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Math-2814 · mathematics · Analysis

Approximation paving for every MASA

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is there a universal C such that, for each $0<\varepsilon <1$, every self-adjoint x in a complex von Neumann algebra M and every MASA $A\subseteq M$ can be strongly approximated by self-adjoint y of norm $\leq 3||x||$, with a partition $(p_i)$ in A into at most $C\varepsilon ^-6$ pieces and a self-adjoint $a\in A$ of norm $\leq ||y||$ satisfying $||\sum p_iyp_i-a||\leq \varepsilon ||y||$? No separability or expectation is assumed.

Short English statement

Is there a universal C such that, for each $0<\varepsilon <1$, every self-adjoint x in a complex von Neumann algebra M and every MASA $A\subseteq M$ can be strongly approximated by self-adjoint y of norm $\leq 3||x||$, with a partition $(p_i)$ in A into at most $C\varepsilon ^-6$ pieces and a self-adjoint $a\in A$ of norm $\leq ||y||$ satisfying $||\sum p_iyp_i-a||\leq \varepsilon ||y||$? No separability or expectation is assumed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2814-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2814

Manuscript 1 (family 300, 2026-09-25): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Approximation paving for every MASA

OpenAI's claim

Every self-adjoint element is claimed to admit strong approximants bounded by three times its norm, norm-pavable over any MASA with O($\varepsilon^{-6}$) pieces and error relative to each approximant's norm. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $0<\varepsilon<1$, OpenAI claims an integer $R(\varepsilon)\le C_{\mathrm{ap}}\varepsilon^{-6}$ for a universal constant, working for every von Neumann algebra $M\subset B(H)$, maximal abelian $A\subset M$, self-adjoint $x\in M$, finite $F\subset H$ and $\delta>0$. There are self-adjoint $y\in M$, $a\in A$ and a unit partition of $R(\varepsilon)$ projections $p_i\in A$ such that \[ \|y\|\le3\|x\|,\quad\|(y-x)\xi\|<\delta\ (\xi\in F),\quad \|a\|\le\|y\|,\quad\left\|\sum_ip_iyp_i-a\right\|\le\varepsilon\|y\|. \] Zero partition projections are allowed. This is paving of a strong-operator approximation $y$, not norm paving of the original $x$; no expectation hypothesis is imposed.

Outline of the argument

Manuscript 1 (family 300, 2026-09-25): Buffered determinant polynomials give measurable one-sided matrix pavings. For the normalizer algebra these transfer through measured equivalence relations, while random diagonal phases treat diffuse complementary kernels. Separable reduction and a three-corner assembly remove the expectation hypothesis; norming vectors and clipping retain the required normalization for the actual approximants.

Formal verification

Family 300 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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