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.

Manuscripts (2)

Approximation Paving over Arbitrary Maximal Abelian Subalgebras

September 25, 2026 · source · cite as \cite{OAI:Approximation-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-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.

Quadratic Strong-Operator Paving over Arbitrary Maximal Abelian Subalgebras

September 25, 2026 · source · cite as \cite{OAI:Quadratic-Strong-Operator-Paving-over-Arbitrary-Maximal-Abelian-Subalgebras-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.

Result 300 · Operator algebras. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.