Hilbert C∗-modules as a subcategory of operator systems and injectivity

In this paper, we study a category whose objects are Hilbert C ∗ -modules and whose morphisms are completely semi- ϕ -maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi- ϕ -maps on Hilbert C ∗ -modules, leading to an analog of the Arveson’s extension theorem for completely semi- ϕ -maps (in contrast with ϕ -maps). This theorem together with previous results suggest that the completely semi- ϕ -maps are proper generalizations of the completely positive maps.