Operator-Valued Matrices with Free or Exchangeable Entries
Résumé
We study matrices whose entries are free or exchangeable noncommutative elements in some tracial $W^*$-probability space. More precisely, we consider operator-valued Wigner and Wishart matrices and give for the associated Cauchy transforms explicit rates of convergence to operator-valued semicircular elements over some subalgebra. Applications to block random matrices with a Wigner or circulant structure and in independent or correlated blocks are also given. Our approach relies on a noncommutative extension of the Lindeberg method and Gaussian interpolation techniques.