A domain is a partially ordered set that is order-theoretically complete (every increasing net has a supremum) and with a good notion of approximation. A C*-algebra is a self-adjoint, closed subalgebra of bounded, linear operators on a Hilbert space.

In 1978, Cuntz showed that there is a connection between these concepts:

The comparison theory of positive elements in a C*-algebra can be encoded in a partially ordered semigroup -- nowadays called the Cuntz semigroup of a C*-algebra. In 2008, Coward-Elliott-Ivanescu showed that the Cuntz semigroup is a domain.

In this talk, I will give an introduction to domains and Cuntz semigroups and then discuss the available categorical notion of ultraproducts, an ad-hoc concept of the Löwenheim-Skolem condition and the search for an underlying model theory.