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Kadison's Similarity Problem

Theorem 1.1: for every unital $C^*$ -algebra $A$ , every complex Hilbert space $H$ and every bounded complex-linear unital homomorphism $\pi:A\to B(H)$ there is a bounded invertible $S$ with $S\pi(\cdot)S^{-1}$ a $*$ -homomorphism. It goes through Theorem 1.2, an absolute-constant commutator bound $\|[Y^{(h)},X]\|\le C\,g_P(Y)\|X\|$ over all von Neumann algebras $P$ , all $Y$ and all matrix amplifications, combined with Kirchberg's derivation criterion. Corollary 1.3: every von Neumann algebra is hyperreflexive with constant at most $2C$ , a statement known to be equivalent to a positive answer (Eleftherakis-Paulsen). No bound on the condition number of $S$ and no value of $C$ is given.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Kadison's Similarity Problem · Kadison similarity problem

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
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Evidence vector

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Named evidence

VibeMathed record: Kadison's Similarity Problem

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the abstract, introduction, Theorem 1.1, Theorem 1.2 and Corollary 1.3, read against Kadison's 1955 problem as cited. The proof was not refereed. The theorem is unrestricted: no separability, nuclearity or normality assumption, arbitrary Hilbert space. Lean-checked on the release's own Comparator challenge KadisonSimilarity together with its solution module, both present at the pinned commit; the challenge is not listed in the release's formalization catalogue, the statement was read here but not independently audited, and the development was not rebuilt here.

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Kadison's Similarity Problem

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Kadison 1955, Amer. J. Math. 77

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Lean Comparator challenge KadisonSimilarity (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.