Approximately Artinian (Noetherian) C*-algebras

In this article, we introduce and study the class of approximately Artinian (Noetherian) C*-algebras, called AR-algebras (AN-algebras), which is a simultaneous generalization of Artinian (Noetherian) C*-algebras and AF-algebras. We study properties such as the ideal property and topological dimension zero for them. In particular, we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite. This extends the Kirchberg’s \(\mathcal{O}_\infty\) $${\mathcal{O}}_{\mathrm{\infty }}$$ -absorption theorem, and implies that a weakly purely infinite C*-algebra is Noetherian iff every its ideal has a full projection.