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 Gₓ 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^* (Gₓ) 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.