A non-stable C*-algebra with an elementary essential composition series
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00523592" target="_blank" >RIV/67985840:_____/20:00523592 - isvavai.cz</a>
Result on the web
<a href="https://www.ams.org/journals/proc/2020-148-05/S0002-9939-2019-14814-0/" target="_blank" >https://www.ams.org/journals/proc/2020-148-05/S0002-9939-2019-14814-0/</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/proc/14814" target="_blank" >10.1090/proc/14814</a>
Alternative languages
Result language
angličtina
Original language name
A non-stable C*-algebra with an elementary essential composition series
Original language description
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. nBy 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$.n
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Proceedings of the American Mathematical Society
ISSN
0002-9939
e-ISSN
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Volume of the periodical
148
Issue of the periodical within the volume
5
Country of publishing house
US - UNITED STATES
Number of pages
15
Pages from-to
2201-2215
UT code for WoS article
000521585500035
EID of the result in the Scopus database
2-s2.0-85082960009