分析学研讨班

We will give an overview of a recent theory of locally convex Hopf algebras, with an emphasis on the duality aspect. After motivating and introducing the theoretical framework, we will describe how Hopf algebras, compact and discrete quantum groups, locally compact groups, Lie groups, as well as their respective duals, can all give rise to examples in various ways. Then the emphasis shifts to the non-locally compact aspects of theory. The main result is that under the very weak assumption that the topology of a topological group $G$ is compactly generated, one can fully describe $G$ using only the locally convex Hopf algebra of continuous functions on it. This goes much beyond the previously well-known locally compact case, and includes interesting examples such as all metrizable groups, all topological groups that admit a cellular decomposition, etc.  On the non-commutative side, we will describe some new interesting topological quantum groups that arise naturally but seem to go beyond the current Kustermans-Vaes framework. Time permitting, exploiting some deep structural resuts related to Hilbert's fifth problem, we shall mention how one can describe a second countable locally compact group using a variant of Bruhat's notion of regular (smooth) functions on it.

The quantum group \mathcal{O}(SL_q(n)) is a non-(co)commutative Hopf algebra dual to the Jimbo-Drinfeld quantum group $\mathcal{U}_q(sl(n))$. It is known that O(SLq(3)) becomes a pre-Hilbert space with the Haar state as the inner product. The matrix coefficients of the Peter-Weyl decomposition of \mathcal{O}(SL_q(n)) are orthogonal with respect to the inner product. However, there is no orthonormal basis on this pre-Hilbert space due to a lack of efficient methods to evaluate the Haar state values of monomials on $\mathcal{O}(SL_q(n)). In this talk, we investigate the Haar state on \mathcal{O}(SL_q(3)). We will define a monomial basis that spans the linear subspace consisting of elements with non-zero Haar state values. Then, we give the Haar state values of those monomials in the basis as rational polynomials in variable q. We will also discuss the method to achieve an orthonormal basis which is proposed by Noumi, Yamada, and Mimachi.

The present talk is devoted to the noncommutative complex analytic geometry of a contractive quantum plane from the formal completion point of view. The formal completion of an Arens-Michael envelope of the quantum plane possesses the same spectrum to be the union of two copies of the complex plane. It turns out that it can be extended up to an Arens-Michael-Fréchet algebra sheaf, which results in the noncommutative analytic space, whose base topological space is the same spectrum. Moreover, that sheaf can be obtained as the deformation quantization of the related commutative analytic space. As the basic tool we use the fibered products of the Fréchet sheaves. To find out a key link between the transversality relation of the noncommutative sections versus to a left Fréchet module we discuss the related topological homology problems, and the related noncommutative Taylor spectrum of the module.

Let $A$ be an infinite dimensional C*-algebra and $1<p<\mathrm{ \infty }$. We compute the Szlenk index of $A$ and $L_p\left(A\right)$, and show that $Sz\left(A\right)= \Gamma \text{'}\left(i\left(A\right)\right)$ and $Dz\left(A\right)=Sz\left(L_p\left(A\right)\right)=\mathrm{ \omega }Sz\left(A\right)=\mathrm{ \omega } \Gamma \text{'}\left(i\left(A\right)\right)$, where $i\left(A\right)$ is the noncommutative Cantor-Bendixson index, $\Gamma \text{'}\left(\mathrm{ \xi }\right)$ is the minimum ordinal number which is greater than $\mathrm{ \xi }$ of the form ${\mathrm{ \omega }}^{\mathrm{ \zeta }}$ for some $\mathrm{ \zeta }$ and we agree that $\Gamma \text{'}\left(\mathrm{ \infty }\right)=\mathrm{ \infty }$ and $\mathrm{ \omega } \cdot \mathrm{ \infty }=\mathrm{ \infty }$. As an application, we compute the Szlenk index [respectively, $w^{ \ast }$-dentability index] of a C*-tensor product $A{ \otimes }_{\mathrm{ \beta }}B$ of non-zero C*-algebras $A$ and $B$ in terms of $Sz\left(A\right)$ and $Sz\left(B\right)$ [respectively, $Dz\left(A\right)$ and $Dz\left(B\right)$]. When $A$ is a separable C*-algebra, we show that there exists $a \in A_h$ such that $Sz\left(A\right)=Sz\left(C^{ \ast }\left(a\right)\right)$ and $Dz\left(A\right)=Dz\left(C^{ \ast }\left(a\right)\right)$, where $C^{ \ast }\left(a\right)$ is the C*-subalgebra of $A$ generated by $a$.

In this talk, we study Fourier multiplier commutators on a twisted crossed product $M ⋊R^d$. We will characterise their Schatten $p$$$-class membership by that of their symbols in the associated Besov space. In addition, we show a formula on the Dixmier trace, which also gives us a characterization of the weak Schatten $p$-class membership of these commutators by a Sobolev space. As an example, our results applied to noncommutative Euclidean Space.

Renyi entropy plays an important role in information theory and large deviation theory. For stochastic processes with good properties as mixing, Renyi entropy rate converges to the Shannon entropy rate. However, in this talk we will give an example to show that for general stationary and ergodic processes, the continuity of Renyi entropy rate breaks down at α=1. This is a joint work with Chengyu Wu, Li Xu and Guangyue Han.

Hypercontractive inequalities are a class of functional inequalities that strengthen the well-known Holder inequalities. These inequalities have found many applications in functional analysis, probability theory, discrete Fourier analysis, theoretical computer science, etc., especially in dealing with extremal problems in the geometry of high-dimensional spaces. In this talk, we will introduce hypercontractive inequalities and their extensions, including the information-theoretic characterization of hypercontractive inequalities, a non-linear strengthening of hypercontractive inequalities, and the opposite version of hypercontractive inequalities, called anti-contractive inequalities.

The first part of the lecture will survey the main facts regarding Connes’ integration, Weyl’s laws for compact operators and their relationships with semiclassical analysis. In particular, we will explain the link between Connes’ integration formula and semiclassical Weyl’s laws. This will include some background on Schatten $p$-classes and the Birman-Schwinger principle. The 2nd part will present new results regarding semiclassical Weyl’s laws and integration formulas for noncommutative manifolds (i.e., spectral triples). This improves and simplifies recent results of McDonald-Sukochev-Zanin and Kordyukov-Sukochev-Zanin.  For the Dirichlet and Neumann problems on Euclidean domains and closed Riemannian manifolds this enables us to recover the semiclassical Weyl’s laws in those settings from old results of Minakshisundaram and Pleijel from the late 40s. For closed manifolds this also allows us to recover the celebrated Weyl’s laws of Birman-Solomyak for negative-order pseudodifferential operators.  A further set of examples is provided by Schrödinger operators associated to sub-Laplacians on sub-Riemannian manifolds, including contact manifolds and Baouendi-Grushin example. Finally, we will explain how this framework enables us to get semiclassical Weyl’s laws for noncommutative tori. This will solve conjectures by Edward McDonald and the speaker.

In this talk, we focus on how we can interpret the actions of the elements in the Gelfand spectrum of a weighted Fourier algebra on connected Lie groups. They can be viewed as evaluations on specific points of the complexification of the underlying Lie group. For this purpose, we need to introduce a particular dense subalgebra of the weighted Fourier algebra, which we call an analytic subalgebra. We first introduce an analytic subalgebra allowing a “local” solution for general connected Lie groups. We will demonstrate that a “global” solution is also possible for simply connected nilpotent Lie groups.

Kuroda’s Theorem is one of the fundamental results in perturbation theory. It states that, if  J is a Banach ideal in  that is not contained in the trace class, then for every self-adjoint operator b on H and every  there exists a diagonal operator d on H such that ∥b − d∥J < ε. In this talk, we will talk about an extension of Kuroda’s theorem for n-tuples of commuting self-adjoint operators affiliated with a semifinite von Neumann algebra , with respect to symmetric spaces associated with 

In this talk we will discuss, on a locally symmetric space of higher rank and infinite volume, the relations between objects in three different topics: the bottom of the L2-spectrum of the Laplace-Beltrami operator, the convergence rate of the discrete subgroup, and the tempered representations.

Amalgamated free products and HNN extensions are the two fundamental objects in the Bass-Serre theory. In this talk, we will introduce some Lp-bounded (1 <p < ∞) Fourier multipliers for HNN extensions, with the help of Khinchine type inequalities on these groups. This work is a continuation of a recent work with Adrián González and Javier Parcet. It is related to the results of Mei, Ricard and Xu who proved Lp-boundedness of the same type of Fourier multipliers on amalgamated free products of von Neumann algebras. 

Two very important concepts arose from Haagerup's highly influential work on the reduced C*-algebra of the free group: weak amenability and the Haagerup property. It is an open problem to determine whether weak amenability with Cowling-Haagerup constant 1 implies the Haagerup property. This is often referred to as Cowling's conjecture. In this talk, I will give an overview of this question, and I will discuss a recent progress in this direction: Every countable weakly amenable group with Cowling-Haagerup constant 1 admits a proper cocycle for a uniformly bounded representation on a subspace of an L₁ space.

Bożejko and Speicher introduced q-Gaussian variables to produce examples of generalized Brownian motions. The resulting von Neumann algebras -- the q-Gaussian algebras -- can be viewed as deformations of the free group factors. It is a very natural question whether these von Neumann algebras are actually isomorphic to the free group factors. Guionnet and Shlyakhtenko introduced the free monotone transport techniques and provided a partial answer: if the number of variables is finite and the parameter q is very small then we do get an isomorphism. There are no results known for infinitely many variables and in my talk I plan to describe a related result about q-Gaussian C*-algebras -- in the infinite case they are not isomorphic to their free counterparts. The von Neumann algebraic case remains open.

Joint work with Matthijs Borst, Martijn Caspers and Mario Klisse.

In this talk, I will present a noncommutative Nevo-Zimmer theorem for actions of lattices in higher rank simple algebraic groups on von Neumann algebras. This extends to the realm of algebraic groups defined over arbitrary local fields the noncommutative Nevo-Zimmer theorem we obtained with Rémi Boutonnet in 2019 for real Lie groups.

I will discuss various applications of the above theorem to topological dynamics, unitary representations and operator algebras. I will also present a noncommutative analogue of Margulis’ factor theorem and discuss its relevance regarding Connes’ rigidity conjecture for group von Neumann algebras of higher rank lattices.

This is based on joint work with Uri Bader and Rémi Boutonnet (arXiv:2112.01337)