On the classification of simple inductive limit $C^*$-algebras. I: The reduction theorem

Abstract

Suppose that A = \lim\limits_{n\to\infty}(A_n = \bigoplus_{i=1}^{t_n} M_{[n,i]}(C(X_{n,i})), \phi_{n,m}) is a simple C^* -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^* -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^* -algebras in this class do not enjoy the real rank zero property.)