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^*_nar_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.