Hannes Thiel, Fields Institute, Toronto
“Recasting the Cuntz category”
The Cuntz semigroup W(A) of a C*-algebra A plays an important role in the structure theory of C*-algebras and the related Elliott classification program. It is defined analogously to the Murray-von Neumann semigroup V(A) by using equivalence classes of positive elements instead of projections.
Coward, Elliott and Ivanescu introduced the category Cu of (completed) Cuntz semigroups. They showed that the Cuntz semigroup of the stabilized C*-algebra is an object in Cu and that this assignment extends to a continuous functor.
We introduce a category W of (pre-completed) Cuntz semigroups such that the original definition of Cuntz semigroups defines a continuous functor from C*-algebras to W. There is a completion functor from W to Cu such that the functor Cu is naturally isomorphic to the completion of the functor W.
If time permits, we will apply this to construct tensor products in W and Cu. (joint work with Ramon Antoine and Francesc Perera).