N ov 1 99 9 PURELY INFINITE , SIMPLE C ∗ - ALGEBRAS ARISING FROM FREE PRODUCT CONSTRUCTIONS , III

In the reduced free product of C∗–algebras, (A, φ) = (A1, φ1) ∗ (A2, φ2) with respect to faithful states φ1 and φ2, A is purely infinite and simple if A1 is a reduced crossed product B ⋊α,r G for G an infinite group, if φ1 is well behaved with respect to this crossed product decomposition, if A2 6= C and if φ is not a trace. The reduced free product construction for C–algebras was invented independently by Voiculescu [11] and, in a more limited sense, Avitzour [1]. (The term “reduced” is to distinguish this construction from the universal or “full” free product of C–algebras.) It is a natural construction in Voiculescu’s free probability theory, (see [12]). Given unital C–algebras Aι with states φι whose GNS representations are faithful, (ι ∈ I), the construction yields (A,φ) = ∗ ι∈I (Aι, φι), where A is a unital C–algebra containing copies Aι →֒ A and generated by ⋃ ι∈I Aι, and where φ is a state on A with faithful GNS representation that restricts to give φι on Aι for every ι ∈ I and such that (Aι)ι∈I is free with respect to φ. Moreover, φ is a trace if and only if every φι is a trace; by [4], φ is faithful on A if and only if φι is faithful on Aι for every ι ∈ I. It is a very interesting open question whether every simple, unital C–algebra must either have a trace or be purely infinite. Purely infinite C–algebras were defined by J. Cuntz [3]. A simple unital C–algebra A is purely infinite if and only if for every positive element x ∈ A there is y ∈ A with yxy = 1. An equivalent condition is that every hereditary C–subalgebra of A contains an infinite projection. Let (A,φ) = (A1, φ1) ∗ (A2, φ2) 1991 Mathematics Subject Classification. 46L05, 46L35. Typeset by AMS-TEX 1