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.

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Abstract. We introduce the decomposition rank, a notion of covering dimension for nuclear C * -algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C * -algebras and that it is closely related to the classification program.