Photo credit: © Patricia Nauta

Francesca Arici

Leiden University

Toeplitz Algebras and Toeplitz Extensions 

This lecture series delves into the world of Toeplitz algebras and Toeplitz extensions, exploring their fundamental properties, applications, and significances in operator algebras and mathematical physics.

We shall start with an introduction to Toeplitz operators, highlighting their origins, classical examples, and role in functional analysis. From there, we advance to the study of Toeplitz algebras, examining their C*-representations. A significant focus will be placed on the concept of Toeplitz extensions. We will discuss the theoretical frame-work of extensions, their topological and algebraic implications, and the role played by Toeplitz extensions in index theory.

Sven Raum

University of Potsdam

Hecke operator algebras

Hecke algebras are certain deformations of group algebras. They are constructed from Coxeter groups – an abstraction of reflection groups. Classically, Hecke algebras are important to understand the representation theory of algebraic groups, but the representation theory of much larger classes of groups acting on buildings can be approached through them. In past years, operator algebraic completions of Hecke algebras have been studied, motivated by questions in cohomology, representation theory and as a ground to test robustness of results obtained for group operator algebras.

In this series of talks, I will first introduce Coxeter groups and Hecke algebras. Subsequently, I will discuss their operator algebraic completions and describe the state-of-the-art concerning the structure and classification of right-angled Hecke operator algebras, which are best amenable to methods in the field. 

Photo credit: © Thorsten Mohr

Moritz Weber

Saarland University

Introduction to compact matrix quantum groups

 Since the 1980s, more and more branches of "quantum mathematics“ have been developed, mainly based on C*-algebras and von Neumann algebras. In this realm, a suitable notion of (quantum) symmetry has been given by Woronowicz. We will introduce to his theory of compact quantum groups, focusing on compact matrix quantum groups, and providing a number of examples such as Wang’s quantum permutation group. 

We will then survey the class of "easy“ quantum groups (introduced by Banica-Speicher in 2009), a quite large class of quantum groups with some combinatorial data. Also, we will mention the latest work on quantum automorphism groups of graphs (with links to quantum information theory). We will mention some interesting C*-algebra questions including simplicity, nuclearity, K-theory, enveloping von Neumann algebras and many open questions.