Negative resolution to the $C^*$-algebraic Tarski problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908324229808128 |
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| author | Elayavalli, Srivatsav Kunnawalkam Schafhauser, Christopher |
| author_facet | Elayavalli, Srivatsav Kunnawalkam Schafhauser, Christopher |
| contents | We compute the $K_1$-group of ultraproducts of unital, simple $C^*$-algebras with unique trace and strict comparison. As an application, we prove that the reduced free group $C^*$-algebras $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily equivalent (i.e., have isomorphic ultrapowers) if and only if $m = n$. This settles in the negative the $C^*$-algebraic analogue of Tarski's 1945 problem for groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Negative resolution to the $C^*$-algebraic Tarski problem Elayavalli, Srivatsav Kunnawalkam Schafhauser, Christopher Operator Algebras Group Theory Logic We compute the $K_1$-group of ultraproducts of unital, simple $C^*$-algebras with unique trace and strict comparison. As an application, we prove that the reduced free group $C^*$-algebras $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily equivalent (i.e., have isomorphic ultrapowers) if and only if $m = n$. This settles in the negative the $C^*$-algebraic analogue of Tarski's 1945 problem for groups. |
| title | Negative resolution to the $C^*$-algebraic Tarski problem |
| topic | Operator Algebras Group Theory Logic |
| url | https://arxiv.org/abs/2503.10505 |