Standard homogeneous C*-algebras as compact quantum metric spaces

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 by $C(X, A)$, 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 to isometrically embed $X$ into the state space of $C(X,A)$. Furthermore, we are able to extend convergence of compact metric spaces in the Gromov–Hausdorff distance to convergence of spaces of matrices over continuous functions on the associated compact metric spaces in Latrémolière’s Gromov–Hausdorff propinquity.