A projection of irrational trace and the Baum–Connes conjecture

The reduced group C*-algebra of a countable group carries a canonical trace, and the Baum-Connes conjecture predicts its K-theory from proper actions. By a theorem of Lück every predicted class has a rational trace. OpenAI constructed in 2026 a finitely generated group with a projection in its reduced group C*-algebra whose trace is the irrational number sum p^(-k_i), so the reduced Baum-Connes conjecture without coefficients fails. The group is a semidirect product of a graphical small cancellation group, whose Cayley graph contains labelled expanders isometrically, with a module over F_p defined by linear codes on the copies of the expanders. A Fourier transform turns a Laplacian-type element into random subgraph Laplacians with a uniform spectral gap, and its kernel projection counts the copies whose random pattern is the prototype. The course is an exposition of OpenAI's proof (2026).

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Traces of projections and the Baum–Connes conjecture
2. Graphical presentations and coned diagrams
3. Thin triangles and simultaneous overlaps
4. Labelled expanders and cycle spaces
5. Codes and the lamp module
6. Fourier decomposition and enabled graphs
7. The spectral gap and an irrational trace

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
• Thin triangles, overlap balls and the residual forest (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• The regular trace and spectral phase (The Kervaire conjecture)
• Plane maps (Van Kampen diagrams)
• Van Kampen's lemma (Van Kampen diagrams)
• C*-algebras, the continuous functional calculus and spectral permanence (course Foundations of von Neumann algebras).
• The regular trace of a group (course The Kervaire conjecture, first lesson).
• Infinite products of probability measures (core course Measure and Integration).
• Graphs, Cayley graphs and group presentations (recalled in the second lesson).

Used in

• The reduced assembly map and the torus class (A torsion-free counterexample to reduced Baum–Connes injectivity)