Maximal ideals of reduced group C*-algebras and Thompson's groups
Kevin Aguyar Brix, Chris Bruce, Kang Li, Eduardo Scarparo
TL;DR
The paper develops a Galois-connection framework between ideal lattices arising from a conditional expectation $P:B\to A$, defining $Ind_P$ and $Coind_P$ to relate ideals in $A$ and $B$. It proves broad equivalences that characterize when maximal ideals are induced and when $Ind_P$ yields a bijection between maximal co-induced ideals of $A$ and maximal ideals of $B$. Applying this to reduced group C*-algebras and Furstenberg boundary stabilizers, it shows that every maximal ideal of $C^*_r(G)$ is induced from a maximal ideal of $C^*(G_x)$ for $x\in\partial_F G$, and that the bijection reduces the maximal-ideal problem to amenable isotropy groups. The Thompson groups T and F are then analyzed: $C^*_r(T)$ always has a unique maximal ideal, and if $F$ is amenable, $C^*_r(T)$ has infinitely many ideals, with several amenability-characterizing equivalences in terms of the ideal structure of $C^*_r(T)$. These results connect dynamical properties on the Furstenberg boundary with the lattice of ideals in both full and reduced group C*-algebras, providing a unified route to understanding maximal ideals via amenable isotropy.
Abstract
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 $G_x$ for the action of $G$ on its Furstenberg boundary, induction gives a bijection between the set of maximal co-induced ideals of $C^*(G_x)$ 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.