On classification of nonunital amenable simple C*-algebras, III: The range and the reduction

Abstract

We prove that every stably projectionless separable simple amenable $C^{\ast }$-algebra in the UCT class has rationally generalized tracial rank one. Following Elliott’s earlier work, we show that the Elliott invariant of any finite separable simple $C^{\ast }$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group (which unifies the unital and nonunital, as well as stably projectionless cases). We also show that, for any given such invariant set, there is a finite separable simple $C^{\ast }$-algebra whose Elliott invariant is the given set, a refinement of the range theorem of Elliott. In the stably projectionless case, modified model $C^{\ast }$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features.