Kirchberg's \mathcal O_2 norm-ultrapower embedding problem

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 \mathcal O_2 answers Kirchberg's norm-ultrapower embedding problem negatively. (Lean)

Manuscript

An explicit obstruction to nuclear norm-ultrapower embeddings

September 23, 2026 · source · cite as \cite{OAI:An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026}

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 B^ω, for any nonzero unital nuclear C^∗-algebra B and any free ultrafilter ω on ℕ. In particular, it does not embed unitally into \mathcal O_2^\omega.

Result 292 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.