Upgraded free independence phenomena for random unitaries

We study upgraded free independence phenomena for unitary elements u₁, u₂, 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 Aⱼ containing uⱼ, 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 ⱼ\'∩ ∏n→ UMₙ(C) in the matrix ultraproduct are freely independent. We then prove free independence of the entire Pinsker algebras Pⱼ containing uⱼ; Pⱼ by definition is the maximal subalgebra containing uⱼ with vanishing $1$-bounded entropy in the sense of Hayes, and Pⱼ contains for instance any amenable algebra containing uⱼ as well as the entire sequential commutation orbit of uⱼ, and it is closed under taking iterated wq-normalizers. Through an embedding argument, we go back and deduce analogous free independence results for MU when M is a free product of Connes embeddable tracial von Neumann algebras Mᵢ, which thus yields a generalization and a new proof of Houdayer--Ioana's results in this case.