Strong-operator paving over arbitrary maximal abelian subalgebras
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Does every self-adjoint operator in every von Neumann algebra admit strong-operator paving over every maximal abelian subalgebra using O(ε⁻²) projections? The projection count must be independent of the inclusion, operator and strong neighborhood.
Why it matters and what counts as progress
Why it matters: A universal quadratic bound would extend strong paving beyond previously treated expected, Cartan and singular cases.
What counts as progress: A bound valid without separability or conditional-expectation assumptions, or an inclusion obstructing arbitrary-masa strong paving.
How the problem developed
- 2015 · Conjecture and partial results [1]
Sorin Popa and Stefaan Vaes Proposed strong-operator paving for every maximal abelian subalgebra of every von Neumann algebra and proved it for several important classes, including singular masas with quadratic paving size.
- 18 Oct 2026 · Proof claim [2]
OpenAI Reported strong-operator paving over arbitrary masas with at most 5×10⁸ε⁻² projections, with the count fixed independently of the inclusion, operator and strong neighborhood.
AI contribution
Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
For every 0<ε<1, every self-adjoint x in any von Neumann algebra M admits strong-operator paving over any maximal abelian subalgebra A with at most 5×10⁸ε⁻² projections. The count is fixed before the inclusion, operator and strong neighborhood, without separability or conditional-expectation assumptions.
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