In this paper, we consider the question whether a unital full free product of
MF algebras with amalgamation over a finite dimensional C*-algebra is an MF
algebra. First, we show that, under a natural condition, a unital full free
product of two separable residually finite dimensional (RFD) C*-algebras with
amalgamation over a finite dimensional C*-algebra is again a separable RFD
C*-algebra. Applying this result on MF C*-algebras, we show that, under a
natual condition, a unital full free product of two MF algebras is again an MF
algebra. As an application, we show that a unital full free product of two AF
algebras with amalgamation over an AF algebra is an MF algebra if there are
faithful tracial states on each of these two AF algebras such that the
restrictions on the common subalgebra agree.
