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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. This is a partial extension of Kirchberg's-absorption theorem and Kirchberg's exact embedding theorem. Finally, we show that each Noetherian AF-algebra has a full finite-dimensional C*-subalgebra.

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