A new look at the category Cu
Francesc Perera (Universitat Autonoma de Barcelona)
Thursday 8th May, 2014 16:00-17:00 Maths 416
The Cuntz semigroup $W(A)$ of a C$^*$-algebra $A$ is an important ingredient, both in the structure theory of C$^*$-algebras, and also in the current format of the Classification Programme. It is defined analogously to the Murray-von Neumann semigroup $V(A)$ by using equivalence classes of positive elements. The lack of continuity of $W(A)$, considered as a functor from the category of C$^*$-algebras to the category of abelian semigroups, led to the introduction (by
Coward, Elliott and Ivanescu) of a new 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 local 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. We also indicate how the category Cu should be recasted, by adding additional axioms.
(This is joint work with Ramon Antoine and Hannes Thiel.)