# On the Decomposition Theorems for C*-algebras

## TL;DR

This paper presents two decomposition theorems for Elliott dimension drop interval algebras, which are crucial for classifying a broad class of $AH$ algebras with no dimension growth.

## Contribution

It introduces new decomposition theorems that advance the classification of $AH$ algebras with the ideal property and no dimension growth.

## Key findings

- Two key decomposition theorems for Elliott dimension drop interval algebras
- Progress towards classifying all $AH$ algebras with no dimension growth
- Foundational results for the structure of $C^*$-algebras in classification theory

## Abstract

Elliott dimension drop interval algebra is an important class among all $C^*$-algebras in the classification theory. Especially, they are building stones of $\mathcal{AHD}$ algebra and the latter contains all $AH$ algebras with the ideal property of no dimension growth. In this paper, we will show two decomposition theorems related to the Elliott dimension drop interval algebra. Our results are key steps in classifying all $AH$ algebras with the ideal property of no dimension growth.