Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

For irrational $\theta$ , let $R_\theta$ be the hyperfinite II $_1$ factor generated by unitaries $U,V$ with $VU=e^{2\pi i\theta}UV$ , and for $f$ on the circle let $T_f=Uf(V)$ . Haagerup-Schultz give nontrivial invariant projections in the factor whenever the Brown measure is not a point mass; for these operators that happens exactly when the Fuglede-Kadison determinant $\exp\int\log|f|\,dm$ is positive. Zhu, Fang and Shi (2017, p. 264) asked: if $f$ is nonzero almost everywhere, $|f|$ is nonconstant and the determinant is zero, must $T_f$ have a nontrivial invariant projection $p\in R_\theta$ , that is, $0<\tau(p)<1$ and $(1-p)T_fp=0$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Operator algebras, II1 factors
Posed by: Z. Zhu, J. Fang and R. Shi, On a class of operators in the hyperfinite II1 factor, Math. Scand. (2017), p. 264, question after Proposition 5.7
Year posed: 2017
Years open: 9y
Solved: 2026-09-27
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 15 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every irrational $\theta\in(0,1)$ there is a continuous $f:\mathbb T\to[0,1]$ vanishing only at $1$ , with $\int\log f\,dm=-\infty$ , such that every projection $p\in R_\theta$ with $(1-p)Uf(V)p=0$ is $0$ or $1$ ; the operator is nonzero and norm-quasinilpotent. The angle is prescribed in advance with no Diophantine condition. Invariant subspaces of the Hilbert-space representation whose projections lie outside the factor are not excluded. An appendix gives a second realization on an infinite product of circles.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against the question as quoted from Zhu-Fang-Shi. The obstruction and transfer arguments were not refereed. Lean-checked on the release's Comparator challenge IrrationalRotation, found through lean/docs/293.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Analysis.IrrationalRotation.Main exists at the pinned commit. Its statement was read here: prescribed_irrational_counterexample gives, for every irrational theta in (0,1), a continuous f with values in [0,1], zero exactly at one point and log integral minus infinity, whose weighted rotation lies in the generated von Neumann algebra, has only trivial invariant projections there, is nonzero, norm-quasinilpotent and has vector Fuglede-Kadison determinant zero. That covers the headline. Not rebuilt here.

Zhu, Fang and Shi's question on invariant projections of weighted irrational rotations

What was actually shown

What the AI did

Verification

Sources

Discussion