Negative resolution to the $C^{\ast }$-algebraic Tarski problem

Abstract

We compute the $K_1$-group of ultraproducts of unital, simple $C^{\ast }$-algebras with unique trace and strict comparison. As an application, we prove that the reduced free group $C^{\ast }$-algebras $C_r^{\ast }(F_m)$ and $C_r^{\ast }(F_n)$ are elementarily equivalent (i.e., have isomorphic ultrapowers) if and only if $m=n$. This settles in the negative the $C^{\ast }$-algebraic analogue of Tarski's 1945 problem for groups.

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