Banach Algebras and Applications

Ciaś, Tomasz

doi:10.1515/9783110602418

Cited by 1 publication

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“…We recall, [25,Proposition 3.1], that this is equivalent to either the left, or the right, topological stable rank of A being 1. We recall (see [17,Lemma 2.1] for example) that having stable rank one implies being Dedekind finite. As shown in [17, Example 2.2], the converse does not hold.…”

Section: Stable Rank Onementioning

confidence: 99%

“…Having stable rank one is not a three-space property (see [25,Examples 4.13]), but in our special situation, we can say something. The following is the Banach-algebraic analogue of the ring-theoretic lemma [17,Lemma 2.10]. We recall that if A is a unital algebra over a field K with multiplicative identity 1 A , and J ⊴ A is a two-sided ideal, then J denotes the unital subalgebra K1 A + J.…”

Section: Stable Rank One As a "Three Space Property"mentioning

confidence: 99%

“…We recall that if A is a unital algebra over a field K with multiplicative identity 1 A , and J ⊴ A is a two-sided ideal, then J denotes the unital subalgebra K1 A + J. Moreover, inv( J) = inv(A) ∩ J (see [17,Lemma 2.4]).…”

Section: Stable Rank One As a "Three Space Property"mentioning

confidence: 99%

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

Daws¹,

Horváth²

2020

Can. J. Math.-J. Can. Math.

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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.

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Section: Stable Rank Onementioning

confidence: 99%

Section: Stable Rank One As a "Three Space Property"mentioning

confidence: 99%