Chain conditions for graph C*-algebras

Abstract

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.

Goldie dimension for C*-algebras

“…Noetherian and/or Artinian C*-algebras as well as C*-algebras with Krull dimension are defined and studied in [20,34,35,40,41]. In this article, we define and study C*-algebras with Goldie dimension as a generalization of all of these classes (see Figure 1), and then extend the main results obtained in [41] and present some new results and applications.…”

“…Furthermore, for a complete-Goldie C*-algebra, Goldie dimension is preserved under Morita equivalence of C*-algebras and passes to ideals and quotients, by Theorem 2.7(i) and Definition 2.22. The first assertion now follow from [40, Lemma 3.3]. The second assertion holds, because in [21, Theorem 2.5], it was shown that if E is a directed graph, then satisfies Condition (K) if and only if the real rank of is zero.…”

“…For the following theorem, we use the concepts in Section 2 of [40]. For more details on the theory of graph C*-algebras, we refer the reader to [3].…”

“…Clearly, every C*-algebra with finitely many closed ideals is Noetherian and Artinian, including simple and finite-dimensional C*-algebras. On the other hand, there are infinitely many mutually non-isomorphic Noetherian (Artinian) C*-algebras with infinitely many closed ideals [20, 40]. Recall that a Noetherian (Artinian) topological space is a space that satisfies the ascending (descending) chain condition for its open subsets.…”

“…This implies that Ell(A) ∼ = Ell(B) if and only if A ∼ = B, by previous assertion.Example 2.21. For every prime number p, let be the graph introduced on p. 496 of[40]. The graph C*-algebra C * (F p+1 ) is Artinian, prime, purely infinite [5, Proposition 5.3], separable and nuclear[3, p. 65].…”