Maximal ideals of reduced group C*-algebras and Thompson’s groups

Given a conditional expectation P P from a C*-algebra B B onto a C*-subalgebra A A , we observe that induction of ideals via P P , together with a map which we call co-induction, forms a Galois connection between the lattices of ideals of A A and B B . Using properties of this Galois connection, we show that, given a discrete group G G and a stabilizer subgroup G x G_x for the action of G G on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of C ∗ ( G x ) C^*(G_x) and the set of maximal ideals of C r ∗ ( G ) C^*_r(G) . As an application, we prove that the reduced C*-algebra of Thompson’s group T T has a unique maximal ideal. Furthermore, we show that, if Thompson’s group F F is amenable, then C r ∗ ( T ) C^*_r(T) has infinitely many ideals.