Kirchberg's embedding problem: does every separable C*-algebra embed in a norm ultrapower of the Cuntz algebra O2?

Theorem 1.1: for $G=\mathbb Z[1/2]^3\rtimes(\mathrm{SL}_3(\mathbb Z)\times\mathbb Z)$ with $(M,k)\cdot v=2^kMv$, the full group C*-algebra $C^*(G)$ is separable and unital and has no unital embedding into $B^\omega$ for any nonzero unital nuclear $B$ and any free ultrafilter $\omega$; in particular none into $\mathcal O_2^\omega$. The obstruction combines property (T) of $\mathbb Z^3\rtimes\mathrm{SL}_3(\mathbb Z)$ with a finiteness lemma for approximate representation types in nuclear algebras. Via Goldbring-Sinclair it follows that no exact or nuclear unital C*-algebra is existentially closed. It says nothing about Connes' embedding problem or the QWEP conjecture.

Theorem 1.1: for $G=\mathbb Z[1/2]^3\rtimes(\mathrm{SL}_3(\mathbb Z)\times\mathbb Z)$ with $(M,k)\cdot v=2^kMv$, the full group C*-algebra $C^*(G)$ is separable and unital and has no unital embedding into $B^\omega$ for any nonzero unital nuclear $B$ and any free ultrafilter $\omega$; in particular none into $\mathcal O_2^\omega$. The obstruction combines property (T) of $\mathbb Z^3\rtimes\mathrm{SL}_3(\mathbb Z)$ with a finiteness lemma for approximate representation types in nuclear algebras. Via Goldbring-Sinclair it follows that no exact or nuclear unital C*-algebra is existentially closed. It says nothing about Connes' embedding problem or the QWEP conjecture.