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

Part of: General and miscellaneous Abstract harmonic analysis Methods of category theory in functional analysis Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  29 June 2020

Matthew Daws  and
Bence Horváth
Matthew Daws
Affiliation:
Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, United Kingdom e-mail: mdaws@uclan.ac.uk
Bence Horváth*
Affiliation:
Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic e-mail: hotvath@gmail.com
*
e-mail: horvath@math.cas.cz
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Abstract

We study ring-theoretic (in)finiteness properties—such as Dedekind-finiteness and proper infiniteness—of ultraproducts (and more generally, reduced products) of Banach algebras.

While 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.

Keywords

Asymptotic sequence algebraBanach algebrabicyclic monoidCuntz semigroupDedekind-finiteŁoś’s Theoremproperly infinitestable rank onesemigroup algebraultraproduct

MSC classification

Primary: 46M07: Ultraproducts
Secondary: 46H99: None of the above, but in this section 16B99: None of the above, but in this section 43A20: $L^1$-algebras on groups, semigroups, etc.
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 5 , October 2021 , pp. 1423 - 1458
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Horváth acknowledges with thanks the funding received from GAČR project 19-07129Y; RVO 67985840 (Czech Republic).

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