Kirchberg's 𝒪₂ norm-ultrapower embedding problem

What OpenAI claims

Constructs an explicit separable unital full group C∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be 𝒪₂ answers Kirchberg's norm-ultrapower embedding problem negatively. (Lean)

Paper

1. An explicit obstruction to nuclear norm-ultrapower embeddings

   23 September 2026 · Preprint, not peer reviewed · PDF

   We give a negative answer to Kirchberg's norm-ultrapower embedding problem. We construct an explicit separable unital full group C∗-algebra that admits no unital embedding into Bω, for any nonzero unital nuclear C∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into 𝒪₂^ω.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.