Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and (Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu

A class of C*-algebras, to be called those of generalized tracial rank one, is introduced. A second class of unital simple separable amenable C*-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable \({\cal Z}\)-stable C*-algebras. A number of results toward the classification of the second class are presented including an isomorphism theorem for a special sub-class of the first class, leading to the general classification of all unital simple s with rational generalized tracial rank one in Part II.

Dans cet article et le prochain, on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.