The transitive algebra problem
An algebra of bounded operators on a Hilbert space is transitive if its only common closed invariant subspaces are and . Arveson (1967) proved that a transitive algebra containing a maximal abelian self-adjoint algebra is strongly dense in , and Douglas and Pearcy (1972) related the general question to hyperinvariant subspaces. Is every unital transitive operator algebra on an infinite-dimensional separable complex Hilbert space strongly (equivalently weakly) dense in ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Operator theory, operator algebras
- Posed by
- Question in the setting of W. B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967); the manuscript states it without attribution
- Year posed
- —
- Years open
- —
- 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
- 46 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Corollary 1.2 of the principal manuscript: the commutant of the constructed nonzero quasinilpotent operator is a proper, strongly closed, unital, transitive complex algebra, so it is not strongly dense in . This answers the question negatively on every infinite-dimensional separable Hilbert space. It is an immediate consequence of the hyperinvariant counterexample, since the commutant of an operator without hyperinvariant subspaces is transitive. The algebra is a commutant, so nothing is said about singly generated transitive algebras, which is the invariant subspace problem.
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 introduction's remark after Corollary 1.2 and the corollary itself, read against the question as stated there. Lean-checked on the release's Comparator challenge BackwardIntertwiners (declaration OAI.BackwardIntertwiners.direct_algebra_corollary, listed in lean/formalization.yaml). Its statement, read here, asserts that the commutant is not the whole operator algebra, is SOT-closed and has no nontrivial closed common invariant subspace, which is the negative answer; strong and weak closures agree for subalgebras, so the weak form follows. Not rebuilt here.