Strong convergence to operator-valued semicirculars

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices η= (ηi,j)i,j ∈ I. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of ηi,j, 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.