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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.

Manuscripts in this family · 1

Sep 23, 2026·paper.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 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

Full abstract