{"id":"openai-math/An-isomorphism-of-the-free-group-factors","pdf":"https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026","tags":["operator-algebras","free-group-factors"],"score":0,"title":"An isomorphism of the free group factors","domain":"Operator algebras","family":"287","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.OA","summary":"We solve the free group factor isomorphism problem affirmatively by proving that $`L(\\mathbb F_2)`$ and $`L(\\mathbb F_3)`$ are isomorphic as tracial von Neumann algebras. The classical free group factor alternative then implies that all interpolated free group factors, including $`L(\\mathbb F_\\infty)`$, are isomorphic and have fundamental group $`\\mathbb R_{\\gt 0}`$.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=I%20got%20really%20impressed%20by%20the%20resolution%20of%20the%20free%20factor%20problem","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"I got really impressed by the resolution of the free factor problem (the von Neumann algebras of free groups are all isomorphic to each other), since I had been wrong about the expected outcome: I thought they rather should be non-isomorphic, since the parallel evidence from measured group theory…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"},{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=I%20was%20initially%20surprised%20when%20I%20saw%20the%20resolution%20to%20the%20free%20group%20factor%20problem","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"I was initially surprised when I saw the resolution to the free group factor problem. I thought that the free group factors ought to be non-isomorphic, and tried hard for a number of years to prove it in this direction. In any case, I am now fully convinced that theorems/proofs have almost…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"},{"url":"https://arxiv.org/abs/2610.19590","date":"2026-10-14","kind":"web","site":"arXiv.org","text":"Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{*}$-algebras generated by free semicircular systems: the isomorphism class of a $C^{*}$-algebra generated by a semicircular $n$-tuple does not depend on…","added":"2026-10-24","title":"On the Isomorphism of Semicircular Algebras","author":"D. Shlyakhtenko","record":"arxiv/2610.19590"}],"formalized":false,"family_title":"Isomorphism of the free group factors"}