Ring-theoretic (in)finiteness in reduced products of Banach algebras

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F21%3A00546805" target="_blank" >RIV/67985840:_____/21:00546805 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.4153/S0008414X20000565" target="_blank" >https://doi.org/10.4153/S0008414X20000565</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4153/S0008414X20000565" target="_blank" >10.4153/S0008414X20000565</a>

Result language
angličtina
Original language name
Ring-theoretic (in)finiteness in reduced products of Banach algebras
Original language description
We study ring-theoretic (in)finiteness properties such as Dedekind-finiteness and proper infiniteness* of ultraproducts (and more generally, reduced products) of Banach algebras.nWhile we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the C∗-algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if “ultrafilter many” of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś’s Theorem, the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of fixed bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for C∗-algebras. Finally, the related notion of having stable rank one is also studied for ultraproducts.
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
2021
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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