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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

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

2021

Kód důvěrnosti údajů

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

Název periodika

Canadian Journal of Mathematics

ISSN

0008-414X

e-ISSN

1496-4279

Svazek periodika

73

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

CA - Kanada

Počet stran výsledku

36

Strana od-do

1423-1458

Kód UT WoS článku

000721260300009

EID výsledku v databázi Scopus

2-s2.0-85107670293