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.

Covering dimension of Cuntz semigroups

“…Given Cu-semigroups S and T , we use S ⊕ T to denote the Cartesian product S × T equipped with elementwise addition and order. It is straightforward to verify that S ⊕ T is a Cu-semigroup and that S ⊕ T is both the product and coproduct of S and T in the category Cu; see also [APT19,Proposition 3.10]. We omit the straightforward proof of the next result.…”