Nontrivially Noetherian $C^*$-algebras

Abstract

We say that a C *

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.…”

“…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.…”

“…Let Prim ( A ) be the set of primitive ideals in a C*-algebra A . Then, Prim ( A ) is a topological space with the Jacobson (or hull-kernel) topology [9, 20]. In [20], it was shown that a C*-algebra is Noetherian (Artinian) if and only if Prim ( A ) is so as a topological space.…”

“…Then, Prim ( A ) is a topological space with the Jacobson (or hull-kernel) topology [9, 20]. In [20], it was shown that a C*-algebra is Noetherian (Artinian) if and only if Prim ( A ) is so as a topological space. Note that the definition of Notherian (and Artinian) C*-algebras differs from the definition of Noetherian (and Artinian) rings, which is defined on the set of one-sided ideals.…”

“…In the latter definition, every Noetherian Banach algebra is finite-dimensional. This is also true (under the new definition based on ideals, rather than left ideals) for commutative C*-algebras, but not in general (see [20, Theorem 3.1]). Note that being Artinian (resp.…”