Ideals Generated by Projections and Inductive Limit $C^*$-Algebras

Abstract

We introduce two classes of inductive limit C * -algebras which generalize the AH algebras: the GAH algebras (GAH stands for generalized AH) and a subclass of it, the strong GAH algebras. We give necessary and sufficientconditions for an ideal of a GAH algebra to be generated by projections which, in particular, gives necessary and sufficient conditions for a GAH algebra to have the ideal property, i.e., any ideal is generated by projections. We prove that if 0→I→A→B→0 is an exact sequence of C * -algebras such that A is a GAH algebra, then A has the ideal property if and only if I and B have the ideal property. We describe the lattice of ideals generated by projections of a strong GAH algebra and also the partially ordered set of the stably cofinite ideals generated by projections of a strong GAH algebra A under the additional assumption that the projections in M ∞ (A) satisfy the Riesz decomposition property. These results generalize some of our previous theorems involving AH algebras.