Von Neumann’s problem for II-1 factors

Von | Dezember 8, 2023

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

Schreibe einen Kommentar

Deine E-Mail-Adresse wird nicht veröffentlicht. Erforderliche Felder sind mit * markiert