I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. Or perhaps one had to tweak the morphisms by specifying "unital" or "contractive". I would appreciate either
- a reference that discusses direct limits of completely positive maps on some category or other, or
- an example (if indeed one exists) of a direct system of ($C^*$-algebras, CP maps) where the limit object is not a $C^*$-algebra? My best guess would be to iterate a CP map from some $A$ to itself, but I don't really know what the limit would look like.