# On the classification of simple inductive limit $C_{∗}$-algebras. I: The reduction theorem

### Guihua Gong

Department of Mathematics University of Puerto Rico, Rio Piedras San Juan, PR 00931-3355, USA

## Abstract

Suppose that $A=n→∞lim (A_{n}=⨁_{i=1}M_{[n,i]}(C(X_{n,i})),ϕ_{n,m})$ 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.)

## Cite this article

Guihua Gong, On the classification of simple inductive limit $C_{∗}$-algebras. I: The reduction theorem. Doc. Math. 7 (2002), pp. 255–461

DOI 10.4171/DM/127