Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

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.Comment: 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 figure