Let A⊂ M be a MASA in a von Neumann algebra, and suppose first that there is a normal conditional expectation from M onto A. The app-paving conjecture. Every MASA in a von Neumann algebra should have the app-paving property, equivalently the so-paving property; in particular, the equivalence between so-pavability and app-pavability should hold for arbitrary MASAs, including those that are not ranges of normal conditional expectations. Moreover, if x∈ M_sa, let n_s(x,varepsilon) denote the so-paving size. Then there should be a universal constant C>0, independent of A⊂ M, such that n_s(x,varepsilon)≤ C varepsilon^-2,qquad forall x∈ M_sa. The first assertion extends the expected paving phenomenon beyond MASAs admitting normal conditional expectations, while the quantitative assertion predicts a uniform quadratic bound on paving size. These claims are presented as expectations in the paper and remain open.

Popa–Vaes app-paving conjecture and universal paving-size bound

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Let $A\subset M$ be a MASA in a von Neumann algebra, and suppose first that there is a normal conditional expectation from $M$ onto $A$ . The app-paving conjecture. Every MASA in a von Neumann algebra should have the app-paving property, equivalently the so-paving property; in particular, the equivalence between so-pavability and app-pavability should hold for arbitrary MASAs, including those that are not ranges of normal conditional expectations. Moreover, if $x\in M_{\mathrm{sa}}$ , let $\mathrm{n}_{\mathrm{s}}(x,\varepsilon)$ denote the so-paving size. Then there should be a universal constant $C>0$ , independent of $A\subset M$ , such that

\mathrm{n}_{\mathrm{s}}(x,\varepsilon)\leq C\varepsilon^{-2},\qquad \forall x\in M_{\mathrm{sa}}.

The first assertion extends the expected paving phenomenon beyond MASAs admitting normal conditional expectations, while the quantitative assertion predicts a uniform quadratic bound on paving size. These claims are presented as expectations in the paper and remain open.

References

Primary source

Sorin Popa and Stefaan Vaes, “Paving over arbitrary MASAs in von Neumann algebras”, arXiv:1412.0631 (2015).

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RemarkAI-assistedClaimed by OpenAI. For every self-adjoint element of a complex von Neumann algebra, claims approximation paving over every maximal abelian subalgebra with a universal O(epsilon^(-6)) projection bound. The norm-paved approximant converges strongly to the original element and has norm at most three times its norm; no separability or expectation assumption is required. The separate strong-paving consequence in this paper requires a normal conditional expectation.See full solutionHide full solution

Claimed by OpenAI. For every self-adjoint element of a complex von Neumann algebra, claims approximation paving over every maximal abelian subalgebra with a universal O(epsilon^(-6)) projection bound. The norm-paved approximant converges strongly to the original element and has norm at most three times its norm; no separability or expectation assumption is required. The separate strong-paving consequence in this paper requires a normal conditional expectation.

For every finite vector set F and delta>0, the approximant y is self-adjoint, norm(y)<=3 norm(x), and norm((y-x)xi)<delta for xi in F. Its norm-paving error is at most $\epsilon\|y\|$ , not an unrestricted norm-paving conclusion for x itself.

Scope relative to this problem: This manuscript claims app-paving over every MASA without separability or normal-expectation assumptions, with O(epsilon^-6) projection count, strongly converging approximants and norm at most 3 times the original. Its separate strong-paving consequence requires a normal conditional expectation. This does not by itself establish the target universal quadratic strong-paving bound or all asserted equivalences for nonexpected MASAs.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf

OpenAI-300-01-Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras.pdf572,733 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For every self-adjoint element, claims strong-operator paving over every maximal abelian subalgebra of a complex von Neumann algebra with a universal O(epsilon^(-2)) projection bound, without separability or conditional-expectation assumptions. Its norm-paving consequence in an Ocneanu ultrapower separately requires a countably decomposable masa and a normal conditional expectation.See full solutionHide full solution

Claimed by OpenAI. For every self-adjoint element, claims strong-operator paving over every maximal abelian subalgebra of a complex von Neumann algebra with a universal O(epsilon^(-2)) projection bound, without separability or conditional-expectation assumptions. Its norm-paving consequence in an Ocneanu ultrapower separately requires a countably decomposable masa and a normal conditional expectation.

The original self-adjoint x is paved after compression by a projection q whose complement is arbitrarily strongly small. Norm paving in the Ocneanu ultrapower is a separate corollary requiring a countably decomposable masa and a normal conditional expectation.

Scope relative to this problem: The manuscript claims strong-operator paving over every MASA with universal O(epsilon^-2) projection count, without separability or conditional expectation. This addresses the target quadratic strong-paving assertion. Its norm-paving consequence in an Ocneanu ultrapower separately requires a countably decomposable MASA and normal conditional expectation; no unrestricted norm-paving or every app/so equivalence is inferred.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf

OpenAI-300-02-Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras.pdf639,497 bytesOpen