Non-simple Purely Innnite C -algebras

A C-algebra A is deened to be purely innnite if there are no characters on A, and if for every pair of positive elements a; b in A, such that b lies in the closed two-sided ideal generated by a, there exists a sequence fr n g in A such that r n ar n ! b. This deenition agrees with the usual deenition by J. Cuntz when A is simple. It is shown that the property of being purely innnite is preserved under extensions, Morita equivalence, inductive limits, and it passes to quotients, and to hereditary sub-C-algebras. It is shown that A O 1 is purely innnite for every C-algebra A. Purely innnite C-algebras admit no traces, and, conversely, an approximately divisible exact C-algebra is purely innnite if it admits no non-zero trace.