A central problem at the intersection of operator algebras and dynamical systems is understanding the ideal structure of reduced crossed products. While great effort has gone in to understand simplicity of the crossed product, sometimes this is too much to ask for. A different approach is needed to venture beyond the simple case.

In this talk, we introduce the $\ell^1$-ideal intersection property for $C^*$-dynamical systems, which says that ideals in the reduced crossed product are detected by the $\ell^1$-algebra sitting inside, giving us a better handle on the ideal structure. We will present classes of discrete groups $\Gamma$ for which $(C(X), \Gamma, \alpha)$ has the $\ell^1$-ideal intersection property for all choices of $C(X)$ and action $\alpha$. As a by-product, we are also able to add to the list of groups $\Gamma$ for which $\ell^1(\Gamma)$ has unique $C^*$-norm.

This is based on joint work with Sven Raum (University of Potsdam).

Bio: Are Austad is a postdoc at the University of Southern Denmark. He obtained his PhD from the Norwegian University of Science and Technology in 2021 under the supervision of Franz Luef and Eduard Ortega. His research interests lie primarily in the intersection between operator algebras, groups and groupoids.