Cuntz semigroups of ultraproduct C∗‐algebras

Abstract

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C ∗ -algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C ∗ -algebras. As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C ∗ -algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

Topics & Concepts