Upgraded free independence phenomena for random unitaries

We study upgraded free independence phenomena for unitary elements $u_1$, $u_2$, \dots representing the large-$n$ limit of Haar random unitaries, showing that free independence extends to several larger algebras containing $u_j$ in the ultraproduct of matrices $\prod_{n \to \mathcal{U}} M_n(\mathbb{C})$. Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras $\mathcal{P}_j$ containing $u_j$. The Pinsker algebra $\mathcal{P}_j$ is the maximal subalgebra containing $u_j$ with vanishing $1$-bounded entropy defined by Hayes; $\mathcal{P}_j$ in particular contains the relative commutant $\{u_j\}' \cap \prod_{n \to \mathcal{U}} M_n(\mathbb{C})$, more generally any unitary that can be connected to $u_j$ by a sequence of commuting pairs of Haar unitaries, and any unitary $v$ such that $v\mathcal{P}_j v^* \cap \mathcal{P}_j$ is diffuse. 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 (in the Connes-embeddable case) a generalization and a new proof of Houdayer--Ioana's results on free independence of approximate commutants. It also yields a new proof of the general absorption results for Connes-embeddable free products obtained by the first author, Hayes, Nelson, and Sinclair.