The Free Group Factors Isomorphism Problem

Let \(L(F_n)\) be the group von Neumann algebra of the free group on \(n\) generators \((2 \leq n \leq \infty)\). Are \(L(F_m)\) and \(L(F_n)\) isomorphic for distinct finite \(m \neq n\)? (It is known that \(L(F_\infty)\) is not isomorphic to any \(L(F_n)\) with \(n\) finite.)

John von Neumann introduced group von Neumann algebras in the 1930s; the free group factors \(L(F_n)\) have been central examples ever since. The isomorphism question is folklore going back decades: all the standard invariants — the fundamental group, Voiculescu's free entropy dimension, Ozawa's solidity — agree on the \(L(F_n)\), so nothing known separates them. The 2020 refutation of the Connes embedding problem (MIP*=RE) reshaped the surrounding landscape without touching this question, and Voiculescu's free probability, built partly to attack it, became a field of its own.