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

TL;DR

This paper investigates ring-theoretic finiteness properties of ultraproducts of Banach algebras, revealing differences from $C^*$-algebras and constructing counterexamples to certain transfer properties.

Contribution

It characterizes when ultraproducts of Banach algebras have finiteness properties and provides counterexamples showing these properties do not always transfer from components.

Findings

Ultraproducts' ring-theoretic properties depend on the entire sequence, not just 'ultrafilter many' components.

Counterexamples show Dedekind-finiteness can differ between components and ultraproducts.

Differences between Banach and $C^*$-algebras in property transfer are explained.

Abstract

We study ring-theoretic (in)finiteness properties -- such as \emph{Dedekind-finiteness} and \emph{proper infiniteness} -- of ultraproducts (and more generally, reduced products) of Banach algebras. Whilst 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 {\L}o\'s'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 \emph{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 \textit{stable rank one} is also studied for ultraproducts.