What was already known
Arveson’s density theorem requires a maximal abelian self-adjoint subalgebra; the construction lacks that hypothesis.
AI involvement
OpenAI is credited as author and attributes the construction and proof artifacts to an internal model; no result-specific autonomous session is documented.
The original OpenAI-authored manuscript and repository provenance attribute the central construction and corollary to its internal model.
Importance
Provides a counterexample to the assertion that every unital transitive operator algebra is strongly dense in B(H).
Limitations
OpenAI reports Lean checking of properness, transitivity and strong closure. No named human check or explanation of the construction is documented. The algebra result is a separate consequence of the operator used in the hyperinvariant-subspace result.
Technical verification
OpenAI's Lean scope note and BackwardIntertwiners comparator statement include transitivity, properness and strong-operator closure of the commutant. The comparator challenge body is a placeholder linked to a separate solution module; the check is source-reported.
Formal checking
The commutant of the constructed nonzero quasinilpotent operator is transitive, proper and strongly closed on each separable infinite-dimensional complex Hilbert space; Source-reported comparison of OAI.Analysis.BackwardIntertwiners.AlgebraCorollary with direct_algebra_corollary; permitted axioms propext, Quot.sound and Classical.choice
- Checker
- Lean 4 kernel and OpenAI Comparator configuration
- Scope
- The commutant of the constructed nonzero quasinilpotent operator is transitive, proper and strongly closed on each separable infinite-dimensional complex Hilbert space
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The model-attributed paper explains why the commutant is transitive, but no identified human explanation was found; Few is the documentation default.
Correction and publication history
Provides a counterexample to the assertion that every unital transitive operator algebra is strongly dense in B(H).
Read this published revision