2014/04/07 by Goldbring, Isaac, Sinclair, Thomas · 1 citation
#FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1404.1861
Kirchberg's Embedding Problem (KEP) asks whether every separable C^* algebra embeds into an ultrapower of the Cuntz algebra O2. 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 O2. 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.