In the 1920s, von Neumann introduced the notion of amenability for groups, and he showed that a group is nonamenable whenever it contains the free group . The question of whether this characterizes (non)amenability became known as von Neumann’s problem, and it was finally answered negatively by Olshanskii in 1980: There exist nonamenable groups that do contain no subgroup isomorphic to .

The group von Neumann algebra is the von Neumann subalgebra of bounded operators on generated by the left regular representation of on . There is a notion of amenability for von Neumann algebras, and Connes showed that is amenable if and only if is. Further, if is a subgroup of , then naturally is a sub-von Neumann algebra of . Thus, if contains , then contains the free group factor as a sub-von Neumann algebra. The analog of von Neumann’s problem in this setting remains open:

Question:If is a nonamenable group, does contain ?

The C*-algebraic version of von Neumann’s problem also remains open:

Question:If is a nonamenable group, does the reduced group C*-algebra contain ?

**Background:** A (discrete) group is said to be *amenable* if it admits a finitely additive, left invariant probability measure. To quote Brown-Ozawa, there are approximately different characterizations of amenability for groups [1].

- [1]N. Brown, N. Ozawa, 𝐶*-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics. (2008). https://doi.org/10.1090/gsm/088.