# Is the injective envelope functorial?

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.

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.

• Thanks. I also got this far. I can show that if $f$ is a $*$-isomorphism, then $\overline{f}$ is also a $*$-isomorphism, but my proof uses a big gun. Dec 1 '21 at 21:56
• The fact that nothing is mentioned in Paulsen's book or any of the foundational papers makes me suspicious. The injective envelope is a very slippery beast. Dec 1 '21 at 21:59
• Indeed, but Paulsen's treatment on the topic lacks some fundamentals (for example, he does not prove/mention that the $C^*$-algebra structure on an injective operator system obtained from the Choi-Effros product is unique, but uses this implicitly later in the chapter). Dec 1 '21 at 22:04
• I know that the above answers the problem in the negative, but (because I'm curious) what if one restricts to unital $\ast$-monomorphisms $f\colon A \to B$? By Hamana's construction of injective envelopes, we obtain an embedding $I(A) \to I(B)$ which extends $f: A \to B \subseteq I(B)$ so a $\ast$-homomorphism exists in this case (which was the obstruction above). I doubt it is unique though. Dec 1 '21 at 22:34
• Oh no, $I(A)$ sits inside $I(B)$ as an operator system, not as a C*-subalgebra in this case! Thanks for clearing that up for me! Dec 1 '21 at 22:48

As mentioned by Chris, injective envelopes are brutal when seen as C$$^*$$-algebras.

Let $$A=\text{UHF}(2^\infty)$$ and $$B$$ the hyperfinite II$$_1$$ factor. Take $$f$$ to be the inclusion map. We have $$I(B)=B$$, while $$I(A)$$ is a wild AW$$^*$$ factor of type III.

If $$g:I(A)\to B$$ is a $$*$$-homomorphism and $$\tau$$ is the trace on $$B$$, then $$\gamma=\tau\circ g$$ is a trace on $$I(A)$$. In a type III AW$$^*$$-factor any projection $$p$$ can be halved, so there exist $$p_1,p_2$$ with $$p=p_1+p_2$$ and $$p\sim p_1\sim p_2$$, which gives us the usual
$$\gamma(p)=\gamma(p_1)+\gamma(p_2)=2\gamma(p),$$ and so $$\gamma(p)=0$$ for any projection $$p$$. Thus $$\gamma=0$$. As $$\tau$$ is faithful, $$g=0$$.

In summary, $$f$$ is a $$*$$-monomorphism that admits no extension to a $$*$$-homomorphism, and in fact the only $$*$$-homomorphism $$I(A)\to I(B)$$ is the zero homomorphism.