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.
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Acknowledgements
We would like to thank T. Banica, M. Neufang, and D. Shlyakhtenko for several useful discussions. S.C. would like to thank his thesis advisor, D.-V. Voiculescu, for his continued guidance and support while completing this project.
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Communicated by Y. Kawahigashi
Research supported by a Discovery grant from NSERC.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Curran, S., Speicher, R. Asymptotic Infinitesimal Freeness with Amalgamation for Haar Quantum Unitary Random Matrices. Commun. Math. Phys. 301, 627–659 (2011). https://doi.org/10.1007/s00220-010-1164-y
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DOI: https://doi.org/10.1007/s00220-010-1164-y