∑math20

Operator algebras · Result 300

Approximation and quadratic strong-operator paving

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

First page of Approximation Paving over Arbitrary Maximal Abelian Subalgebras

Approximation Paving over Arbitrary Maximal Abelian Subalgebras

OpenAI · September 25, 2026

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.

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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}
}
First page of Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

OpenAI · September 25, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.