On finiteness properties of Noetherian (Artinian) C*-algebras.

In this article, we present a trichotomy (a division into three classes) on Noetherian and Artinian C*-algebras and obtain some structural results about Noetherian (and/or Artinian) C*-algebras. We show that every Noetherian, purely infinite and σ-unital C*-algebra A is generated as a C*-ideal by a single projection. We show that if A is a purely infinite, nuclear, separable, Noetherian and Artinian C*-algebra, then A ≅ A ⊗ Z ≅ A ⊗ O ∞ . This is a partial extension of Kirchberg's O ∞ -absorption theorem and Kirchberg's exact embedding theorem. Finally, we show that each Noetherian AF-algebra has a full finite-dimensional C*-subalgebra. [ABSTRACT FROM AUTHOR]