A torsion-free counterexample to the Kadison–Kaplansky conjecture
Kadison and Kaplansky conjectured that the reduced group C*-algebra of a torsion-free group contains no projections other than 0 and 1. OpenAI constructed in 2026 a finitely generated torsion-free group with a projection whose canonical trace lies strictly between 0 and 1/2. The group is defined by the closed walks of a voltage cover of an infinite layered expander, the tower, with voltages in a torsion-free nilpotent group; a weighted adjacency matrix over its group ring has a separated spectral band of small trace, and compression after a free product gives the scalar projection. The tower is built from streams and gated permutations and pruned so that long paths are recovered from their words; the voltages realize a model operator assembled from Schwartz idempotents of twisted group algebras, with one random draw per edge. The course is an exposition of OpenAI's proof (2026).
18 lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- The Kadison–Kaplansky conjecture and scalar compression
- Separated bands and equivariant traces
- Streams, slots and slot evaluations
- Gated permutations and the two towers
- The mean incidence and its gap
- Expansion of the realized incidence
- Pruning and recovery of long paths
- Voltage covers and embedded copies
- Thin triangles, overlap balls and the residual forest
- Torsion-freeness
- A word detector in unitriangular matrices
- An affine nilpotent group
- Schwartz idempotents and lattice projections
- The magnetic factor and the weighted trace
- The model operator and finite tests
- Masks, lists and one draw per edge
- Girth and centered moments
- From the cover band to the Cayley projection
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Idempotents and integral traces (The Bass trace conjecture for group rings)
- Order, local units and quotients of C*-algebras · Foundations of von Neumann algebras
- The spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
- Hermite functions, tempered distributions and the Schwartz kernel theorem · Open Mathematics Courses (Index theory of elliptic operators — Hermite and Schwartz tools)
- Graphical presentations and coned diagrams (A projection of irrational trace and the Baum–Connes conjecture)
- Traces of projections and the Baum–Connes conjecture (A projection of irrational trace and the Baum–Connes conjecture)
- Directly finite group algebras (Kaplansky's direct-finiteness conjecture)
- Quasitraces and a uniform additivity defect (Kaplansky's quasitrace problem)
- Group algebras and zero divisors (A torsion-free group algebra with zero divisors)
- Plane maps (Van Kampen diagrams)
- Van Kampen's lemma (Van Kampen diagrams)
- C*-algebras, the continuous functional calculus and the spectral theorem (course Foundations of von Neumann algebras).
- Traces of projections and graphical presentations (course A projection of irrational trace and the Baum–Connes conjecture).
- Hermite functions and the Schwartz space (course Index theory of elliptic operators).
- Graphs, walks and Cayley graphs; basic probability with independent random variables.