Countable Saturation of Corona Algebras

We present unified proofs of several properties of the corona of σ-unital C*-algebras such as AA-CRISP, SAW*, being sub-σ-Stonean in the sense of Kirchberg, and the conclusion of Kasparov’s Technical Theorem. Although our results were obtained by considering C*algebras as models of the logic for metric structures, the reader is not required to have any knowledge of model theory of metric structures (or model theory, or logic in general). The proofs involve analysis of the extent of model-theoretic saturation of corona algebras. Résumé. Nous présentons des démonstrations unifiées de plusieurs propriétés de la corona des C*-algebras σ-unitales tel qu’AA-CRISP, SAW*, étant sous-σ-Stonean dans le sens de Kirchberg, et la conclusion du théorème technique de Kasparov. Bien que nos résultats aient été obtenus en considérant les C*-algebras comme modèles de la logique pour les structures métriques, le lecteur n’est pas requis d’avoir aucune connaissance de la théorie des modèles des structures métriques (ou la théorie des modèles, ou de la logique en général). Les démonstrations impliquent l’analyse de l’ampleur de la saturation modèle-théorétique des algebres de corona. We shall investigate the degree of countable saturation of coronas (see Definition 1.1 and paragraph following it). This property is shared by ultraproducts associated with nonprincipal ultrafilers on N in its full form. The following summarizes our main results. All ultrafilters are nonprincipal ultrafilters on N. Theorem 1. Assume a C*-algebra M is in one of the following forms: (1) the corona of a σ-unital C*-algebra, (2) an ultraproduct of a sequence of C*-algebras, (3) an ultrapower of a C*-algebra, (4) ∏ nAn/ ⊕ nAn, for unital C*-algebras An, (5) the relative commutant of a separable subalgebra of an algebra that is in one of the forms (1)–(4). Then M satisfies each of the following: (6) It is SAW* (7) It has AA-CRISP (asymptotically abelian, countable Riesz separation property), Date: February 12, 2012. 1991 Mathematics Subject Classification. 46L05, 03C65. Partially supported by NSERC. .