Locally universal C⁎-algebras with computable presentations

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.