On Unital Full Amalgamated Free Products of Quasidiagonal $C^*$-Algebras

In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal $C^*$-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal $C^*$-algebras with amalgamation over a finite-dimensional $C^*$-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional $C^*$-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.