Ober­se­mi­nar Free Pro­ba­bi­lity

In this research seminar we treat topics ranging from free probability and random matrix theory to combinatorics, operator algebras, functional analysis and quantum groups.

Talks

Mi, 4.11., 16:15 - Daniel Gromada (Saarbrücken), Group-theoretical graph categories

In the talk, we are going to recall the definition of group-theoretical categories of partitions [Raum–Weber '14] and skew categories of
partitions [Maaßen '18]. We will then generalize those structures into the framework of graph categories (in the sense of Mančinska–Roberson) and introduce some kind of group-theoretical description also here.  We also discuss the quantum groups associated to such categories. The talk is based on a recent preprint arXiv:2009.06998.


Mi, 11.11., 16:15 - Hannes Thiel (Münster/Dresden), Diffuse traces and Haar unitaries

A Haar unitary (with respect to a given tracial state) is a unitary such that every nonzero power of the unitary and its adjoint has vanishing trace. We show that a tracial state admits a Haar unitary if and only if it is diffuse (the unique extension to a normal tracial state on the enveloping von Neumann algebra vanishes on every minimal projection), if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient.

It follows that a unital C*-algebra has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. In particular, every tracial state on an infinite-dimensional, simple C*-algebra admits a Haar unitary.

I will sketch a proof of this result and present some applications to group C*-algebras and reduced free products.


Mi, 18.11. + Mi, 25.11. + Mi, 2.12., 14:00 (sharp) - Loop models meet diagram algebras

(A series of talks, details to be announced)


Mi, 9.12. - Nicolas Faroß (Saarbrücken), Towards a Concrete Model for the Quantum Permutation Group