Analysis and Probability Seminar

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).

If time permits, I will also report on some ongoing work on ideal intersection properties for spectral interpolation triples coming from locally compact groups.