Uniformly Based Cuntz semigroups and approximate intertwinings

TL;DR

This paper develops a uniform approach to approximating elements in Cuntz semigroups and introduces an approximate intertwining theory, with applications to classifying unitaries in AF-algebras.

Contribution

It introduces a new framework for uniform approximation in Cu-semigroups and develops an approximate intertwining theory, extending previous metrics and classification results.

Findings

Defined a semimetric on Cu-morphisms for uniform approximation.

Developed an approximate intertwining theory for Cu.

Applied the theory to classify unitaries in unital AF-algebras.

Abstract

We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a Cu-semigroup by a rapidly increasing sequence. This approximation induces a semimetric on the set of Cu-morphisms, generalizing Cu-metrics that had been constructed in the past for some particular cases. Further, we develop an approximate intertwining theory for the category Cu. Finally, we give several applications such as the classification of unitary elements of any unital AF-algebra by means of the functor Cu.