Leibniz Seminorms for “matrix Algebras Converge to the Sphere”

In an earlier paper of mine relating vector bundles and Gromov–Hausdorff distance for ordinary compact metric spaces, it was crucial that the Lipschitz seminorms from the metrics satisfy a strong Leibniz property. In the present paper, for the now noncommutative situation of matrix algebras converging to the sphere (or to other spaces) for quantum Gromov–Hausdorff distance, we show how to construct suitable seminorms that also satisfy the strong Leibniz property. This is in preparation for making precise certain statements in the literature of high-energy physics concerning “vector bundles” over matrix algebras that “correspond” to monopole bundles over the sphere. We show that a fairly general source of seminorms that satisfy the strong Leibniz property consists of derivations into normed bimodules. For matrix algebras our main technical tools are coherent states and Berezin symbols.