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.

我们研究了 Banach 代数的超积（更一般地说，是约积）的环论（in）有限性特性——例如Dedekind 有限性和真无限性。

虽然我们根据代数的基本序列来描述超乘积何时具有这些环论性质，但我们发现，与 $C^*$ -代数设置，当且仅当代数的基础序列的“超滤许多”具有相同的属性时，超积具有环论有限性属性通常是不正确的。这似乎违反了 Łoś 定理的连续模型理论对应；不是的原因是对于一般的巴拿赫代数，我们考虑的环论性质不能通过考虑固定界代数的有界子集来验证。对于 Banach 代数，我们构造了反例来证明，例如，每个分量 Banach 代数不能是 Dedekind 有限而超积是 Dedekind 有限的，我们解释了为什么这样的反例对于 $C是不可能的^*$ - 代数。最后，具有稳定排名第一的相关概念 还研究了超产品。