On Kirchberg's Embedding Problem

Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. 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 $\mathcal{O}_2$. 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 II$_1$ factor.Comment: 42 pages; final version to appear in the Journal of Functional Analysi

On the structural theory of II1\rm II_1 factors of negatively curved groups

Ozawa showed that for any i.c.c., hyperbolic group, the associated group factor is solid. Developing a new approach that combines some methods of Peterson, Ozawa and Popa, and Ozawa, we strengthen this result by showing that these factors are strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana, we show that profinite actions of lattices in Sp(n,1), n>1, are virtually W*-superrigid.Comment: Fina version; to appear as such in Annales Scientifiques de l'EN