgroups. This, among other related results, makes it apparent that the structure of the amenable subgroups of $G$ contains important information about $C^*_r(G)$. For a quantum group $\mathbb{G}$, an appropriate analogue of a subgroup is a coideal $N\subseteq L^\infty(\mathbb{G})$. We will present notions of relative amenability, amenability, and coamenability for coideals of discrete and compact quantum groups motivated by ``relativizations'' of amenability and coamenability of a subgroup of a group. We will discuss the known relationships between these formally distinct notions and their relevance to certain properties of the reduced $C^*$-algebras of discrete quantum groups.
- Léonard Cadilhac Noncommutative maximal inequality for ergodic averages on amenable groups I will present a joint work with Simeng Wang in which we prove a maximal inequality of weak type (1,1) for some sequences of ergodic averages on amenable groups. A motivation for this result is to obtain a Birkhoff-type pointwise convergence theorem in this setting. After introducing the subject, the talk will focus on the proof and its two distinct aspects : a harmonic analytic one relying on noncommutative Calderon-Zygmund theory and a geometric one where we construct particular tilings of amenable groups.
- Mikael de la Salle Stability, maximal inequalities and non-local games I will report on a few things that I have understood while studying the recent resolution of Connes embedding problem by Ji-Natarajan-Vidick-Yuen-Wright. I will in particular insist on two aspects that might be of interest to the audience : a connection with non-commutative maximal inequalities, and with stability for group representations.
- Charles Duquet Characterisation of unital positive dilatable Schur multipliers I will present new characterisation of dilatation on non-commutative $L_p$-spaces. For any unital, positive Schur multiplier $T_M$ on $B(\ell_2(\mathbb N))$, we have equivalences between $T_M$ is dilatable for a $p\in (1,\infty)$, TM is dilatable for all 1< $p$ < $\infty$, $T_M$ is absolutely dilatable and a particular writing for the matrix $M$.
- Javier Parcet Balanced Fourier truncations and $\mathrm{X}_p$ inequalities We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (motivated by its consequences in metric geometry) to discrete groups. As application, we establish new forms of metric $\mathrm{X}_p$ inequalities in group von Neumann algebras. Joint work with I. Cano-Marmol and J. Conde-Alonso.
- Sutanu Roy Braided free unitary quantum groups Braided quantum groups naturally arise in the study of semidirect products of quantum groups. Braided compact quantum groups are certain $C^*$-bialgebras in a (nontrivial) monoidal category of unital $C^*$-algebras equipped with an action of a quasitriangular quantum group. In this talk I shall present a braided version of the free unitary quantum groups. The contents of this talk is the part of a joint work with Suvrajit Bhattacharjee and Soumalya Joardar.
- Adam Skalski Classifying $C^*$-algebras with the K-theory: (Hecke) algebras of Coxeter groups will first discuss in very general terms how one can attempt to distinguish unital C*-algebras using their K-theoretic invariants. Then I will present a specific example of group C*-algebras of Coxeter groups and their deformations, showing both the power and limitations of K-theoretic techniques in a concrete context. Based on joint work with Sven Raum.
- Runlian Xia Fourier multipliers coming from group actions on tree-like structures In this talk, we introduce a generalised Cotlar identity and a geometric form of some $L_p$-bounded Fourier multipliers for groups acting on $\mathbb R$-trees. This class of groups includes free groups, amalgamated free products, HNN extensions, totally ordered groups and many others. The pioneering work in this direction is due to Mei and Ricard who proved $L_p$-boundedness of Hilbert transforms on free group von Neumann algebras using a Cotlar identity. This work is followed by the paper of Mei, Ricard and Xu focusing on more general Fourier multipliers.
- Zhendong Xu From the Littlewood-Paley-Stein inequality to the Burkholder-Gundy inequality 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 to a martingale which relates the Littlewood $G$-function with the matingale square function pointwise. This can be also be done in the vector valued and noncommutative cases.
- Kai Zeng Schatten properties of commutators Given a quantum tori $\mathbb T_\theta$, we can define the commutators of the multiplication operators by $x$ on $L_2(\mathbb T_\theta)$ with the Riesz transforms. In this talk, we will characterize their Schatten properties in terms of appatenance of $x$ to some Besov classes. Futhermore, we will extend this characterisation to the more general case where Riesz transforms replaced by an arbitrary Calderon-Zygmund operator. To date, these new results treat the quantum differentiability in the strictly noncommutative setting.