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Math-2836 · mathematics · Analysis

Universal strong Kadison–Kastler stability for von Neumann algebras

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $\varepsilon >0$, is there $\delta >0$ such that any two unital von Neumann algebras on the same complex Hilbert space, with the same identity and two-sided unit-ball Hausdorff distance below $\delta $, are conjugate by a unitary u satisfying $||u-1||<\varepsilon $? The bound must be independent of both algebras and of the Hilbert space.

Short English statement

For every $\varepsilon >0$, is there $\delta >0$ such that any two unital von Neumann algebras on the same complex Hilbert space, with the same identity and two-sided unit-ball Hausdorff distance below $\delta $, are conjugate by a unitary u satisfying $||u-1||<\varepsilon $? The bound must be independent of both algebras and of the Hilbert space.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2836-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2836

Manuscript 1 (family 289, 2026-09-23): 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.

Universal strong Kadison–Kastler stability for von Neumann algebras

OpenAI's claim

For any prescribed small unitary displacement, the paper claims one uniform Kadison--Kastler tolerance guaranteeing conjugacy of close unital von Neumann algebras on any complex Hilbert space. 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 $\varepsilon>0$, OpenAI claims a single $\delta(\varepsilon)>0$ valid for every complex Hilbert space $H$ and all unital von Neumann algebras $M,N\subset B(H)$ with common identity: if their norm-unit-ball Hausdorff distance $d(M,N)<\delta$, then \[ \exists u\in B(H):\quad u^*u=uu^*=I_H,\quad uMu^*=N,\quad\|u-I_H\|<\varepsilon. \] No separability, type or injectivity hypothesis is imposed on these von Neumann algebras.

Outline of the argument

Manuscript 1 (family 289, 2026-09-23): For finite factors, free-word estimates control amplification while boundary averaging corrects nearby unitaries into a measurable cocycle. Double ergodicity makes the resulting correspondence multiplicative and yields a spatial implementation. Properly infinite algebras are handled through matrix amplification and modular preparation; alignment of centers and limits over separable restrictions assemble the universal conclusion.

Formal verification

For family 289, the scope page documents the following selections. The selection gives a tolerance depending only on the requested distance of an implementing unitary from identity, for two close common-identity von Neumann algebras. The separate one-way near-inclusion and general norm-closed examples are outside it. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: StrongKadisonKastler.lean. Declarations: OAI.KadisonKastler.universal_strong_stability.

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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