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$\mathrm{C}^{*}$-algebras hereditarily containing nonzero, square-zero elements

Part of: Selfadjoint operator algebras Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  04 November 2024

Massoud Amini Open the ORCID record for Massoud Amini [Opens in a new window] ,
George A. Elliott  and
Mohammad Rouzbehani Open the ORCID record for Mohammad Rouzbehani [Opens in a new window]
Massoud Amini
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, 14115-134, Iran e-mail: mamini@modares.ac.ir mamini@ipm.ir.
George A. Elliott
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada e-mail: elliott@math.toronto.edu
Mohammad Rouzbehani*
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, 1417614411, Iran
*
e-mail: rouzbehani.m.math@gmail.com
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Abstract

We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.

Keywords

C*-algebrasquare-zero elementsfiniteness properties

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L85: Noncommutative topology 46H10: Ideals and subalgebras

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1081 - 1091
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was partly supported by a grant from the Institute for Research in Fundamental Sciences (IPM) (Grant No. 4027151). The second author was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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