Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

Details Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

2009-11-30

arxiv.org/abs/0912.0025v2

Mathematics

Operator Algebras

Added reference [13], and replaced Lemma 3.7 by a stronger result from that paper. Minor change to the statement of Theorem 4.

Scientific paper

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.

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