On Kirchberg's embedding problem

Kirchberg's Embedding Problem (KEP) asks whether every separable C⁎ algebra embeds into an ultrapower of the Cuntz algebra O2O2. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C⁎ algebras. Along the way, we establish a connection between existentially closed C⁎ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper concludes with a discussion of the model theory of O2O2. Several results in this last section are proven using some technical results concerning tubular embeddings, a notion first introduced by Jung for studying embeddings of tracial von Neumann algebras into the ultrapower of the hyperfinite II1 factor.