Non-simple purely infinite C *-algebras

A C *-algebra A is defined to be purely infinite 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 { r n } in A such that r * n ar n → b . This definition agrees with the usual definition by J. Cuntz when A is simple. It is shown that the property of being purely infinite 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 ∞ is purely infinite for every C *-algebra A . Purely infinite C *-algebras admit no traces, and, conversely, an approximately divisible exact C *-algebra is purely infinite if it admits no nonzero trace.