数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2485-2497.doi: 10.1007/s10473-024-0623-z

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APPROXIMATELY ARTINIAN (NOETHERIAN) C*-ALGEBRAS

Mohammad ROUZBEHANI1, Massoud AMINI2,3, Mohammad B. ASADI1,†   

  1. 1. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran;
    2. Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran;
    3. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran
  • 收稿日期:2023-04-25 修回日期:2023-12-26 发布日期:2024-12-06
  • 通讯作者: † Mohammad B. ASADI, E-mail: mamini@ipm.ir
  • 作者简介:Mohammad ROUZBEHANI, E-mail: rouzbehani.m.math@gmail.com; mb.asadi@ut.ac.ir; Massoud AMINI, E-mail: mamini@modares.ac.ir
  • 基金资助:
    The first and third authors' research were supported by grants from INSF (98029498, 99013953). The second author's research was partly supported by a grant from IPM (96430215).

APPROXIMATELY ARTINIAN (NOETHERIAN) C*-ALGEBRAS

Mohammad ROUZBEHANI1, Massoud AMINI2,3, Mohammad B. ASADI1,†   

  1. 1. School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran;
    2. Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran;
    3. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran
  • Received:2023-04-25 Revised:2023-12-26 Published:2024-12-06
  • Contact: † Mohammad B. ASADI, E-mail: mamini@ipm.ir
  • About author:Mohammad ROUZBEHANI, E-mail: rouzbehani.m.math@gmail.com; mb.asadi@ut.ac.ir; Massoud AMINI, E-mail: mamini@modares.ac.ir
  • Supported by:
    The first and third authors' research were supported by grants from INSF (98029498, 99013953). The second author's research was partly supported by a grant from IPM (96430215).

摘要: In this article, we introduce and study the class of approximately Artinian (Noetherian) C*-algebras, called AR-algebras (AN-algebras), which is a simultaneous generalization of Artinian (Noetherian) C*-algebras and AF-algebras. We study properties such as the ideal property and topological dimension zero for them. In particular, we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite. This extends the Kirchberg's $\mathcal{O}_\infty$-absorption theorem, and implies that a weakly purely infinite C*-algebra is Noetherian iff every its ideal has a full projection.

关键词: C*-algebra, graph C*-algebra, Artinian, Noetherian, inductive limit, (weakly) purely infinite, ideal property

Abstract: In this article, we introduce and study the class of approximately Artinian (Noetherian) C*-algebras, called AR-algebras (AN-algebras), which is a simultaneous generalization of Artinian (Noetherian) C*-algebras and AF-algebras. We study properties such as the ideal property and topological dimension zero for them. In particular, we show that a faithful AR or AN algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite. This extends the Kirchberg's $\mathcal{O}_\infty$-absorption theorem, and implies that a weakly purely infinite C*-algebra is Noetherian iff every its ideal has a full projection.

Key words: C*-algebra, graph C*-algebra, Artinian, Noetherian, inductive limit, (weakly) purely infinite, ideal property

中图分类号: 

  • 46L05