Relative topological principality and the ideal intersection property for groupoid C*-algebras
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910684141322240 |
|---|---|
| author | Eagle, Chris J. Goerke, Gavin Laca, Marcelo |
| author_facet | Eagle, Chris J. Goerke, Gavin Laca, Marcelo |
| contents | We introduce the notion of relative topological principality for a family $\{H_α\}$ of open subgroupoids of a Hausdorff étale groupoid $G$. The C*-algebras $C^*_r(H_α)$ of the groupoids $H_α$ embed in $ C^*_r(G)$ and we show that if $G$ is topologically principal relative to $\{H_α\}$ then a representation of $C^*_r(G)$ is faithful if and only if its restriction to each of the subalgebras $C^*_r(H_α)$ is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_03204 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative topological principality and the ideal intersection property for groupoid C*-algebras Eagle, Chris J. Goerke, Gavin Laca, Marcelo Operator Algebras 46L55 We introduce the notion of relative topological principality for a family $\{H_α\}$ of open subgroupoids of a Hausdorff étale groupoid $G$. The C*-algebras $C^*_r(H_α)$ of the groupoids $H_α$ embed in $ C^*_r(G)$ and we show that if $G$ is topologically principal relative to $\{H_α\}$ then a representation of $C^*_r(G)$ is faithful if and only if its restriction to each of the subalgebras $C^*_r(H_α)$ is faithful. This variant of the ideal intersection property potentially involves several subalgebras, and gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As applications we prove a uniqueness theorem for Toeplitz C*-algebras of left cancellative small categories that generalizes a recent result of Laca and Sehnem for Toeplitz algebras of group-embeddable monoids, and we also discuss and compare concrete examples arising from integer arithmetic. |
| title | Relative topological principality and the ideal intersection property for groupoid C*-algebras |
| topic | Operator Algebras 46L55 |
| url | https://arxiv.org/abs/2312.03204 |