Maximal ideals of reduced group C*-algebras and Thompson's groups
(2024)- 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 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... (More) - 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/2dda43f1-bffd-4722-a328-a365f0924707
- author
- Brix, Kevin Aguyar LU ; Bruce, Chris ; Li, Kang and Scarparo, Eduardo
- organization
- publishing date
- 2024
- type
- Working paper/Preprint
- publication status
- published
- subject
- DOI
- 10.48550/arXiv.2403.13645
- language
- English
- LU publication?
- yes
- id
- 2dda43f1-bffd-4722-a328-a365f0924707
- date added to LUP
- 2024-04-22 11:01:50
- date last changed
- 2024-05-15 10:04:18
@misc{2dda43f1-bffd-4722-a328-a365f0924707,
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 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).<br style="font-family: "Lucida Grande", Helvetica, Arial, sans-serif; font-size: 13.608px;"/>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.}},
author = {{Brix, Kevin Aguyar and Bruce, Chris and Li, Kang and Scarparo, Eduardo}},
language = {{eng}},
note = {{Preprint}},
title = {{Maximal ideals of reduced group C*-algebras and Thompson's groups}},
url = {{http://dx.doi.org/10.48550/arXiv.2403.13645}},
doi = {{10.48550/arXiv.2403.13645}},
year = {{2024}},
}