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

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