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On the classification of simple inductive limit $$C^*$$-algebras. I: The reduction theorem. (English) Zbl 1024.46018
Summary: Suppose that $A = \lim\limits_{n\to\infty}\left(A_n = \bigoplus_{i=1}^{t_n} M_{[n,i]}(C(X_{n,i})), \phi_{n,m}\right)$ 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).

MSC:
 46L35 Classifications of $$C^*$$-algebras
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