Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-algebras for the Choi-Effros product.
Given a unital $*$-morphism $f: A \to B$, is it true that there exists a unique unital $*$-morphism $\overline{f}: I(A) \to I(B)$ that extends $f$?
Once the above question is answered positively (if the answer is positive), the following will probably be easy:
Is this construction functorial? I.e. is $I(-)$ a functor from the category of unital $C^*$-algebras to the category of unital $C^*$-algebras (with morphisms unital $*$-homomorphisms?
A reference is more than enough for me to be satisfied with an answer.
Best Answer
One can view $A$ and $B$ as sitting completely isometrically inside their injective envelopes $I(A)$ and $I(B)$. Then by injectivity a unital *-homomorphism (or more generally a unital completely positive map) $f:A\rightarrow B\subseteq I(B)$ extends to a unital completely positive map $\overline f:I(A) \rightarrow I(B)$.
[Edit: this should work]
Paulsen in this paper, Proposition 3.5, points out that any C$^*$-algebra containing $K(H)$ has injective envelope $B(H)$. Then $A = K(H) + \mathbb C I$ has $I(A) = B(H)$.
Consider the $*$-homomorphism $f:A\rightarrow \mathbb C$ given by $f(k+\alpha I) = \alpha$. Note that $I(\mathbb C) = \mathbb C$ and that any state of the Calkin algebra $B(H)/K(H)$ precomposed with the quotient map $q:B(H)\rightarrow B(H)/K(H)$ extends the map $f$. Therefore, $\overline f$ need not be a $*$-homomorphism or unique.