Does there exist a connection between contractive completely positive map and surjective map

Question

If $\psi:A \rightarrow M_n(\mathbb{C})$ is a c.c.p map.What is the relationship between c.c.p maps and surjective maps?Can we deduce that $\psi$ is a surjective map?If not,does there a close connection between a c.c.p map and a homomorphism?

Answer 1

No, not at all. Take any state $f$ on $A$, and then $a\longmapsto f(a)\,I$ is ucp and it is as far from surjective as it can be (well, more properly, the zero map is also ccp; my example is unital, at least).

As for $*$-homomorphisms, they are all ccp (easy exercise, that should definitely be done if it is not obvious to you).