An explicit obstruction to nuclear norm-ultrapower embeddings

We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C${}^{\ast }$-algebra that admits no unital embedding into B${}^{\omega }$, for any nonzero unital nuclear C${}^{\ast }$-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into ${\mathcal{O}}_2^{\omega }$.