One of the most important outstanding problems in von Neumann algebras asks if the group von Neumann algebra of the free group on two generators, denoted by $L(F_2)$, is isomorphic to the group von Neumann algebra of the free group on infinitely many generators, denoted by $L(F_{\infty})$. Recently, S. Popa established a roadmap for showing the nonisomporshism of the aforementioned von Neumann algebras. The first step of this this roadmap is to establish the so called Mean Value property (abbreviated as MV-property) for $L(F_2)$.

In this talk, I shall describe the proof of the result that $L(F_2)$ has the MV-property, thereby establishing the first step of Popa's roadmap. I shall discuss Popa's roadmap in detail, and describe the MV-property, along with the proposed solution. This talk is based on a recent joint work with Prof. Jesse Peterson.