Summary: Consider the reduced free product of \(C^*\)-algebras, \((A,\varphi)= (A_1,\varphi_1)* (A_2,\varphi_2)\), with respect to states \(\varphi_1\) and \(\varphi_2\) that are faithful. If \(\varphi_1\) and \(\varphi_2\) 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, \(K_0(A)^+\), of the \(K_0\)-group of \(A\) consists of those elements \(g\in K_0(A)\) for which \(g= 0\) or \(K_0(\varphi)(g)> 0\). Thus, the ordered group \(K_0(A)\) is weakly unperforated.
If, on the other hand, \(\varphi_1\) or \(\varphi_2\) is not a trace and if a certain condition weaker than the Avitzour conditions holds, then \(A\) is properly infinite.