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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.
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If you want to talk cp maps, I guess you'll have to stay within operator systems, rather than operator spaces. – Martin Argerami Apr 8 '12 at 1:36
Oops, sorry about that! Edited appropriately. – Dave Gaebler Apr 8 '12 at 2:12
I'm a little puzzled about your question, as the image of a C$^*$-algebra through a CP map is not necessarily a C$^*$-algebra. – Martin Argerami Apr 20 '12 at 2:44
Yes, but sometimes it is. So there do exist inductive systems of CP maps between $C^*$-algebras. If you take the limit of such a system in the category of operator spaces, you will get an operator space that may or may not be a $C^*$-algebra, so that the limit might not exist with the category of CP maps between $C^*$-algebras. Or at least I think that's what I've been told...I'm looking to pin down the details. – Dave Gaebler Apr 24 '12 at 17:42

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