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

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.

A purely infinite Cuntz-like Banach $*$-algebra with no purely infinite ultrapowers

“…We thus focus on ultrapowers in this paper. As in [7], and perhaps not surprisingly from the perspective of continuous model theory, we find that an ultrapower (A) U is purely infinite if and only if it satisfies a "metric" form of the definition, where we have some sort of norm control. From this perspective, it is unsurprising to find that the fact that purely infinite C * -algebras have purely infinite ultrapowers follows from such norm control always being available.…”

“…Introduction. We continue our study of infiniteness properties of Banach algebras, and how these interact with reduced products, in the continuous model theory sense, which we initiated in [7]. Recall that an idempotent p in an algebra A is infinite if it is (algebraically Murray-von Neumann) equivalent to a proper sub-idempotent of itself.…”

“…Recall that an idempotent p in an algebra A is infinite if it is (algebraically Murray-von Neumann) equivalent to a proper sub-idempotent of itself. One prominent property which we did not study in [7] is that of being purely infinite, which for simple rings could be defined by saying that every left ideal contains an infinite idempotent. We discuss this notion, and the literature surrounding it, in Section 1.2 below.…”

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We continue our investigation, from [7], of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a C * -algebra is purely infinite if and only if any of its ultrapower is.

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Perturbations of surjective homomorphisms between algebras of operators on Banach spaces

“…We recall a folklore result, a stronger version of which was proved by Zemánek in [30,Lemma 3.1]. A self-contained elementary proof can be found in [8,Lemma 2.8].…”