Haagerup’s commutative-determination problem for weight normality

Determine whether a weight on a von Neumann algebra is normal whenever its restriction to every unital abelian von Neumann subalgebra is normal.

Background

The paper identifies this as Haagerup’s Problem 1.11. For a weight Φ\Phi on a von Neumann algebra MM, condition (H) requires that Φ\Phi be normal on every unital abelian von Neumann subalgebra of MM. The unresolved issue is whether this commutative normality condition forces normality of Φ\Phi on the entire, possibly noncommutative, algebra.

The paper proves affirmative answers in several special cases, including strongly or strictly semifinite weights and weights on finite von Neumann algebras, but leaves the general problem unresolved.

References

This runs into Haagerup's big open Problem 1.11 in , namely whether normality of weights is commutatively determined.

— Quantum integrals  (2610.06771 - Blecher et al., 5 Oct 2026) in Section 1, Introduction; Section 6, Corollary 6.3 and surrounding discussion

Is a weight on a von Neumann algebra \sigma-normal provided its restriction to every commutative von Neumann subalgebra is \sigma-normal (equivalently, by Lemma~\ref{lem:abelian-sequential-lsc}, sequentially weak* lower semicontinuous)?

— Quantum integrals  (2610.06771 - Blecher et al., 5 Oct 2026) in Section 6, subsection “Back to the general case,” immediately following Proposition 6.2