Projections in free product C*-algebras
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Consider the reduced free product of C*-algebras, (A,ϕ)=(A_1,ϕ_1)*(A_2,ϕ_2), with respect to states ϕ_1 and ϕ_2 that are faithful. If ϕ_1 and ϕ_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(ϕ)(g)>0. Thus, the ordered group K_0(A) is weakly unperforated.
If, on the other hand, ϕ_1 or ϕ_2 is not a trace and if a certain condition weaker than the Avitzour conditions hold, then A is properly infinite.