# A revised augmented Cuntz semigroup

## TL;DR

This paper revises the augmented Cuntz semigroup construction to improve its functorial properties across all C*-algebras, enhancing its utility in classification problems.

## Contribution

It introduces a new version of the augmented Cuntz semigroup that is stable, continuous, and split exact for all C*-algebras, broadening its applicability.

## Key findings

- The revised construction is a stable, continuous, split exact functor.
- It applies to the entire category of C*-algebras, not just those of stable rank one.
- Enhances classification techniques for inductive limits of noncommutative CW complexes.

## Abstract

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: We show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.