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Math-2391 · mathematics · Algebra

Rigidity of Completely Close $C^*$-Algebras

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For $C^*$-algebras $A,B\subseteq B(H)$, what structural properties and invariants are forced by small completely bounded Kadison–Kastler distance $d_{\mathrm{cb}}(A,B)$, and must sufficiently close algebras be isomorphic?

Formulation references: [S1], [S2].

Short English statement

For $C^*$-algebras $A,B\subseteq B(H)$, what structural properties and invariants are forced by small completely bounded Kadison–Kastler distance $d_{\mathrm{cb}}(A,B)$, and must sufficiently close algebras be isomorphic?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2391-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2391

Manuscript 1 (family 289, 2026-09-23): The primary manuscript establishes uniform near-identity unitary conjugacy of sufficiently two-sided close unital von Neumann algebras. The theorem gives spatial isomorphism and near-identity conjugacy for the von Neumann, common-identity subcase. This problem also quantifies over general C*-algebras and asks about their structural invariants at completely bounded distance; that full class is not covered.

Rigidity of Completely Close \$C^*\$-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. 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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