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

Srivatsav Kunnawalkam Elayavalli; Christopher Schafhauser

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

Operator Algebras 2025-04-18 v2 Group Theory Logic

Authors: Srivatsav Kunnawalkam Elayavalli , Christopher Schafhauser

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.

Keywords

c*-algebras c*-algebra

Cite

@article{arxiv.2503.10505,
  title  = {Negative resolution to the $C^*$-algebraic Tarski problem},
  author = {Srivatsav Kunnawalkam Elayavalli and Christopher Schafhauser},
  journal= {arXiv preprint arXiv:2503.10505},
  year   = {2025}
}

Comments

Comments welcome. In v2, minor edits including adding references, fixed typos

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

Abstract

Keywords

Cite

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