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Suppose that A is a C*-algebra for which [Formula: see text], where [Formula: see text] is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then [Formula: see text] if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterize when A is of real rank zero.

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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