On finiteness properties of Noetherian (Artinian) C*-algebras

Pourgholamhossein, Mahmood; Rouzbehani, Mohammad; Amini, Massoud

doi:10.1080/03081087.2020.1733458

Cited by 2 publications

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“…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.…”

Section: Introductionmentioning

confidence: 87%

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Goldie dimension for C*-algebras

Rouzbehani,

Amini,

Asadi

2024

Proceedings of the Edinburgh Mathematical Society

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In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

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Section: Introductionmentioning

confidence: 87%

“…Note that being Artinian (resp. Noetherian) is stable under taking hereditary C*-subalgebras and it passes to closed ideals and quotients, and also it is preserved under extension and Morita equivalence (see [34, Lemma 2.2 and Corollary 2.5] and [35, Proposition 1.1]).…”

Section: Goldie Dimension For C*-algebrasmentioning

confidence: 99%

Goldie dimension for C*-algebras

Rouzbehani,

Amini,

Asadi

2024

Proceedings of the Edinburgh Mathematical Society

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Krull dimension for C*-algebras

2022

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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.

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On finiteness properties of Noetherian (Artinian) C*-algebras

Cited by 2 publications

References 25 publications

Goldie dimension for C*-algebras

Goldie dimension for C*-algebras

Krull dimension for C*-algebras

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