Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

TL;DR

This paper proves that certain large random matrices with entries in a unital C*-algebra become asymptotically free with amalgamation over the algebra when conjugated by Haar quantum unitary matrices, extending to an infinitesimal setting.

Contribution

It establishes asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary matrices, generalizing previous results and highlighting differences from classical Haar unitaries.

Findings

Asymptotic freeness with amalgamation over al for quantum unitaries

Extension to infinitesimal freeness of Belinschi-Shlyakhtenko

Counterexample showing failure for classical Haar unitaries in infinite-dimensional case

Abstract

We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N \times N$ matrices with entries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcal B$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haar distributed quantum unitary random matrix with entries independent from $\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ and $B_N$ are asymptotically free with amalgamation over $\mathcal B$. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko. We provide an example which demonstrates that this example may fail for classical Haar unitary random matrices when the algebra $\mathcal B$ is infinite-dimensional.