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.



Is the injective envelope functorial?
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](https://www.ams.org/journals/tran/2011-363-09/S0002-9947-2011-05203-7/S0002-9947-2011-05203-7.pdf), 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.


