# Cuntz semigroups of ultraproduct C*-algebras

## TL;DR

This paper establishes the categorical properties of Cuntz semigroups for ultraproducts of C*-algebras, showing they form a bicomplete category and aligning their structure with ultraproducts of individual semigroups, with applications to K-theory and simplicity.

## Contribution

It proves the bicompleteness of the category of abstract Cuntz semigroups and demonstrates that scaled Cuntz semigroups of ultraproducts match the ultraproducts of individual semigroups, advancing the understanding of their structure.

## Key findings

- The category of abstract Cuntz semigroups is bicomplete.
- Scaled Cuntz semigroup of an ultraproduct equals the ultraproduct of scaled Cuntz semigroups.
- Characterization of when ultraproducts are simple and criteria for order properties.

## 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.