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

Morphisms of Cuntz-Pimsner algebras from completely positive maps