On the classification of simple inductive limit $C^{\ast }$-algebras. I: The reduction theorem

Abstract

Suppose that

is a simple $C^{\ast }$-algebra, where $X_{n,i}$ are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that $A$ can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit $C^{\ast }$-algebras is classified by the Elliott invariant – consisting of the ordered K-group and the tracial state space – in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the $C^{\ast }$-algebras in this class do not enjoy the real rank zero property.)