A purely infinite Cuntz-like Banach *-algebra with no purely infinite ultrapowers

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00556606" target="_blank" >RIV/67985840:_____/22:00556606 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.jfa.2022.109488" target="_blank" >https://doi.org/10.1016/j.jfa.2022.109488</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jfa.2022.109488" target="_blank" >10.1016/j.jfa.2022.109488</a>

Jazyk výsledku

angličtina

Název v původním jazyce

A purely infinite Cuntz-like Banach *-algebra with no purely infinite ultrapowers

Popis výsledku v původním jazyce

We continue our investigation, from [10], of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a C⁎-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a “Cuntz-like” Banach ⁎-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algebra is a naturally occurring analogue of the Cuntz algebra, and of the Lp-analogues introduced by Phillips. However, our proof of being purely infinite is combinatorial, but direct, and so differs from existing proofs. We show that there are non-zero traces on our algebra, which in particular implies that our algebra is not isomorphic to any of the Lp-analogues of the Cuntz algebra.

Název v anglickém jazyce

A purely infinite Cuntz-like Banach *-algebra with no purely infinite ultrapowers

Popis výsledku anglicky

We continue our investigation, from [10], of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a C⁎-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a “Cuntz-like” Banach ⁎-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algebra is a naturally occurring analogue of the Cuntz algebra, and of the Lp-analogues introduced by Phillips. However, our proof of being purely infinite is combinatorial, but direct, and so differs from existing proofs. We show that there are non-zero traces on our algebra, which in particular implies that our algebra is not isomorphic to any of the Lp-analogues of the Cuntz algebra.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GJ19-07129Y" target="_blank" >GJ19-07129Y: Metody lineární analýzy v operátorových algebrách a naopak</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

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

Název periodika

Journal of Functional Analysis

ISSN

0022-1236

e-ISSN

1096-0783

Svazek periodika

283

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

36

Strana od-do

109488

Kód UT WoS článku

000792625900004

EID výsledku v databázi Scopus

2-s2.0-85127493049