Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917430695034880 |
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| author | Martínez, Diego Szakács, Nóra |
| author_facet | Martínez, Diego Szakács, Nóra |
| contents | We give the first examples of étale (non-Hausdorff) groupoids $\mathcal G$ whose $C^*$-algebras contain singular elements that cannot be approximated by singular elements in $\mathcal C_c(\mathcal G)$. We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential $C^*$-algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions Martínez, Diego Szakács, Nóra Operator Algebras Rings and Algebras 46L55, 46L06, 20M18 We give the first examples of étale (non-Hausdorff) groupoids $\mathcal G$ whose $C^*$-algebras contain singular elements that cannot be approximated by singular elements in $\mathcal C_c(\mathcal G)$. We provide two examples: one is a bundle of groups, and the other a minimal and effective groupoid constructed from a self-similar action on an infinite alphabet. Moreover, we also prove that the Baum--Connes assembly map for the first example is not surjective, not even on the level of its essential $C^*$-algebra. |
| title | Algebraic singular functions are not always dense in the ideal of $C^*$-singular functions |
| topic | Operator Algebras Rings and Algebras 46L55, 46L06, 20M18 |
| url | https://arxiv.org/abs/2510.01947 |