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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

Motivated by operator algebraic questions, von Neumann introduced in 1936 the class of (von Neumann) regular rings as a suitable algebraic analogue for von Neumann algebras.  The underlying idea behind this introduction was that solving the regular ring translation of a von Neumann algebraic problem can provide significant insight on how to proceed in the analytic case. Von Neumann's idea has been extremely fruitful, and has produced several important results both in operator algebras and homological algebra. From a C*-algebraist perspective, regular rings can also be used as test cases in the study of real rank zero C*-algebras (which, in a very concrete sense, are "von Neumann-like").

In this talk I will give an introduction to the so-called "realization problem", a long-standing open problem in the theory of regular rings. In an attempt to hide that this is a homological algebra talk, I will explain how the problem relates to well-known problems for real rank zero C*-algebras that date back to Rørdam's famous examples. Having explained the current state of the realization problem, I will discuss some new results that are part of ongoing joint work with Pere Ara, Joan Bosa, Laurent Cantier and Enrique Pardo.

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