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A class of $C^*$-algebras, to be called those of generalized tracial rankone, is introduced, and classified by the Elliott invariant. A second class ofunital simple separable amenable $C^*$-algebras, those whose tensor productswith UHF-algebras of infinite type are in the first class, to be referred to asthose of rational generalized tracial rank one, is proved to exhaust allpossible values of the Elliott invariant for unital finite simple separableamenable ${\cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a specialsub-class of those $C^*$-algebras are presented. This provides the basis forthe classification of $C^*$-algebras with rational generalized tracial rank onein Part II.

A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one

A classification of finite simple amenable ${\cal Z}$-stable $C^$-algebras, I: $C^$-algebras with generalized tracial rank one