Kirchberg's \mathcal O_2 norm-ultrapower embedding problem
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
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^(∗)-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^ω.
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