Title: Upgraded free independence phenomena for random unitaries

Abstract: We study upgraded free independence phenomena for unitary elements $u_1$, $u_2$, \dots in a matrix ultraproduct constructed from the large-$n$ limit of Haar random unitaries. Using a uniform asymptotic freeness argument and volumetric analysis, we establish freeness of several much larger algebras $\mathcal{A}_j$ containing $u_j$, which sheds new light on the structural properties of matrix ultraproducts, as well as free products of tracial von Neumann algebras. First, motivated by Houdayer and Ioana's results on free independence of approximate commutants in free products, we show that the commutants $\{u_j\}'\cap \prod_{n\to \mathcal{U}}\mathbb{M}_n(\mathbb{C})$ in the matrix ultraproduct are freely independent. We then prove free independence of the entire Pinsker algebras $\mathcal{P}_j$ containing $u_j$; $\mathcal{P}_j$ by definition is the maximal subalgebra containing $u_j$ with vanishing $1$-bounded entropy in the sense of Hayes, and $\mathcal{P}_j$ contains for instance any amenable algebra containing $u_j$ as well as the entire sequential commutation orbit of $u_j$, and it is closed under taking iterated wq-normalizers. Through an embedding argument, we go back and deduce analogous free independence results for $\mathcal{M}^{\mathcal{U}}$ when $\mathcal{M}$ is a free product of Connes embeddable tracial von Neumann algebras $\mathcal{M}_i$, which thus yields a generalization and a new proof of Houdayer--Ioana's results in this case.