Noncommutative Geometry and Topology in Prague

Ring-theoretic (in)finiteness in ultraproducts of Banach algebras

We say that a unital ring is “Dedekind-finite” (or “directly finite” or “DF”) if every left-invertible element is also right-invertible. In other words, a ring is DF if and only if the only idempotent which is algebraically Murray–von Neumann equivalent to the unit of the ring is the unit itself. A related notion is that of the so-called “proper infiniteness” and “pure infiniteness”. These notions are well-studied in non-commutative ring theory, as well as in the theory of C*-algebras, but not that much in the Banach algebraic setting.

In our talk we outline how these properties are preserved under taking ultraproducts of Banach algebras, and vice versa. As one might expect, in the general Banach algebraic setting the situation differs quite a bit to the C*-algebraic one. Time permitting we say a few words about the related notion of having “stable rank one” and connections to the area of Continuous Model Theory. The talk is based on joint work with Matthew Daws (UCLAN, United Kingdom).

[1]  M. Daws and B. Horváth. Ring-theoretic (in)finiteness in reduced products of Banach algebras. Canad. J. Math. 73(5):1423–1458 (2021) DOI:10.4153/S0008414X20000565.
[2]  M. Daws and B. Horváth. A purely infinite Cuntz-like Banach ∗-algebra with no purely infinite ultrapowers. J. Funct. Anal. 283(1):Id/No 109488 (2022) DOI:10.1016/j.jfa. 2022.109488.