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

Abstract

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 stable case. 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. We also show that every stably projectionless separable simple amenable $C^{\ast }$-algebra in the UCT class has rationally generalized tracial rank one.

193 pages, the revision removes the condition of stable rank one in the main theorem (it has been divided into two papers). 2021/12: This is the first part of the original paper. The second part of the revision is also ready and we will try to post it at the same time

References in corpus (6)

• Completely positive maps of order zero
• Quasitraces on exact C*-algebras are traces
• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
• The nuclear dimension of C*-algebras
• A classification of finite simple amenable Z-stable C*-algebras, II, --C*-algebras with rational generalized tracial rank one
• Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

Cited by in corpus (2)

• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• Equivariant property Gamma and the tracial local-to-global principle for C*-dynamics