Locally universal C*-algebras with computable presentations

Abstract

The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra ${\mathcal{O}}_2$. In an effort to provide a negative solution to the KEP and motivated by the recent refutation of the Connes Embedding Problem, we establish two computability-theoretic consequences of a positive solution to KEP. Both of our results follow from the a priori weaker assumption that there exists a locally universal C*-algebra with a computable presentation.

12 pages. This paper supercedes an earlier paper by the second and third author (arXiv:2107.14010). First draft; comments welcome!