Upgraded free independence phenomena for random unitaries

• Published in: arXiv.org 2024-06
• Main Authors: Jekel, David, Elayavalli, Srivatsav Kunnawalkam
• Format: Article
• Language: English
• Subjects: Volumetric analysis
• ISSN: 2331-8422

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.