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Thank so much for this answer! It really does seem that ccp maps cannot be expected to have closed range. I'll proceed to close the question. Again, thank you.
@RobertIsrael if you ever find the time and energy to provide a sketch of the proof that'd be great, because that'd mean that unital cp maps, for instance, do have closed range. Thank you as well
@YemonChoi that, erm, makes a lot of sense. What I meant was whether injective cp maps of norm 1 are isometric, but that is also false. Thank you for pointing it out.
