Completions of monoids with applications to the Cuntz semigroup

TL;DR

This paper develops a categorical framework for completing ordered monoids, specifically applied to the Cuntz semigroup of C*-algebras, facilitating computations of stabilized invariants.

Contribution

It introduces a functorial, explicit completion process for ordered monoids, connecting Cuntz semigroups of algebras and their stabilizations.

Findings

Provides a categorical completion method for monoids

Establishes functoriality and uniqueness of the construction

Applies framework to compute stabilized Cuntz semigroups

Abstract

We provide an abstract categorical framework that relates the Cuntz semigroups of the C$^*$-algebras $A$ and $A\otimes \mathcal{K}$. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C$^*$-algebras.