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 $O_2$. 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.