An extension of compact operators by compact operators with no nontrivial multipliers
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F18%3A00499152" target="_blank" >RIV/67985840:_____/18:00499152 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.4171/JNCG/316" target="_blank" >http://dx.doi.org/10.4171/JNCG/316</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4171/JNCG/316" target="_blank" >10.4171/JNCG/316</a>
Alternative languages
Result language
angličtina
Original language name
An extension of compact operators by compact operators with no nontrivial multipliers
Original language description
We construct a noncommutative, separably represented, type I and approximately finite dimensional $C^*$-algebra such that its multiplier algebra is equal to its unitization. This algebra is an essential extension of the algebra $mathcal K(ell_2(mathfrak{c}))$ of compact operators on a nonseparable Hilbert space by the algebra $mathcal K(ell_2)$ of compact operators on a separable Hilbert space, where $mathfrak{c}$ denotes the cardinality of continuum. Although both $mathcal K(ell_2(mathfrak{c}))$ and $mathcal K(ell_2)$ are stable, our algebra is not. This sheds light on the permanence properties of the stability in the nonseparable setting. Namely, unlike in the separable case, an extension of a nonseparable $C^*$-algebra by $mathcal K(ell_2)$ does not have to be stable. Our construction can be considered as a noncommutative version of Mrówka’s $Psi$-space, a space whose one point compactification is equal to its Cech–Stone compactification and is induced by a special uncountable family of almost disjoint subsets of $mathbb{N}$.
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
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Noncommutative Geometry
ISSN
1661-6952
e-ISSN
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Volume of the periodical
12
Issue of the periodical within the volume
4
Country of publishing house
CH - SWITZERLAND
Number of pages
27
Pages from-to
1503-1529
UT code for WoS article
000453796600009
EID of the result in the Scopus database
2-s2.0-85061322081