INFINITE NON-SIMPLE C^{∗}-ALGEBRAS: ABSORBSION OF THE CUNTZ ALGEBRA O_{∞}

EBERHARD KIRCHBERG MIKAEL RORDAM

Abstract. The first named author has given a classification theorem for separable, nuclear, C^{∗}-algebras A that absorb the Cuntz algebra O_{∞}. (We say that A absorbs O_{∞} if A is isomorphic to A ⊗ O_{∞}.) Motivated by this theorem, we investigate here if one can intrinsically characterize which C^{∗}-algebras absorb O_{∞}. This leads us to three different notions of pure infiniteness of a C^{∗}-algebra, 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_{∞} 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: purely infinite, that extends the well-known property for simple C^{∗}-algebras. The weaker condition of the three, called weakly purely infinite, 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.