Séminaire

Takahiro Hasebe (Hokkaido University) : Lifting operators on a Hilbert space nicely to the tensor product or free product Hilbert space Abstract: The basic five notions of independences in noncommutative probability (tensor, free, boolean, monotone, antimonotone) have canonical operator models on Hilbert spaces. All these models are based on constructing a "good composite Hilbert space" (mostly, the tensor product or free product Hilbert spaces) and a "good lifting" of operators to the composite space. In classical probability, this corresponds to the well known construction of independent random variables on the product of probability spaces. In this talk, we axiomatize "good liftings" and classify them. The subsequent talk, given by Malte Gerhold, discusses applications to finding new noncommutative independences for random vectors. This talk is based on a joint work with Malte Gerhold and Michael Ulrich.

Malte Gerhold (Université de la Sarre) : Classification of two-faced independences Abstract: Two-faced independences are independence relations for pairs of noncommutative random variables. An important example is bifree independence, which models the relation between left and right regular representation of the free group in the canonical tracial state. Around 2000, in works of Speicher, Ben Ghorbal & Schürmann, and Muraki, a complete classification of "single-faced" independences was obtained: the only independences in this case are boolean, tensor, free, monotone and anti-monotone independence. I report on the current status of the classification program for two (or multi-faced) independences and we will focus on the interplay between the lifting approach (discussed in the preceding talk, given by Takahiro Hasebe) and the combinatorial "cumulant" approach to multi-faced independences. Based on joint work with Takahiro Hasebe, Michaël Ulrich, and Philipp Varšo.

Oleg Szehr (Swiss AI Lab IDSIA): New insights on the asymptotic behavior of Jacobi polynomials and applications Abstract: Although a recurrent tool in applied asymptotic analysis, the asymptotic behavior of Jacobi polynomials with varying parameters is frequently good for a surprise. We introduce a new, simple and robust approach to this topic and demonstrate that certain "established" formulas in the literature yield inaccurate results. Using our method we identify previously unknown forms of asymptotic behavior and we investigate how this affects applications of Jacobi polynomial asymptotics: 1) We show how the Fourier coefficients of powers of Blaschke factors can be written in terms of Jacobi polynomials and we discuss their asymptotic behavior. 2) We discuss the asymptotics of quantum random walks on 1D spin systems using Jacobi polynomials, where we observe a new form of asymptotic behavior. This talk is based on: https://www.sciencedirect.com/science/article/pii/S0021904522000041

Hao Zhang : Schatten class membership of commutators Abstract: Commutators are well-known as generalization of Hankel type operators. In this talk, we are concerned about the Schatten class membership of commutators involving nondegenerate singular integral operators. We describe their Schatten class membership in terms of Besov spaces. Our approach is based on the complex median method, which is an extension of Lerner’s median method and is applicable to complex-valued functions.

Alexander Volberg : A_2 conjecture fails for matrix weights because 3/2 >1. Abstract: How to estimate the norm of the Hilbert transform (or other singular operators) in space [katex]L^2(w)[/katex] via weight [katex]w[/katex] This classical question appeared from probability (i.e. from the regularity theory of stationary stochastic processes). Initial answer was given by works of Hunt—Muckenhoupt—Wheeden and Helson—Szeg\”o in the 1970’s. But the sharp estimates are from 2000’s due to Petermichl, Volberg, Hytönen. The answer (obtained first for Ahlfors—Beurling transform, then for the Hilbert transform, and then for all Calder\’on—Zygmund operators) is that the bound is linear in [katex] A_2[/katex] Hunt—Muckenhoupt—Wheeden characteristic of weight [katex]w[/katex] . This was a solution of the so-called [katex] A_2[/katex] conjecture. But what if the stationary process in question is a vector one (as Wiener and Masani asked)? Then the weight [katex] W[/katex] is a matrix weight. It was a long standing problem to prove that the same linear estimate holds. However, recently we, Domelevo—Petermichl—Treil—Volberg, showed that this is not the case, [katex]A_2[/katex] conjecture fails, and the right sharp estimate is [katex] [W]_{A_2}^{3/2}[/katex] .

Amaury Freslon (Laboratoire de Mathématiques d’Orsay) : A la recherche du mouvement brownien quantique Résumé : La théorie des processus aléatoires sur les groupes quantiques compacts est établie depuis maintenant plusieurs décennies, mais notre compréhension des exemples concrets reste très partielle. Par exemple, dans le cas des groupes quantiques de matrices - qui généralisent les groupes de Lie - il n'y a pas à ce jour de notion générale de mouvement Brownien. Je présenterai une première approche à ce problème basée sur la notion de processus Gaussien due à Schürmann, qui mène à des questions topologiques intrigantes. J'en présenterai ensuite une seconde, due à Cipriani, Franz et Kula, qui mène à des exemples très concrets dont ont peut explorer les propriétés asymptotiques.

Tao Mai (Baylor University, Texas) : An extension of Hilbert transform to hyperbolic groups Abstract: The classical Hilbert transform has a natural analogue on the nonabelian free groups by decomposing the free group into disjoint subsets according to the first letter of the reduced words. Mei and Ricard prove that such a transform (decomposition) is unconditional with respect to the noncommutative Lp norm associated with the free group von Neumann algebras for all 1<p<\infty. I plan to talk about a possible extension of the Lp unconditionality of such “transforms” to the general hyperbolic groups.

Estelle Basset (LmB) : The Point of Continuity Property in some Lipschitz-free spaces Abstract: The Point of Continuity Property (PCP) is a property that characterizes whether the weak topology and the norm topology agree at some points on the subspaces of a Banach space. We will introduce Lipschitz-free spaces, which are specific Banach spaces in which the PCP is expressed in a simpler way. After defining an index named "weak-fragmentability index" which measures "at which point" a Banach space has the PCP, we will see that there exist Lipschitz-free spaces with arbitrarly high weak-fragmentability index, and what are the consequences of this result.