All nonabelian free group factors are isomorphic at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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All nonabelian free group factors are isomorphic

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	infinite matrices	Levels:	1
Category:	Operator algebras	Lean version:	YES! ✔
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Isomorphism of the free group factors. Resolves the free group factor isomorphism problem: $L(\mathbb F_2)\cong L(\mathbb F_3)$, and hence all interpolated free group factors, including $L(\mathbb F_\infty)$, are isomorphic. Their common factor has fundamental group $\mathbb R_{\gt 0}$.

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LEVEL 1 An isomorphism of the free group factors

released 2026-09-23 | 2 theorems · 10 lemmas · 16 proofs · 9,471 words | PLAY LEVEL 1 » (pdf)

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 $\mathbb R_{\gt 0}$.

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