A non-stable C*-algebra with an elementary essential composition series

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

A non-stable C*-algebra with an elementary essential composition series

Popis výsledku v původním jazyce

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

Název v anglickém jazyce

A non-stable C*-algebra with an elementary essential composition series

Popis výsledku anglicky

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

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Proceedings of the American Mathematical Society

ISSN

0002-9939

e-ISSN

—

Svazek periodika

148

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

15

Strana od-do

2201-2215

Kód UT WoS článku

000521585500035

EID výsledku v databázi Scopus

2-s2.0-85082960009