Projections in free product C*-algebras

Consider the reduced free product of C*-algebras, (A,\phi)=(A_1,\phi_1)*(A_2,\phi_2), with respect to states \phi_1 and \phi_2 that are faithful. If \phi_1 and \phi_2 are traces, if the so-called Avitzour conditions are satisfied, (i.e. A_1 and A_2 are not ``too small'' in a specific sense) and if A_1 and A_2 are nuclear, then it is shown that the positive cone of the K_0-group of A consists of those elements g in K_0(A) for which g=0 or K_0(\phi)(g)>0. Thus, the ordered group K_0(A) is weakly unperforated. If, on the other hand, \phi_1 or \phi_2 is not a trace and if a certain condition weaker than the Avitzour conditions hold, then A is properly infinite.