Chain conditions for  C *-algebras coming from Hilbert  C *-modules

Pourgholamhossein, Mahmood; Rouzbehani, Mohammad; Amini, Massoud

doi:10.1016/s0252-9602(18)30806-3

Cited by 6 publications

(3 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…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%

“…where [9, Example II.8.2.2(ii)]. According to [34, Proposition 3.4(ii)], every Noetherian, purely infinite and separable C*-algebra has the ideal property (i.e., its ideals are generated by thier projections). Moreover, the ideal property and purely infinite are preserved under inductive limits (see [30, Proposition 2.3] and [23, Proposition 4.18]).…”

Section: Goldie Dimension For C*-algebrasmentioning

confidence: 99%

See 2 more Smart Citations

Goldie dimension for C*-algebras

Rouzbehani,

Amini,

Asadi

2024

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

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.

show abstract