"Geometry and Analysis on Groups" Research Seminar
Exotic group \(C^*\)-algebras of automorphism groups of trees.
Tim de Laat (Münster)
With every locally compact group, we can associate two natural \(C^*\)-algebras: the universal and the reduced group \(C^*\)-algebra. It is well known that these algebras coincide if and only if the group is amenable. In general, there can be many \(C^*\)-algebras, called exotic group \(C^*\)-algebras, which lie "between" these two. After an introduction to this relatively young topic, I will explain a strategy to determine families of exotic group \(C^*\)-algebras of a group \(G\) from \(L^p\)-integrability properties of matrix coefficients of unitary representations of \(G\). I will explain how to apply this strategy in the setting of automorphism groups of trees and, if time permits, in the setting of simple Lie groups with real rank one. This is based on joint work with Dennis Heinig and Timo Siebenand.