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Math-2379 · mathematics · Algebra

Kadison–Kastler Conjecture for Separable $C^*$-Algebras

Status Partially solved · unsolvedmath.comLLM status Open unverified

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 a separable Hilbert space $H$, does there exist $\delta>0$ such that every pair of separable $C^*$-algebras $A,B\subset B(H)$ with $d(A,B)<\delta$ are spatially isomorphic?

Formulation references: [S1], [S2].

Short English statement

For a separable Hilbert space $H$, does there exist $\delta>0$ such that every pair of separable $C^*$-algebras $A,B\subset B(H)$ with $d(A,B)<\delta$ are spatially isomorphic?

Sources

Catalogue entry

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

Contributions 1

Math-2379-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2379

Manuscript 1 (family 289, 2026-10-05): For every epsilon the paper constructs separable C*-algebras at two-sided norm distance below epsilon with no implementing unitary at all, directly refuting the universal spatial C*-algebra Kadison–Kastler assertion.

Manuscript 2 (family 289, 2026-09-23): The theorem gives small-unitary conjugacy for the von Neumann algebra subclass; the problem quantifies over arbitrary separable norm-closed C*-algebras, whose weak closures do not determine their norm-closed subalgebras.

Kadison–Kastler Conjecture for Separable \$C^*\$-Algebras

OpenAI's claim

Manuscript 1 (family 289, 2026-10-05): Arbitrarily close, norm-separable unital C*-algebras with a shared von Neumann closure are claimed to exist without any unitary conjugacy, on one separable complex Hilbert space. Manuscript 2 (family 289, 2026-09-23): 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.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $\varepsilon>0$, OpenAI claims a separable complex Hilbert space $H$ and norm-separable unital $C^*$-algebras $A,B\subset B(H)$ with the same identity such that \[ d_{\mathrm{KK}}(A,B)<\varepsilon,\qquad A''=B'',\qquad u Au^*\ne B\text{ for every unitary }u\in B(H). \] Here $d_{\mathrm{KK}}$ is the Hausdorff distance between norm unit balls, and $A'',B''$ are von Neumann closures. This is the norm-separable $C^*$-algebra spatial-conjugacy assertion; it does not contradict the separate von Neumann theorem.
Mathematical claim 2.
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, 2026-10-05): Close free-group deformations yield two inclusions distinguished by a relative tensor-norm identity. A convergent-sequence algebra turns this positional difference into an invariant under every spatial conjugacy: central coordinate projections force permutations and coordinatewise normalizers. The tensor identity survives such normalizers and finite-family norm limits, ruling out conjugacy despite the small gap.

Manuscript 2 (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. Manuscript 1 is not among the papers selected by that page, so the family link does not formalize its main conclusion. It does not establish formal coverage of the complete problem clause. The documented statement covers the main claim of manuscript 2. It does not establish formal coverage of the complete problem clause. 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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