Browse › Operator algebras › 300 Approximation and quadratic strong-operator paving
Approximation Paving over Arbitrary Maximal Abelian Subalgebras
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove the approximation-paving conjecture of Popa and Vaes for every maximal abelian subalgebra of a complex von Neumann algebra. For each $0\lt \varepsilon\lt 1$, every self-adjoint operator is a strong limit of self-adjoint operators of norm at most three times its norm, each admitting a norm paving with error at most ε times the approximant's own norm. The number of projections is at most $C\varepsilon^{-6}$ for a universal constant C. No separability or conditional-expectation hypothesis is required.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
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]
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
- Emergent Mind: family 300
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026,
author = {{OpenAI}},
title = {{Approximation Paving over Arbitrary Maximal Abelian Subalgebras}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026/Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-September-25-2026.pdf}{OAI:Approximation-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:
- Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras · September 25, 2026