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Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras
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Abstract
We prove the quadratic strong-operator paving conjecture of Popa and Vaes. For every $0\lt \varepsilon \lt 1$, every self-adjoint element of a von Neumann algebra admits strong-operator paving over each maximal abelian subalgebra with at most $5\times10^8\varepsilon ^{-2}$ projections. The bound is uniform over representations and requires no separability or conditional-expectation assumption.
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Related prior literature mentioned by others
- Sorin Popa and Stefaan Vaes, Paving over arbitrary MASAs in von Neumann algebras (arXiv:1412.0631; Analysis & PDE 8, 2015). The AI Proved What? episode identifies this as the source of the paving conjectures addressed by both manuscripts and explains its earlier results for special classes of MASAs. [family-level]
- Mohan Ravichandran and Nikhil Srivastava, Asymptotically Optimal Multi-Paving (arXiv:1706.03737, 2017). The AI Proved What? episode discusses this finite-dimensional quadratic paving result as an input to the strong-operator argument and compares its block bounds with those claimed in the release. [family-level]
Possibly relevant literature (found by the agent, not checked)
- S. Popa, On the paving size of a subfactor (arXiv:2210.04396, 2022; to appear in Pure Appl. Math. Q. per arXiv comments). Proves paving bounds of order ε^{-2} (times a factor ⌈4[M:N]ε^{-2}⌉) for finite-index II_1 subfactors and defines "paving size" invariants, relevant to the paper's discussion of ε^{-2} paving counts.
Digests and expositions
- Latent Seminar, Paving arbitrary MASAs: strong topology and quadratic bounds (7 October 2026). This exposition compares the two paving statements and outlines the normalizer decomposition, measured coloring, moment transfer and approximation construction, explicitly limiting its review to selected proof mechanisms. [family-level]
- AI Proved What?, Every MASA, quadratically? The Popa–Vaes paving conjecture (episode 300, October 2026). An animated introduction with a written transcript explains the distinction between approximation and strong-operator paving, the claimed block counts and their historical background. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 300
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026,
author = {{OpenAI}},
title = {{Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/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}{OAI:Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026}},
year = {2026}
}Family 300: Approximation and quadratic strong-operator paving
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with $O(\varepsilon^{-2})$ blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.
Other papers in this family:
- Approximation Paving over Arbitrary Maximal Abelian Subalgebras · September 25, 2026