Quantifier elimination in C*-algebras

Christopher J. Eagle, Ilijas Farah, Eberhard Kirchberg, Alessandro Vignati

Published 2015-02-02Version 1

Conjecturally, the only C*-algebras that admit elimination of quantifiers in continuous logic are $\mathbb{C}, \mathbb{C}^2$, $C($Cantor space$)$ and $M_2(\mathbb{C})$. Concentrating on the noncommutative case, we prove that $M_2(\mathbb C)$ is the only exact noncommutative C*-algebra whose theory admits elimination of quantifiers and that every other example is purely infinite and simple.