Von Neumann’s problem for II-1 factors

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 $\mathbb{F}_2$ . 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 $\mathbb{F}_2$ .

The group von Neumann algebra $L(G)$ is the von Neumann subalgebra of bounded operators on $\ell^2(G)$ generated by the left regular representation of $G$ on $\ell^2(G)$ . There is a notion of amenability for von Neumann algebras, and Connes showed that $G$ is amenable if and only if $L(G)$ is. Further, if $H$ is a subgroup of $G$ , then $L(H)$ naturally is a sub-von Neumann algebra of $L(G)$ . Thus, if $G$ contains $\mathbb{F}_2$ , then $L(G)$ contains the free group factor $L(\mathbb{F}_2)$ as a sub-von Neumann algebra. The analog of von Neumann’s problem in this setting remains open:

Question: If $G$ is a nonamenable group, does $L(G)$ contain $L(\mathbb{F}_2)$ ?

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

Question: If $G$ is a nonamenable group, does the reduced group C*-algebra $C^*_{\text{red}}(G)$ contain $C^*_{\text{red}}(\mathbb{F}_2)$ ?

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 $10^{10^{10}}}$ 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.

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