An extension of compact operators by compact operators with no nontrivial multipliers

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

An extension of compact operators by compact operators with no nontrivial multipliers

Popis výsledku v původním jazyce

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 $mathbb{N}$.

Název v anglickém jazyce

An extension of compact operators by compact operators with no nontrivial multipliers

Popis výsledku anglicky

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 $mathbb{N}$.

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

2018

Kód důvěrnosti údajů

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

Název periodika

Journal of Noncommutative Geometry

ISSN

1661-6952

e-ISSN

—

Svazek periodika

12

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

27

Strana od-do

1503-1529

Kód UT WoS článku

000453796600009

EID výsledku v databázi Scopus

2-s2.0-85061322081