Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra O∞
✍ Scribed by Eberhard Kirchberg; Mikael Rørdam
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 425 KB
- Volume
- 167
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
The first named author has given a classification of all separable, nuclear C g -algebras A that absorb the Cuntz algebra O . . (We say that A absorbs O . if A is isomorphic to A é O . .) Motivated by this classification we investigate here if one can give an intrinsic characterization of C g -algebras that absorb O . . This investigation leads us to three different notions of pure infiniteness of a C g -algebra, all given in terms of local, algebraic conditions on the C g -algebra. The strongest of the three properties, strongly purely infinite, is shown to be equivalent to absorbing O . for separable, nuclear C g -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 g -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 g -algebra. The three conditions may be equivalent for all C g -algebras, and we prove this to be the case for C g -algebras that are either simple, of real rank zero, or approximately divisible.