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.
- Pub Date:
- July 2021
- Mathematics - Operator Algebras
- 29 pages - (Revision after submission to International Journal of Mathematics: dfn of uniform basis has been adjusted for a better fit, e.g, the Cuntz sg of the CAR algebra has a uniform basis and proof that Lsc(X,S) admits a uniform basis, where dimX =1, has been corrected)