Given a compact metric space X and a unital C*-algebra A, we introduce a family of seminorms on the C*-algebra of continuous functions from X to A, denoted C(X, A), induced by classical Lipschitz seminorms that produce compact quantum metrics in the sense of Rieffel if and only if A is finite-dimensional. As a consequence, we are able isometrically embed X into the state space of C(X,A). Without the finite-dimensional assumption on A, we are still able show that all of these seminorms satisfy a Leibniz-type rule and have dense domains, and provide finite diameter for their associated Monge Kantorovich metrics. Finally, we apply these quantum metrics to provide convergence of families of homogeneous C*-algebras and finite-dimensional approximations in the quantum Gromov-Hausdorff propinquity of F. Latré-molière. (Joint work with Tristan Bice (IMPAN), arXiv: 1711.08846)
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