The transitive algebra problem on Hilbert space
Follow the latest progress by email.
Must every unital transitive operator algebra on a separable infinite-dimensional complex Hilbert space be strongly dense in B(H)? Transitivity means there is no nontrivial common closed invariant subspace.
Why it matters and what counts as progress
Why it matters: The question tests how far density theorems extend without a maximal abelian self-adjoint subalgebra.
What counts as progress: A general density theorem or a proper strongly closed unital transitive algebra, with its invariant-subspace and closure properties proved.
How the problem developed
- 1967 · Partial result [1]
William B. Arveson Proved strong density for transitive algebras containing a maximal abelian self-adjoint algebra, a restricted case of the general problem.
- 1971 · Open problem [2]
John B. Conway Recorded that no proper weakly closed transitive subalgebra of B(H) was known at the time and proved a density result under an additional condition.
- 27 Sept 2026 · Reported claim [3]
OpenAI The preprint constructs, on every separable infinite-dimensional complex Hilbert space, an operator with a proper strongly closed transitive commutant.
AI contribution
Backward intertwiners and a transitive commutant
On every separable infinite-dimensional complex Hilbert space, the commutant of the constructed operator is a proper unital complex algebra, closed in the strong operator topology and transitive.
Sources
- Arveson, A density theorem for operator algebras
- Conway, On algebras of operators with totally ordered lattice of invariant subspaces
- OpenAI backward-intertwiner paper
- Backward intertwiners and a transitive commutant
- OpenAI mathematics release
- Lean scope for the transitive algebra
- Transitive-algebra comparator statement
- Comparator solution mapping for transitive algebra
Website timeline
Publication dates and editorial changes.
OpenAI reports a proper transitive strongly closed operator algebra