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We consider amalgamated unital full free products of the form A1*D A2, where A1, A2 and D are finite-dimensional C*-algebras and there are faithful traces on A1 and A2 whose restrictions to D agree. We provide several conditions on the matrices of partial multiplicities of the inclusions D ↪ A1 and D ↪ A2 that guarantee that the C*-algebra A1*D A2 is primitive. If the ranks of the matrices of partial multiplicities are one or all entries are 0 or ≥ 2, we prove that the algebra A1*D A2 is primitive if and only if it has a trivial center.

Conditions for primitivity of unital amalgamated full free products of finite-dimensional C*-algebras