A nonstable 𝐶*-algebra with an elementary essential composition series

Author:

Ghasemi Saeed,Koszmider Piotr

Abstract

A C C^* -algebra A \mathcal {A} is said to be stable if it is isomorphic to A K ( 2 ) \mathcal {A} \otimes \mathcal {K}(\ell _2) . Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C C^* -algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra A \mathcal {A} of B ( 2 ) \mathcal {B}(\ell _2) , which is the inductive limit of length ω 1 \omega _1 of its separable stable ideals I α \mathcal {I}_\alpha ( α > ω 1 \alpha >\omega _1 ) satisfying I α + 1 / I α K ( 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for each α > ω 1 \alpha >\omega _1 , while A \mathcal {A} is not stable. The sequence ( I α ) α ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} is the GCR composition series of A \mathcal {A} which in this case coincides with the Cantor–Bendixson composition series as a scattered C C^* -algebra. A \mathcal {A} has the property that all of its proper two-sided ideals are listed as I α \mathcal {I}_\alpha ’s for some α > ω 1 \alpha >\omega _1 , and therefore the family of stable ideals of A \mathcal {A} has no maximal element.

By taking A = A K ( 2 ) \mathcal {A}’=\mathcal {A}\otimes \mathcal {K}(\ell _2) we obtain a stable C C^* -algebra with analogous composition series ( J α ) α > ω 1 (\mathcal {J}_\alpha )_{\alpha >\omega _1} whose ideals J α \mathcal {J}_\alpha are isomorphic to I α \mathcal {I}_\alpha for each α > ω 1 \alpha >\omega _1 . In particular, there are nonisomorphic scattered C C^* -algebras whose GCR composition series ( I α ) α ω 1 (\mathcal {I}_\alpha )_{\alpha \leq \omega _1} satisfy I α + 1 / I α K ( 2 ) \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2) for all α > ω 1 \alpha >\omega _1 , for which the composition series differs first at α = ω 1 \alpha =\omega _1 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference32 articles.

1. A note on the Akemann-Doner and Farah-Wofsey constructions;Bice, Tristan;Proc. Amer. Math. Soc.,2017

2. 𝐶*-algebras with and without ≪-increasing approximate units;Bice, Tristan;J. Math. Anal. Appl.,2019

3. Traces on simple AF 𝐶*-algebras;Blackadar, Bruce E.;J. Functional Analysis,1980

4. Encyclopaedia of Mathematical Sciences;Blackadar, B.,2006

5. Inductive limits of finite dimensional 𝐶*-algebras;Bratteli, Ola;Trans. Amer. Math. Soc.,1972

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. On $\mathbb R$-embeddability of almost disjoint families and Akemann–Doner C$^*$-algebras;Fundamenta Mathematicae;2021

2. A non-diagonalizable pure state;Proceedings of the National Academy of Sciences;2020-12-16

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3