Kadison's Similarity Problem
Kadison (1955) asked whether every bounded unital homomorphism from a unital -algebra into the bounded operators on a Hilbert space is similar to a -homomorphism, that is, whether there is a bounded invertible such that is a -homomorphism. Known cases included nuclear algebras (Bunce, Christensen), cyclic representations and algebras without tracial states (Haagerup) and II factors with property (Christensen). Haagerup showed similarity is equivalent to complete boundedness, and Kirchberg showed the problem is equivalent to the derivation problem. Is every bounded representation of a -algebra on Hilbert space similar to a -representation?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Operator algebras
- Posed by
- Richard V. Kadison, On the orthogonalization of operator representations, Amer. J. Math. 77 (1955)
- Year posed
- 1955
- Years open
- 71y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 58 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every unital -algebra , every complex Hilbert space and every bounded complex-linear unital homomorphism there is a bounded invertible with a -homomorphism. It goes through Theorem 1.2, an absolute-constant commutator bound over all von Neumann algebras , all and all matrix amplifications, combined with Kirchberg's derivation criterion. Corollary 1.3: every von Neumann algebra is hyperreflexive with constant at most , a statement known to be equivalent to a positive answer (Eleftherakis-Paulsen). No bound on the condition number of and no value of is given.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
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.