In this article, we introduce and study the notion of Krull dimension for C*-algebras. We show that every C*-algebra with Krull dimension contains an essential ideal that is a finite direct sum of critical ideals. We show that a C*-algebra with Krull dimension has finite-dimensional center, and conclude that every graph C*-algebra with Krull dimension has real rank zero, and is
𝒪
∞
{\mathcal{O}_{\infty}}
-stable in the purely infinite case.
We also study the (weak) ideal property for critical C*-algebras.
In this article, we study chain conditions for graph C*-algebras. We show that there are infinitely many mutually non isomorphic Noetherian (and Artinian) purely infinite graph C*-algebras with infinitely many ideals. We prove that if E is a graph, then {C^{*}(E)} is a Noetherian (resp. Artinian) C*-algebra if and only if E satisfies condition (K) and each ascending (resp. descending) sequence of admissible pairs of E stabilizes.
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