Infinite non-simple $C^*$-algebras: absorbsion of the Cuntz algebra $O_\infty$
Eberhard Kirchberg and Mikael Rordam
The first named author has given a classification theorem for separable, nuclear, C*-algebras A that absorb the Cuntz algebra $O_\infty$. (We say that A absorbs $O_\infty$ if A is isomorphic to $A \otimes O_\infty$.) Motivated by this theorem, we investigate here if one can intrinsically characterize which C*-algebras absorb $O_\infty$. This leads us to three different notions of pure infiniteness of a C*-algebras, all given in terms of local, algebraic conditions on the C*-algebra. The stronger of the three properties, called strongly purely infinite, is shown to be equivalent to absorbing $O_\infty$ for all separable, nuclear C*-algebras, that either are stable or have an approximate unit consisting of projections. In a previous paper we studied an intermediate, and perhaps more natural, condition: pure infiniteness, that extends the well-known property for simple C*-algebras. The weaker condition of the three, called weak pure infiniteness, is shown to being equivalent to the absence of quasitraces in an ultrapower of the C*-algebra. The three conditions may be equivalent for all C*‑algebras, and we prove that this is the case in the special cases where the C*-algebra is either simple, of real rank zero, or approximately divisible.