Infinite non-simple C*-algebras:: Absorbing the Cuntz algebra φ∞

The first named author has given a classification of all separable, nuclear C*-algebras A that absorb the Cuntz algebra O-infinity. (We say that A absorbs O-infinity if A is isomorphic to A circle times O-infinity.) Motivated by this classification we investigate here if one can give an intrinsic characterization of C*-algebras that absorb O-infinity. This investigation 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 strongest of the three properties, strongly purely infinite, is shown to be equivalent to absorbing O-infinity. for separable, nuclear C*-algebras that either are stable or have an approximate unit consisting of projections. In a previous paper (2000, Amer. J. Math. 122, 637-666), we studied an intermediate, and perhaps more natural, condition: purely infinite, that extends a well known property for simple C*-algebras. The weakest condition of the three, weakly purely infinite, is shown to be 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 this to be the case for C*-algebras that are either simple, of real rank zero, or approximately divisible. (C) 2002 Elsevier Science (USA).