Maximal ideals of reduced 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 connec...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Brix, Kevin Aguyar, Bruce, Chris, Kang, Li, Scarparo, Eduardo
Format: Article
Language:English
Subjects:
Algebra
Group theory
Lattices
Subgroups
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2331-8422