A group is said to be C*-simple if its reduced C*-algebra is simple. A celebrated result by Kalantar and Kennedy says that a group G is C*-simple if and only if the action of G on its Furstenberg boundary is free. I will present on recent work in which we extend this result, proving that even the maximal ideal structure of the reduced C*-algebra of a discrete group is governed by the action of G on its Furstenberg boundary: given a point x in the Furstenberg boundary of G, we prove that there is a bijection between maximal ideals of the reduced C*-algebra of G and maximal co-induced ideals of the C*-algebra of the stabiliser subgroup of x. Interestingly, our result reduces the problem of computing maximal ideals in reduced group C*-algebras to computing ideals in C*-algebras of amenable group.

This is joint work with Kevin Aguyar Brix, Kang Li, and Eduardo Scarparo.