14.9 (1999)

(Well-known problem). Let $W^*(F_k)$ denote the von Neumann algebra of the free group of rank $k \in \{2, 3, \dots, \boldsymbol{\aleph}_0\}$. Is $W^*(F_k)$ isomorphic to $W^*(F_l)$ for $k \neq l$?

For a group $G$, recall that $W^*(G)$ is an appropriate completion of the group algebra $\mathbb{C}G$ (see e. g. S. Sakai, $C^*$-algebras and $W^*$-algebras, Springer, 1971, in particular Problem 4.4.44). It is known that either $W^*(F_k) \cong W^*(F_l)$ for all $k, l \in \{2, 3, \dots, \boldsymbol{\aleph}_0\}$, or that the $W^*(F_k)$ are pairwise non-isomorphic (F. Radulescu, Invent. Math., 115 (1994), 347–389, Corollary 4.7).