Ian Charlesworth, Cardiff University
"Permutation matrices, graph independence over the diagonal, and consequences"
Graph products were first introduced by Green in the context of groups, giving a mixture of direct and free products. They have recently been studied in the contexts of operator algebras and of non-commutative probability theory by M\l{}otkowski, Caspers and Fima, Speicher and Wysocza\'nski, and others. It is interesting to ask how properties of a family of von Neumann algebras are witnessed in a graph product; while free products and tensor products are well understood, their interplay can be quite subtle in this more general setting. With Collins, I showed how conjugation by random unitary matrices in a tensor product of matrix algebras creates asymptotic graph independence, when the unitaries are independent and uniformly distributed but only on particular subalgebras. In this talk, after spending some time introducing the setting, I will discuss how techniques inspired by the work of Au, C\'ebron, Dahlqvist, Gabriel, and Male can be used to make a similar statement about random permutations leading to asymptotic graph independence over the diagonal subalgebra; the combinatorial techniques required involve some interesting subtleties which are not apparent at first glance. I will also discuss some consequences for von Neumann algebras. For example, suppose that $(M_v)_v$ is a collection of finite dimensional algebras. $M_v$ can be embedded into a larger matrix algebra in such a way that it is constant on the diagonal, and the standard matrix units of $M_v$ are embedded as elements whose entries are roots of unity. Then if $M$ is a graph product of the $M_v$, we can find matricial approximations of a generating set which enjoy the same properties, and this in turn allows us to show (using techniques of Shlyakhtenko) that if the if $M$ is diffuse and algebra generated by the $M_v$ within $M$ has vanishing first $L^2$ Betti number then $M$ is strongly 1-bounded in the sense of Jung. This is joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson.
This seminar will be held both online and in person:
- Room: MC 5479
- Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09