2 N ov 1 99 9 Projections in free product C ∗ – algebras , II

Let (A,φ) be the reduced free product of infinitely many C–algebras (Aι, φι) with respect to faithful states. Assume that the Aι are not too small, in a specific sense. If φ is a trace then the positive cone of K0(A) is determined entirely by K0(φ). If, furthermore, the image of K0(φ) is dense in R, then A has real rank zero. On the other hand, if φ is not a trace then A is simple and purely infinite. Introduction Let I be a set having at least two elements and, for every ι ∈ I, let Aι be a unital C –algebra with a state, φι, whose GNS representation is faithful. Their reduced free product, (A,φ) = * ι∈I (Aι, φι) (1) was introduced by Voiculescu [20] and independently (in a more restricted way) by Avitzour [1]. Thus A is a unital C–algebra with canonical, injective, unital –homomorphisms, πι : Aι → A, and φ is a state on A such that φ◦πι = φι for all ι. It is the natural construction in Voiculescu’s free probability theory (see [21]), and Voiculescu’s theory has been vital to the study of these C–algebras. In [12], for reduced free product C–algebras A as in (1), when all the φι are faithful, we investigated projections in A and the related topic of positive elements in K0(A). The behaviour we discovered, under mild conditions specifying that the Aι are not too small, depended broadly on whether φ is a trace, (i.e. on whether all the φι are traces). If φ is a not trace then by [12] A is properly infinite. It remained open whether A must be purely infinite. (Some special classes of reduced free product C–algebras have in [13] and [9] been shown to be purely infinite.) When φ is a trace, then it follows from [12] that