Maximal ideals of reduced group C*-algebras and Thompson's groups
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866911243848122368 |
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| author | Brix, Kevin Aguyar Bruce, Chris Li, Kang Scarparo, Eduardo |
| author_facet | Brix, Kevin Aguyar Bruce, Chris Li, Kang Scarparo, Eduardo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximal ideals of reduced group C*-algebras and Thompson's groups Brix, Kevin Aguyar Bruce, Chris Li, Kang Scarparo, Eduardo Operator Algebras Group Theory 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. |
| title | Maximal ideals of reduced group C*-algebras and Thompson's groups |
| topic | Operator Algebras Group Theory |
| url | https://arxiv.org/abs/2403.13645 |