Upgraded free independence phenomena for random unitaries

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 uⱼ in the ultraproduct of matrices ∏ n → U M n ( C ) ∏ n → U Mₙ(C) . Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras P j Pⱼ containing u j uⱼ . The Pinsker algebra P j Pⱼ is the maximal subalgebra containing u j uⱼ with vanishing 1 1 -bounded entropy (see Hayes [Int. Math. Res. Not. IMRN 1 (2018), pp. 57–137]); P j Pⱼ in particular contains the relative commutant <mml:math xmlns:mm