Kadison–Kaplansky projection conjecture for reduced group C*-algebras
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The question is whether a finitely generated torsion-free discrete group has a scalar projection e in its reduced group C*-algebra whose canonical trace lies strictly between zero and one-half.
Why it matters and what counts as progress
Why it matters: A nontrivial projection would show that torsion-freeness alone does not force projection rigidity in a reduced group C*-algebra.
What counts as progress: A nontrivial scalar projection in a reduced group C*-algebra of a torsion-free group, or a proof excluding all such projections.
How the problem developed
- 2022 · Partial result [1]
Johan Öinert and Stefan Wagner Published criteria for the Kadison–Kaplansky conjecture on group C*-algebras of certain central extensions involving amenable groups.
- 23 Sept 2026 · Counterexample claim [2]
OpenAI Released a manuscript claiming a finitely generated torsion-free discrete group whose reduced group C*-algebra contains a scalar projection with trace strictly between zero and one-half.
AI contribution
A torsion-free counterexample to the Kadison–Kaplansky projection conjecture
A finitely generated torsion-free discrete group has a scalar projection e in its reduced group C*-algebra whose canonical trace lies strictly between zero and one-half.
Sources
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Publication dates and editorial changes.
OpenAI reports a torsion-free Kadison–Kaplansky counterexample