Reduction to dimension two of local spectrum for $AH$ algebra with ideal property

TL;DR

This paper demonstrates that certain $AH$ algebras with the ideal property can be represented as inductive limits of simpler subalgebras with dimension at most two, simplifying their analysis.

Contribution

It proves that $AH$ algebras with the ideal property can be decomposed into inductive limits of subhomogeneous algebras with dimension two or less.

Findings

$AH$ algebras with ideal property are inductive limits of dimension drop and 2D matrix algebras.

The decomposition simplifies the structure and analysis of these algebras.

The result applies to $C^{*}$-algebras with uniformly bounded dimension.

Abstract

A $C^{*}$-algebra $A$ has ideal property if any ideal $I$ of $A$ is generated as a closed two sided ideal by the projections inside the ideal. Suppose that the limit $C^{*}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has ideal property. In this paper, we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension drop interval algebras and matrix algebras over 2-dimensional spaces with torsion $H^{2}$ groups.