Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

Pimsner introduced the C*-algebra O-x generated by a Hilbert bimodule X over a Ci-algebra sl. We look for additional conditions that X should satisfy in order to study the simplicity and, more generally, the ideal structure of when X is finite projective. We introduce two conditions, "(I)-freeness" and "(II)-freeness," stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math. 56, 1980, 251-268) and J. Cuntz (Invent. Math. 63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of "Cuntz-Krieger bimodules." If X satisfies this condition the C*-algebra O-x does not depend on the choice of the generators when A is Faithfully represented. As a consequence, if X is ill-free and A is X-simple, then O-x is simple. In the case of Cuntz-Krieger algebras O-A, X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then O-x is p.i.; if A is non nuclear then O-x is nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal structure of O-x. (C) 1998 Academic Press.