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D. Voiculescu (2) proved that a standard family of independent random unitary k ×k matrices and a constant k ×k unitary matrix is asymptotically free as k → ∞. This result was a key ingredient in Voiculescu's proof (3) that his free entropy is additive when the variables are free. In this paper, we give a very elementary proof of a more detailed version of this result (2). We have not yet recaptured Voiculescu's strengthened version (4).

An elementary proof of Voiculescu's asymptotic freeness for random unitary matrices