We study upgraded free independence phenomena for unitary elements u 1 u₁, u 2, u₂, … representing the large- n n limit of Haar random unitaries, showing that free independence extends to several larger algebras containing u j u₉ in the ultraproduct of matrices ∏ n → U M n (C) ₍ ₔ M₍ (C). Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras P j P₉ containing u j u₉. The Pinsker algebra P j P₉ is the maximal subalgebra containing u j u₉ with vanishing 1 1 -bounded entropy (see Hayes Int. Math. Res. Not. IMRN 1 (2018), pp. 57–137) ; P j P₉ in particular contains the relative commutant <mml: math xmlns: mm
Jekel et al. (Thu,) studied this question.