# Swedish Operator Algebra and Noncommutative Geometry Workshop

​This miniworkshop gathers the operator algebraically inclined groups in Gothenburg, Stockholm and Lund. It is organized by Lyudmila Turowska (Gothenburg), Sven Raum (Stockholm) and Magnus Goffeng (Lund). Feel free to contact the organizers for more information.

### Schedule:

 12:00 – 13:00 Lunch 13:30 – 14:10 Christian Voigt (Glasgow)Infinite quantum permutations Break 14:20 – 15:00 Jan Gundelach (Gothenburg)On embeddings into the $L^p$-Cuntz algebra $\mathcal{O}_2^p$ 15:00 – 15:30 Coffee 15:30 – 16:10 Sven Raum (Stockholm)Detecting ideals in groups C*-algebras of lattice in Lie groups Break 16:20 – 17:00 Magnus Fries (Lund)Higher order differential operators and relative K-cycles Break 17:10 – 17:50 Julian Krantz (Münster)K-theory of non-commutative Bernoulli-Shifts From 18:15 Dinner, please contact Magnus Goffeng if you want to join

### Abstracts:

Christian Voigt (Glasgow)
Infinite quantum permutations
I will discuss an approach to study quantum symmetries of infinite graphs. This leads to new examples of discrete quantum groups, in analogy to the quantum symmetry groups of Wang and Banica/Bichon. For finite sets and graphs, the resulting quantum groups can in fact be naturally viewed as discretizations of the latter (compact) quantum groups. Along the way I will present a number of concrete examples, and highlight some open problems.

Jan Gundelach (Gothenburg)
On embeddings into the $L^p$-Cuntz algebra $\mathcal{O}_2^p$
Kirchberg’s celebrated embedding theorem states that every unital simple separable nuclear C*-algebra embeds unitally into the Cuntz algebra $\mathcal{O}_2$. This result is fundamental in the classification of purely infinite C*-algebras by K-theory. For the $L^p$-version of Kirchberg’s embedding theorem, however, we observe a totally different behaviour. Despite $\mathcal{O}_2^p$ and all of its tensor powers being unital simple separable nuclear $L^p$-operator algebras as well, we prove that not even $\mathcal{O}_2^p\otimes_p \mathcal{O}_2^p$ embeds into $\mathcal{O}_2^p$. In this talk, I outline the core argument of the proof which is to relate unital isometric embeddings to embeddings of the topological full groups of the groupoid models.

Sven Raum (Stockholm)
Detecting ideals in groups C*-algebras of lattice in Lie groups
A major problem at the intersection of operator algebras and unitary representation theory is to determine the ideal structure of reduced group C*-algebras. Besides explicit computations, which can only be carried out for a very limited class of groups, two notions have been most studied: C*-simplicity and C*-uniqueness. While the former describes the absence of any non-trivial ideals in the reduced group C*-algebra, the latter describes groups for which the l1-convolution algebra carries a unique C*-norm. We introduce the l1-ideal intersection property, which interpolates between the previous notions. Our main result says that every lattice in a connected Lie group has the l1-ideal intersection property. The key novelty of the proof is to combine the theory of twisted groupoid C*-algebras and C*-simplicity with structure results about amenable subgroups of lattices in Lie groups. This is based on joint work in progress with Are Austad (University of Southern Denmark).

Magnus Fries (Lund)
Higher order differential operators and relative K-cycles
Geometric K-cycles are defined using first order differential operators and was the motivation for the unbounded formulation of K-homology and KK-theory. In this talk, I will present unbounded K-cycles that are modeled after higher order differential operators. These K-cycles can be made work with relative K-homology and can therefore be used in calculations of the boundary map in the six-term exact sequence of K-homology. The geometric example behind this is elliptic differential operators on a manifold with boundary where the boundary map maps to the K-homology of the geometric boundary.

Julian Kranz (Münster)
K-theory of non-commutative Bernoulli-Shifts
The non-commutative Bernoulli-Shift of a discrete group G on a unital C*-algebra A is the shift action of G on an infinite tensor product of G-many copies of A. I will explain how the K-theory of the associated crossed product can be computed under relatively mild assumptions on G and A. As an application, we will obtain a K-theory formula for C*-algebras of wreath products of amenable groups. This is joint work with S. Chakraborty, S. Echterhoff and S. Nishikawa.

Kategori Workshop
Plats: MV:L15, Chalmers tvärgata 3
Tid: 2022-11-17 13:30
Sluttid: 2022-11-17 17:50

Sidansvarig Publicerad: on 16 nov 2022.