Uniformly based Cuntz semigroups and approximate intertwinings

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.

The unitary Cuntz semigroup on the classification of non-simple C⁎-algebras

“…The (proof of the) uniqueness result obtained in [3] roughly shows that converse holds whenever A is a unital AF-algebra. More particularly, it stipulates that Cu distinguishes unitary elements of unital AF-algebras.…”

“…Throughout the paper, we will make use of the discrete Cu-semimetric for practical reasons. We recall that these two metrics are topologically equivalent as shown in [3,Proposition 5.5].…”

“…We take the opportunity to remark that the proof relies on [3, Lemma 5.7] that has been proven for inductive sequences (instead of inductive systems) and hence the result should be narrowed down to the separable case. Nevertheless, [3,Lemma 5.7] can be extended to any inductive systems by a similar proof using the following characterization of inductive limits in the category Cu.…”

“…Then for any n ∈ N, there exists a partial Cu-morphism Theorem 3.7] for the result and [3] for details on uniform bases for Cu-semigroups. )…”

The Cuntz semigroup of unital commutative AI-algebras

“…Remark 4. 6 Let 𝑆 be an Lsc-like Cu-semigroup. Since the semigroup generated by ↓ 𝑒 is dense in 𝑆, 𝑒 is an order unit.…”