Isomorphisms of free group factors

OpenAI's theorem (2026) that the free group factors L(F_n), 2 ≤ n < ∞, are all isomorphic. The first part proves OpenAI's rank step L(F_n) ≅ L(F_{n+1}) for n ≥ 3: free independence and freely generating Haar tuples, a bounded logarithm of the extra generator and a norm bound for free sums that does not depend on the number of summands; trace-preserving polynomial flows with displacement bounds; a small cocycle with a prescribed word value, through free prefixes, Catalan moments and cutoff inversion; and a limit with adaptive error budgets whose tuple generates the whole factor. The second part proves Voiculescu's corner formula, that the corner of L(F_n) cut down by a projection of trace ½ is L(F_{4n−3}), and deduces L(F_2) ≅ L(F_3): the semicircle law, a freeness criterion on the full Fock space, free semicircular families, 2×2 matrices with semicircular and circular entries that are free from matrices over the other variables, and the polar decomposition of a circular element. The third part extends the corner formula to every trace 1/k, with k×k semicircular matrices, and shows that every positive rational number lies in the fundamental group of L(F_2).

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

1. Free independence and Haar tuples
2. Trace-preserving polynomial flows
3. A small cocycle with a prescribed word value
4. Absorbing a free generator
5. Semicircular elements on the full Fock space
6. Matrices with semicircular and circular entries
7. Polar decomposition of a circular element
8. Corners of free group factors
9. Corners of rational trace

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Cauchy's theorem for cycles and its consequences · Foundations of von Neumann algebras
• Kaplansky's density theorem and its consequences · Foundations of von Neumann algebras
• The double commutant theorem · Foundations of von Neumann algebras
• The spectral theorem for bounded self-adjoint operators · Foundations of von Neumann algebras
• The Stone–Weierstrass theorem for functions vanishing at infinity · Open Mathematics Courses (Function algebras and approximation)
• Haar unitaries, freeness and tracial ultraproducts (Generators of II₁ factors)
• Classification of injective factors · Injective factors
• Integration for a trace, the commutation theorem, and applications · Traces and noncommutative integration
• Haar unitaries and projections in II₁ factors (course Generators of II₁ factors); group von Neumann algebras of ICC groups (course Injective factors); the double commutant and Kaplansky density theorems, the polar decomposition, the spectral theorem and Banach-valued Riemann integrals (Foundations of von Neumann algebras); the trace-preserving conditional expectation (course Traces and noncommutative integration); the Stone–Weierstrass and Weierstrass approximation theorems (course Function algebras and approximation); Taylor's theorem (core course Real Analysis I).