Upgraded free independence phenomena for random unitaries

We study upgraded free independence phenomena for unitary elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 1"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">u_{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 2 comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u_{2},</mml:annotation> </mml:semantics> </mml:math> </inline-formula> … representing the large- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> limit of Haar random unitaries, showing that free independence extends to several larger algebras containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript j"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">u_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the ultraproduct of matrices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript n right-arrow script upper U Endscripts upper M Subscript n Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∏ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">U</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\prod _{n \to \mathcal {U}} M_{n}(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Using a uniform asymptotic freeness argument and volumetric analysis, we prove free independence of the Pinsker algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript j"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {P}_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript j"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">u_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The Pinsker algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript j"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {P}_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the maximal subalgebra containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript j"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">u_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with vanishing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bounded entropy (see Hayes [Int. Math. Res. Not. IMRN 1 (2018), pp. 57–137]); <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript j"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {P}_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in particular contains the relative commutant <inline-formula content-type="math/mathml"> <mml:math xmlns:mm