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 × N matrices with entries in a unital C*-algebra $${\mathcal B}$$ which have limiting $${\mathcal B}$$ -valued distributions as N → ∞, and U N is a N × 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 result may fail for classical Haar unitary random matrices when the algebra $${\mathcal B}$$ is infinite-dimensional.