On classification of non-unital amenable simple C*-algebras, III, the range and the reduction

TL;DR

This paper advances the classification of simple C*-algebras by extending the Elliott invariant framework to include non-unital and stably projectionless cases, providing a comprehensive range and reduction theorem.

Contribution

It introduces a unified description of the Elliott invariant for all finite nuclear dimension simple C*-algebras, including non-unital and stably projectionless cases, and constructs models with generalized tracial rank one.

Findings

Elliott invariant can describe all such C*-algebras.

Constructed models are of generalized tracial rank one.

Every stably projectionless algebra in the UCT class has rationally generalized tracial rank one.

Abstract

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-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 $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features. We also show that every stably projectionless separable simple amenable $C^*$-algebra in the UCT class has rationally generalized tracial rank one.