Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

Abstract

We consider the limiting distribution of $U_N{A}_N{U}_N^{\ast }$ and $B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N\times N$ matrices with entries in a unital C${}^{\ast }$-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 $U_N{A}_N{U}_N^{\ast }$ 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.

Added reference [13], and replaced Lemma 3.7 by a stronger result from that paper. Minor change to the statement of Theorem 4.6. 25 pages, 3 figures

References in corpus (3)

• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
• De Finetti theorems for easy quantum groups
• Infinitesimal non-crossing cumulants and free probability of type B