Gelfand pairs, spherical functions and exotic group C*-algebras

Abstract: For a non-amenable group $G$, there can be many group C*-algebras that lie naturally between the universal and the reduced C*-algebra of $G$. These are called exotic group C*-algebras. After a short introduction, I will explain that if $G$ is a simple Lie group or an appropriate locally compact group acting on a tree, the $L^p$-integrability properties of different spherical functions on $G$ (relative to a maximal compact subgroup) can be used to distinguish between exotic group C*-algebras. This recovers results of Samei and Wiersma. Additionally, I will explain that under certain natural assumptions, the aforementioned exotic group C*-algebras are the only ones coming from $G$-invariant ideals in the Fourier-Stieltjes algebra of $G$.

This is based on joint work with Dennis Heinig and Timo Siebenand.

group theory

Audience: researchers in the discipline

Symmetry in Newcastle