Operator Theory Seminar

Feb 25, 2020

01:30 PM - 02:30 PM

Van Allen Hall, 309

30 North Dubuque Street, North Liberty, IA 52317

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Assistant Professor Brent Nelson; Department of Mathematics; Michigan State University 

“A random matrix approach to absorption in free products” 

Abstract: Consider a tracial free product von Neumann algebra $\mathcal{M}=\mathcal{P}*\mathcal{Q}$, where $P$ and $Q$ can be embedded into $\mathcal{R}^\omega$. Using random matrix theory, one can show that $\mathcal{P}$ contains any von Neumann subalgebra $\mathcal{N}\subset\mathcal{M}$ which intersects $\mathcal{P}$ diffusely and has 1-bounded entropy zero. Here the 1-bounded entropy of $\mathcal{N}$ is a modification of the free entropy dimension that—roughly speaking—measures how many matrix approximations $\mathcal{N}$ 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, some aspects of its proof, and a few of its consequences. This is based on joint work with Ben Hayes, David Jekel, and Thomas Sinclair. 

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