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

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
A purely infinite Cuntz-like Banach *-algebra with no purely infinite ultrapowers
Original language description
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.
Czech name
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Czech description
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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

Project
<a href="/en/project/GJ19-07129Y" target="_blank" >GJ19-07129Y: Linear-analysis techniques in operator algebras and vice versa</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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