Matricial bridges for "Matrix algebras converge to the sphere"

In the high-energy quantum-physics literature one finds statements such as "matrix algebras converge to the sphere". Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. In the present paper, as preparation of discussing similar statements for convergence of "vector bundles" over matrix algebras to vector bundles over spaces, we introduce and study suitable matrix-norms for matrix algebras and spaces. Very recently Latremoliere introduced an improved quantum Gromov-Hausdorff-type distance between quantum metric spaces. We use it throughout this paper. To facilitate the calculations we introduce and develop a general notion of "bridges with conditional expectations".