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arXiv · 2503.10505

Negative resolution to the $C^*$-algebraic Tarski problem

Srivatsav Kunnawalkam Elayavalli·Christopher Schafhauser
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Abstract

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.

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Srivatsav Kunnawalkam Elayavalli, Christopher Schafhauser. 2025-03-13. Negative resolution to the $C^*$-algebraic Tarski problem. https://arxiv.org/abs/2503.10505

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