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تسجيل مستخدم جديدRing-theoretic (in)finiteness in reduced products of Banach algebras
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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}oss 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.
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