---
abstract: |
    In this paper, we consider the real rank zero ${\mathrm{C}}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.

    [*Keywords:*]{} decomposition, real rank zero, Elliott-Thomsen algebra.\
    [*2000 MR Subject Classification:*]{} Primary 46L80; Secondary 19K14, 19K33, 19K35.
author:
- |
    Zhichao Liu\
    [**]{}
title: 'A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes'
---

Introduction {#introduction .unnumbered}
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The problem people we considered is to classify ${\mathrm{C}}^*$-algebras by their K-theoretical data. The first result of this kind was the classification of inductive limits of sequence of finite direct sums of matrix algebras (called AF algebras)[@Ell1]. This result was extended in [@Ell2]. It replaced the matrix algebras by matrix algebras over the unit circle and restricted the limit of ${\mathrm{C}}^*$-algebras to have real rank zero. In [@EG], [@EGS] and [@DG], a classification was given to classes of separable nuclear ${\mathrm{C}}^*$-algebras of real rank zero.

The decomposition theory plays an important role in the classification theorem. It will have some similar forms and reflect the property of the connecting homomorphisms. If $A$ is 