The spatial Kadison-Kastler problem for close separable C*-algebras
Kadison and Kastler (1972) initiated the study of close operator algebras. For C*-algebras, Johnson showed implementing unitaries need not be close to the identity and Choi-Christensen gave close non-isomorphic algebras, but only non-separable ones. Christensen, Sinclair, Smith, White and Winter proved that sufficiently close separable nuclear C*-algebras on a separable Hilbert space are spatially isomorphic, and discussed the separable spatial formulation: is there a such that any two norm-separable C*-algebras on a separable Hilbert space with Kadison-Kastler distance below satisfy for some unitary ?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Operator algebras; perturbations of C*-algebras
- Posed by
- E. Christensen, A. M. Sinclair, R. R. Smith, S. A. White and W. Winter, The spatial isomorphism problem for close separable nuclear C*-algebras, PNAS 107 (2010)
- Year posed
- 2010
- Years open
- 16y
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 22 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every there are unital norm-separable C*-algebras on a separable Hilbert space with and but no unitary with . The construction uses sequence algebras over a deformation of the free-group regular representation (Pytlik-Szwarc family) and a tensor-norm obstruction. It does not show and are non-isomorphic, imposes no nuclearity (the CSSWW positive theorem needs it), and does not affect von Neumann algebras.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the separable spatial question as the paper states it. It claims, for every , unital norm-separable C*-algebras on a separable Hilbert space with common identity, Kadison-Kastler distance below and equal bicommutants, that are not unitarily conjugate. Proof not refereed; not formalized. The paper says it gives no abstract non-isomorphism and no von Neumann counterexample.