The Kadison-Kaplansky Projection Conjecture
For a torsion-free discrete group , let be its reduced group -algebra. The Kadison-Kaplansky conjecture asserts that contains no projections other than and . Pimsner and Voiculescu proved it for free groups, and it follows from surjectivity of the coefficient-free Baum-Connes map, so it holds for amenable and hyperbolic groups. Does every torsion-free discrete group have a projectionless reduced -algebra?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Operator algebras; group C*-algebras
- Posed by
- Richard Kadison, from Irving Kaplansky's question on idempotents in simple C*-algebras
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 50 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
There is a finitely generated torsion-free group and a projection with , so . It is built from a matrix projection with small unnormalized trace over a graphical small-cancellation group, compressed to a scalar projection by free products. Not shown: a counterexample to the algebraic Kaplansky idempotent conjecture (the group ring has no nontrivial idempotents), or a finitely presented group.
What the AI did
The release README says the 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, and that some outputs build on earlier model results. 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. The manuscript cites a further release manuscript (OpenAI, 2026, on Bass-type trace results) for the group ring having no nontrivial idempotents.
Verification
No independent mathematician has checked this yet. Checked here: the main theorem was read against the projection form of the conjecture as the manuscript states it. The construction was not refereed. No Lean formalization is listed. The paper separates this from the algebraic idempotent conjecture: the group ring of its group has no nontrivial idempotents, by a companion release result.