An isomorphism of the free group factors
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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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…
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…
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100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
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…