Title: Group actions on Lp spaces and non-commutative Lp spaces

Abstract: I will discuss group actions by isometries on Banach spaces, and mainly their rigidity aspects such as generalizations of Kazhdan's property (T). I will focus principally on Lp spaces and non-commutative Lp spaces, trying to insist on the differences. There will be a few results (joint with Amine Marrakchi) and many questions.

Slides Video

Title: Martingale inequalities and Fourier multipliers

Abstract: Probabilistic methods play an important role in the study of boundedness properties of wide classes of Fourier multipliers. In particular, as evidenced in many papers, a martingale approach can be applied efficiently in the investigation of various sharp estimates for first- and second-order Riesz transforms, as well as for the Beurling-Ahlfors operator, an important object in the theory of quasiconformal mappings. The purpose of the talk is to discuss noncommutative analogues of the above interplay. We will show how certain martingale Lp-inequalities lead to the corresponding estimates for Fourier multipliers on a class of group von Neumann algebras. The talk will be based on joint works with Tomasz Galazka, Yong Jiao, Lian Wu and Yahui Zuo.

Video slides

Title: Decomposable Operators, noncommutative Fourier and Schur multipliers

Abstract: A theorem by Haagerup tells that a completely bounded mapping T from a C-star-algebra A into B(H) can be put into the off-diagonal corner of a 2x2 matrix amplification from M_2(A) to M_2(B(H)), which is moreover completely positive. Then T is called decomposable. From there, one can deduce further nice properties of cb mappings related to cpos mappings. Also the case of T mapping between noncommutative Lp-spaces has been studied, e.g. by Pisier, Junge and Ruan. Here, complete boundedness and decomposability fall apart. We study decomposability for Schur and noncommutative Fourier multipliers on discrete groups. Finally, we indicate what can and can't be said for Fourier multipliers on locally compact groups. This is joint work with Cédric Arhancet (Albi, France).

Video Slides

Title: Maximal inequality for averages on the lamplighter group

Abstract: I will present some ongoing joint work with S. Wang where we study ergodic properties of actions of amenable groups on noncommutative measure spaces. This builds on a recent article of G. Hong, B. Liao and S. Wang that establishes ergodic theorems of almost uniform convergence for groups satisfying a doubling condition. Following the theory of actions on classical measure spaces, a key ingredient to their approach is a maximal inequality for Følner averages on the group which is at the origin of the doubling hypothesis. We exhibit new examples of groups for which this inequality holds by combining a variational principle introduced to the noncommutative setting by G. Hong and B. Xu and Følner tilings.

Video Slides

Title: Curvature-dimension conditions for symmetric quantum Markov semigroups

Abstract: 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 helpful in proving many functional inequalities. In this talk, I will first review several notions around lower Ricci curvature bounds in the noncommutative setting and present our work on complete gradient estimates. Then 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).

Slides Video

Title: Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

Abstract: The classical Calderón-Zygmund decomposition is a fundamental tool that helps one study endpoint estimates for singular integral operators near L1. In this talk, we shall study semicommutative extensions of it both in the doubling and the nondoubling context. The novelties of our approach when compared to previous results involve the smoothness of the Calderón-Zygmund kernels that we can treat and the family of measures on the underlying space. We shall discuss these aspects in detail. This is joint work with L. Cadilhac and J. Parcet.

Video Slides

Title: Spectral multipliers for sub-Laplacians: recent developments and open problems

Abstract: I will present some old and new results about the $L^p$ functional calculus for sub-Laplacians $L$ on Lie groups and more general sub-Riemannian manifolds. It has been known for a long time that, under fairly general assumptions on the sub-Laplacian and the underlying geometry, an operator of the form $F(L)$ is bounded on $L^p$ ($1<p<\infty$) whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of sufficiently large order. The problem of determining the minimal smoothness assumptions, however, remains widely open and will be the focus of our discussion.

Sildes Video

Title: Schatten properties of quantum derivatives on quantum tori

Abstract: The core ingredients of the quantized calculus, introduced by A. Connes, are a separable Hilbert space $H$, a unitary self-adjoint operator $F$ on $H$ and a $C^*$-algebra $\mathcal{A}$ represented on $H$ such that for all $a \in \mathcal{A}$ the commutator $[F,a]$ is a compact operator on $H$. Then the quantized differential of $a \in \mathcal{A}$ is defined to be the operator $\mathbf{d} a = i[F,a]$. I will talk about the characterizations of the Schatten properties of quantum derivatives on quantum tori $\mathbb{T}_\theta ^d$ by Sobolev or Besov spaces.

Slides Video