PROJECTIONS IN FREE PRODUCT C-ASTERISK-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, K-0(A)(+), of the K-0-group of A consists of t hose elements g is an element of 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 th e other hand, phi(1) or phi(2) is not a trace and if a certain conditi on weaker than the Avitzour conditions holds, then A is properly infin ite.