Free Products of C ∗ -Algebras

Abstract

Small (spatial) free of C*-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras C,*(GI * G2) where G1, G2 are discrete groups. The freeproduct analogs of noncommutative Bernoulli shifts are discussed. 0. Introduction. Let K be a field. Consider the category of unital algebras over K. It is well known that this category admits coproducts: free of algebras [2]. Heuristically, the free of algebras is the algebra generated by them, with no relations except for the identification of unit elements. If K = C, the complex numbers, and we consider unital * -algebras, we can easily define a * -operation on the free products. Let A, B be unital C*-algebras, and A * B their free product, which is a unital *-algebra. The question arises: in what ways may one define a pre-C* norm on A * B that extends the norms on A and B? Guided by analogy with tensor products, we expect to have a choice among many pre-C* norms, giving rise to many C* free products of A and B. One natural norm is 1c I I = sup{ 1IT(c)IH: ST * -representation of A * B). The * -representations of A * B are in 1-1 correspondence with pairs of * representations of A and B, which act on the same Hilbert space. Let A * B be the completion of A * B in this norm. It is easy to see that this construction defines a coproduct in the category of C*-algebras, and that A * B is the free product of A and B, analogous to the biggest tensor A 0 B. If G 1, G2 are discrete groups we obtain C*(G1) * C*(G2) C*(GI * G2) where G1 * G2 is the free group, and this is analogous to the relation C*(G1) 0 C*(G2) C*(G1 X G2). This paper is motivated by the question: Is there a C*-product, A*B, in analogy to the smallest tensor product, A * B, satisfying a relation Cr*(GI * G2) Cr*(GI) * Cr*(G2) Received by the editors January 28, 1981 and, in revised form, May 19, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46L55. (1 982 American Mathematical Society 0002-9947/82/00001022/$04.25