Abstract:

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(F₂), is isomorphic to the group von Neumann algebra of the free group on infinitely many generators, denoted by L(F_∞). Recently, S. Popa established a roadmap for showing the nonisomorphism of L(F₂) and L(F_∞). The first step of the proposed roadmap is to establish the so-called mean value property (abbreviated MV-property) for L(F₂).

In this talk I shall describe the proof of the result that L(F₂) has the MV-property, thereby establishing the first step of Popa's roadmap. This talk is based on a recent joint work with Prof. Jesse Peterson.