Simplicity and the Stable Rank of Some Free Product C∗–algebras

A necessary and sufficient condition for the simplicity of the C∗– algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C∗–algebras are proved. Introduction The reduced free product of C∗–algebras with respect to given states was introduced independently by Voiculescu [19] and Avitzour [2]. It is the appropriate construction associated to Voiculescu’s free probability theory (see [19], [21]). The motivating example concerns reduced group C∗–algebras. For a discrete group G, its reduced group C∗–algebra is generated by the left regular representation of G on l(G) and is denoted C∗ r (G). Its canonical tracial state (which is the vector state associated to the characteristic function of the identity element of G) is written τG. Then for discrete groups G1 and G2, the reduced free product construction yields (C∗ r (G1), τG1) ∗ (C∗ r (G2), τG2) = (C∗ r (G), τG), where G = G1 ∗G2 is the free product of groups. Voiculescu’s definition of freeness is an abstraction of some essential facets of the relationship between the copies of C∗ r (G1) and C ∗ r (G2) embedded in C ∗ r (G), with respect to the trace τG. The reduced free product of C∗–algebras can be described with respect to freeness as follows. Let A1 and A2 be unital C∗–algebras with states φ1 and φ2, respectively whose associated GNS representations are faithful. Then the reduced free product of (A1, φ1) and (A2, φ2) is the (unique) unital C∗–algebra A and state φ with unital embeddings Aι ↪→ A such that (1) the GNS representation associated to φ is faithful on A; (2) φ|Aι = φι; (3) A1 and A2 are free with respect to φ; (4) A is generated by A1 ∪A2. We denote this by (A, φ) = (A1, φ1) ∗ (A2, φ2). (1) It is further known [19] (or see [21, 2.5.3]) that φ is a trace if φ1 and φ2 are traces. Moreover, by [7], φ is also faithful on A if φ1 and φ2 are faithful. The reduced free product thus provides a multitude of constructions of C∗– algebras, about which some results are known (see [19], [2], [11], [10], [9]). For Received by the editors January 21, 1997. 1991 Mathematics Subject Classification. Primary 46L05, 46L35. c ©1999 American Mathematical Society 1