Morphisms of Cuntz-Pimsner algebras from completely positive maps

We introduce positive correspondences as right C*-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the C*-correspondence (Enchilada) category as a "retract". Kasparov's KSGNS construction provides a semi-functor from this semi-category onto the C*-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz-Pimsner algebras, and we describe classes of examples arising from projections on C*-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.