Krull dimension for C*-algebras

Abstract

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.

Goldie dimension for C*-algebras

“…Therefore, A (and also each of its ideals) has an ideal that is a prime C*-algebra. Now, similar to the first paragraph of the proof of Theorem 2.8 in [41], we can show that A contains an essential ideal which is a finite direct sum of prime C*-algebras, and so has Goldie dimension.…”

“…Here, we work with (closed) two-sided ideals instead of one-sided ideals (as in the algebraic setting). We show that C*-algebras with Goldie dimension share some basic properties of C*-algebras with Krull dimension (obtained in [41]) and their local multiplier algebra and primitive ideal space can be described. We show that the Goldie dimension of a C*-algebra (if exists) is the same as the dimension of the center of its local multiplier algebra.…”

“…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.…”

“…Recall that, the Krull dimension of a C*-algebra measures how close the C*-algebra is to being Artinian (see [41,Definition 2.1], for a detailed definition). The proof of the following lemma is similar to that of Theorem 2.8 in [41].…”

“…The paper is organized as follows. In § 2, we define a notion of the Goldie dimension for C*-algebras (Definition 2.2), which is shown to be well-defined (Lemma 2.1), and conclude that every C*-algebra with Goldie dimension has a closed essential ideal which is a finite direct sum of prime C*-algebras (extending [41,Theorem 2.8]). We study the perseverance of the Goldie dimension under passing to hereditary C*-subalgebras, extensions and Morita equivalence.…”