Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916663351312384 |
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| author | Kwaśniewski, Bartosz K. Meyer, Ralf Prasad, Akshara |
| author_facet | Kwaśniewski, Bartosz K. Meyer, Ralf Prasad, Akshara |
| contents | We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff étale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable finiteness of the reduced and essential C*-algebras. If the reduced C*-algebra of a twisted groupoid is simple and the type semigroup satisfies a weak version of almost unperforation, then the C*-algebra is either stably finite or purely infinite. We apply our theory to Cartan inclusions. We calculate the type semigroup for the possibly non-Hausdorff groupoids associated to self-similar group actions on graphs and deduce a dichotomy for the resulting Exel-Pardo algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17190 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras Kwaśniewski, Bartosz K. Meyer, Ralf Prasad, Akshara Operator Algebras 46L55, 20M18, 22A22 We develop a theory of type semigroups for arbitrary twisted, not necessarily Hausdorff étale groupoids. The type semigroup is a dynamical version of the Cuntz semigroup. We relate it to traces, ideals, pure infiniteness, and stable finiteness of the reduced and essential C*-algebras. If the reduced C*-algebra of a twisted groupoid is simple and the type semigroup satisfies a weak version of almost unperforation, then the C*-algebra is either stably finite or purely infinite. We apply our theory to Cartan inclusions. We calculate the type semigroup for the possibly non-Hausdorff groupoids associated to self-similar group actions on graphs and deduce a dichotomy for the resulting Exel-Pardo algebras. |
| title | Type semigroups for twisted groupoids and a dichotomy for groupoid C*-algebras |
| topic | Operator Algebras 46L55, 20M18, 22A22 |
| url | https://arxiv.org/abs/2502.17190 |