Simplicity of algebras and $C^*$-algebras of self-similar groupoids
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914603658641408 |
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| author | Aakre, Josiah |
| author_facet | Aakre, Josiah |
| contents | Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19735 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplicity of algebras and $C^*$-algebras of self-similar groupoids Aakre, Josiah Rings and Algebras Operator Algebras Many previously studied path algebras or self-similar group algebras may be viewed as Steinberg algebras of self-similar groupoids. By way of inverse semigroup algebras, we characterize when the Steinberg algebra of a self-similar groupoid is simple. We show that the simplicity of the reduced $C^*$-algebra of a contracting self-similar groupoid coincides with the simplicity of the Steinberg algebra. As an aside, we show that simplicity of the two algebras sometimes depends only on the skeleton of the self-similar groupoid acting on a strongly connected graph. Finally, we apply our methods to examples including a self-similar groupoid akin to multispinal self-similar groups and a self-similar groupoid built from the well-known Basilica group. |
| title | Simplicity of algebras and $C^*$-algebras of self-similar groupoids |
| topic | Rings and Algebras Operator Algebras |
| url | https://arxiv.org/abs/2510.19735 |