Asymptotic Infinitesimal Freeness with Amalgamation for Haar Quantum Unitary Random Matrices

Abstract

We consider the limiting distribution of UNANUN* and BN (and more general expressions), where AN and BN are N × N matrices with entries in a unital C*-algebra B which have limiting B-valued distributions as N → ∞, and UN is a N × N Haar distributed quantum unitary random matrix with entries independent from B. Under a boundedness assumption, we show that UNANUN* and BN are asymptotically free with amalgamation over 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 B is infinite-dimensional. © 2010 The Author(s).