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

Srivatsav Kunnawalkam Elayavalli, Christopher Schafhauser

TL;DR

The paper addresses the C*-algebraic analogue of Tarski's problem by computing the $K_1$-group of ultraproducts of unital, simple $C^*$-algebras with unique trace and strict comparison, and applying this to distinguish reduced free group algebras at the level of elementary theory. A central technical advance is showing that the map $\bar{\eta}: K_1(A_\omega) \to K_1(A)_\omega$ is an isomorphism after a $2\times 2$ amplification, achieved via trace-kernel extensions and absorption techniques together with stable rank considerations. This enables a full analysis of $K_1$ under products and ultraproducts, leading to $K_1(C^*_r(F_n)_\omega) \cong (\mathbb{Z}^n)_\omega$ and ultimately to a negative resolution of the C*-algebraic Tarski problem: $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily equivalent only when $m=n$. The results illuminate the model-theoretic rigidity of free group C*-algebras and connect structural properties like strict comparison and corona-factorization with elementary invariants.

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.

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

TL;DR

The paper addresses the C*-algebraic analogue of Tarski's problem by computing the -group of ultraproducts of unital, simple -algebras with unique trace and strict comparison, and applying this to distinguish reduced free group algebras at the level of elementary theory. A central technical advance is showing that the map is an isomorphism after a amplification, achieved via trace-kernel extensions and absorption techniques together with stable rank considerations. This enables a full analysis of under products and ultraproducts, leading to and ultimately to a negative resolution of the C*-algebraic Tarski problem: and are elementarily equivalent only when . The results illuminate the model-theoretic rigidity of free group C*-algebras and connect structural properties like strict comparison and corona-factorization with elementary invariants.

Abstract

We compute the -group of ultraproducts of unital, simple -algebras with unique trace and strict comparison. As an application, we prove that the reduced free group -algebras and are elementarily equivalent (i.e., have isomorphic ultrapowers) if and only if . This settles in the negative the -algebraic analogue of Tarski's 1945 problem for groups.

Paper Structure

This paper contains 8 sections, 11 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cuntz subsequivalence and comparison
  4. Ultraproducts and the $K_1$-group
  5. The trace-kernel extension
  6. Absorption
  7. Proof of Theorem \ref{['thm:k1-inj']}
  8. Proof of Theorem \ref{['thm:tarski']}

Key Result

Theorem 1.1

If $m, n \in \mathbb N \cup \{\infty\}$ are such that $C^*_r(F_m)$ and $C^*_r(F_n)$ are elementarily equivalent, then $m = n$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3: Lin:sr1
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5: cf. CGSTW1
  • ...and 13 more