On the Classification of Simple Inductive Limit $C^*$-Algebras, I: The Reduction Theorem

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.)

2000 Mathematics Subject Classification:

Keywords and Phrases:

Full text: dvi.gz 319 k, dvi 922 k, ps.gz 1251 k, pdf 1362 k.

Home Page of DOCUMENTA MATHEMATICA