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.
This paper argues that the unitary Cuntz semigroup, introduced in [9] and termed Cu 1 , contains crucial information regarding the classification of non-simple C * -algebras. We exhibit two (non-simple) C * -algebras that agree on their Cuntz semigroups, termed Cu, and their K 1 -groups and yet disagree at level of their unitary Cuntz semigroups. In the process, we establish that the unitary Cuntz semigroup contains rigorously more information about non-simple C * -algebras than Cu and K 1 alone.
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 Cusemigroup by a rapidly increasing sequence. This approximation induces a semimetric on the set of Cumorphisms, 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.
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