# The category Cu. Which maps are the correct ones? *-homomorphisms or cpc order zero maps?

### Joan Bosa (University of Glasgow)

Friday 14th March, 2014 16:50-17:35 504 Joseph Black Building

#### Abstract

In this talk we focus on the fact that the map induced by a cpc order zero φ:AB in the category Cu does not preserve the compactly containment relation. In particular, these kinds of maps are not in the category Cu, so that, in general, they may not be used in the classification of C*-algebras via the Cuntz Semigroup. Nevertheless, there is a subclass of these maps which preserves the relation, and so they can be used in the above mentioned classification. Our main result characterizes these maps via the positive element induced by the description of cpc order zero maps shown in [Winter, W. and Zacharias, J. Completely positive maps of order zero. Münster J. Math., 2, 2009, 311-324.].