Isomorphism problem for free group factors

Determine whether the group von Neumann algebras L(𝔽_r) and L(𝔽_s) are isomorphic for r≠s, and more broadly classify isomorphism classes of free group factors.

Background

The paper highlights fundamental open questions in the structure theory of von Neumann algebras associated with free groups. Despite extensive paper, even basic isomorphism questions for free group factors remain unresolved.

This problem is central to the classification of II_1 factors and has deep connections with free probability, random matrices, and operator algebras.