Classification of Simple C-Algebras: Inductive Limits of Matrix Algebras Over One-Dimensional Spaces

In this article, we will give a complete classification of simple C^*-algebras which can be written as inductive limits of algebras of the form A_n=⊕_i=1^k_nM_[n,i](C(X_n,i)), where X_n,i are arbitrary variable one-dimensional compact metrizable spaces. The results unify and generalize the previous results for the case X_n,i=S¹ and for the case of X_n,i being trees. We obtain our classification results by reducing the case of general one-dimensional spaces to the case of circles. The techniques in this paper play important roles in the study of the case of higher-dimensional spaces.