ROOM THREE: A random matrix approach to absorption in free products
Consider a tracial free product von Neumann algebra M=P∗Q, where P and Q can be embedded into Rω. Using random matrix theory, one can show that P contains any von Neumann subalgebra N⊂M which intersects P diffusely and has 1-bounded entropy zero. Here the 1-bounded entropy of N is a modification of the free entropy dimension that—roughly speaking—measures how many matrix approximations N has. This quantity is known to be a von Neumann algebra invariant and in particular is zero for all amenable algebras. Consequently, as a corollary of the aforementioned absorption result, one obtains a novel proof of Popa’s famous theorem that the generator MASA in a free group factor is maximal amenable. In this talk I will discuss this result and some aspects of its proof. This is based on joint work with Ben Hayes, David Jekel, and Thomas Sinclair.