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Given a compact metric space X and a unital AF algebra A equipped with afaithful tracial state, we place quantum metrics on the tensor product of C(X)and A given established quantum metrics on C(X) and A from work with Bice andLatr\'emoli\`ere. We prove the inductive limit of C(X) tensor A given by A is ametric limit in the Gromov-Hausdorff propinquity. We show that our quantummetric is compatible with the tensor product by producing a Leibniz rule onelementary tensors and showing the diameter of our quantum metric on the tensorproduct is bounded above the diameter of the Cartesian product of the quantummetric spaces. We provide continuous families of C(X) tensor A which extendsour previous results with Latr\'emoli\`ere on UHF algebras.

Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra

Quantum metrics on the tensor product of a commutative C-algebra and an AF C-algebra