Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: Voiculescu showed that independent Haar random unitaries $U_1^{(n)}$, $U_2^{(n)}$, \dots are almost surely asymptotically freely independent in the large-$n$ limit; thus, in particular, the normalized trace of any product $(U_{i_1}^{(n)})^{\pm 1} \dots (U_{i_k}^{(n)})^{\pm 1}$ of the $U_j^{(n)}$'s and their adjoints converges almost surely to zero unless the product reduces to the identity by cancellations, as a word in the free group. We show that the free independence relation passes from the $U_j^{(n)}$'s to other elements that commute with them. Namely, if $B_j^{(n)}$ is another random matrix such that $B_j^{(n)}$ and $U_j^{(n)}$ almost surely asymptotically commute as $n \to \infty$, then the elements $B_1^{(n)}$, $B_2^{(n)}$, \dots, are asymptotically freely independent. The proof is based on probabilistic/volume arguments akin to those in von Neumann's work on almost commuting matrices and Voiculescu's free entropy theory. We then describe how to formulate and generalize this result in the context of von Neumann algebras, using ultraproducts and free entropy theory. Our general result also recovers (important subcases) of the results of Houdayer and Ioana on approximate commutants in free products, and earlier results from my joint work with Hayes, Nelson, and Sinclair on maximal algebras with vanishing 1-bounded entropy in free products.

This is based on joint work with Srivatsav Kunnawalkam Elayavalli.

Prerequisites: Some knowledge of matrix algebras, operators on Hilbert space, and free groups.