We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices η= (η₈, ₉) ₈, ₉ ₈. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of η₈, ₉, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a B-valued semicircular family and generators of the base algebra B. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.
Jekel et al. (Tue,) studied this question.
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