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


Authors: Saeed Ghasemi and Piotr Koszmider
Journal: Proc. Amer. Math. Soc. 148 (2020), 2201-2215
MSC (2010): Primary 03E05, 03E75, 46L05, 46M40
DOI: https://doi.org/10.1090/proc/14814
Published electronically: November 19, 2019
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Abstract: A $ C^*$-algebra $ \mathcal {A}$ is said to be stable if it is isomorphic to $ \mathcal {A} \otimes \mathcal {K}(\ell _2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable $ 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 $ \mathcal {A}$ of $ \mathcal {B}(\ell _2)$, which is the inductive limit of length $ \omega _1$ of its separable stable ideals $ \mathcal {I}_\alpha $ ( $ \alpha <\omega _1$) satisfying $ \mathcal {I}_{\alpha +1}/\mathcal {I}_\alpha \cong \mathcal {K}(\ell _2)$ for each $ \alpha <\omega _1$, while $ \mathcal {A}$ is not stable. The sequence $ (\mathcal {I}_\alpha )_{\alpha \leq \omega _1}$ is the GCR composition series of $ \mathcal {A}$ which in this case coincides with the Cantor-Bendixson composition series as a scattered $ C^*$-algebra. $ \mathcal {A}$ has the property that all of its proper two-sided ideals are listed as $ \mathcal {I}_\alpha $'s for some $ \alpha <\omega _1$, and therefore the family of stable ideals of $ \mathcal {A}$ has no maximal element.

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


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Additional Information

Saeed Ghasemi
Address at time of publication: Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic
Email: ghasemi@math.cas.cz

Piotr Koszmider
Affiliation: Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Email: piotr.koszmider@impan.pl

DOI: https://doi.org/10.1090/proc/14814
Received by editor(s): December 6, 2017
Received by editor(s) in revised form: January 6, 2019, January 6, 2019, and August 15, 2019
Published electronically: November 19, 2019
Additional Notes: The research of the second author was partially supported by grant PVE Ciência sem Fronteiras - CNPq (406239/2013-4).
Communicated by: Adrian Ioana
Article copyright: © Copyright 2019 American Mathematical Society