Vol. 7, No. 2, 2022

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

Guihua Gong and Huaxin Lin

Vol. 7 (2022), No. 2, 279–384

We prove that every stably projectionless separable simple amenable C-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-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 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.

classification of simple $C^*$-algebras
Mathematical Subject Classification
Primary: 46L35
Secondary: 46L05
Received: 25 April 2021
Revised: 31 January 2022
Accepted: 17 February 2022
Published: 13 September 2022
Guihua Gong
College of Mathematics and Information Science
Hebei Normal University
Shijiazhuang, Hebei
Department of Mathematics
University of Puerto Rico
Rio Piedras, PR
United States
Huaxin Lin
Department of Mathematics
East China Normal University
Department of Mathematics
University of Oregon
Eugene, OR
United States