Reference request: norm of completely positive map between C*-algebras is attained along approximate identity Let $A, B$ be C*-algebras, with $e_\lambda$ an approximate identity in $A$. I'm fairly certain that if $\varphi : A\to B$ is a (completely) positive map, then $\|\varphi\| = \lim_\lambda\|\varphi(e_\lambda)\|$. I could've sworn I had references for this result, but I can't seem to find it anywhere. I checked Blackadar's book "Operator Algebras", but it appears only to include the result that a unital linear map $\varphi : A\to B$ is positive if and only if it is contractive, which is not what I'm looking for.
I've found a reference for the unital case: if $\phi : A\to B$ is a positive map between unital C*-algebras, then $\|\phi\| = \|\phi(1)\|$. This can be found in Paulsen's "Completely Bounded Maps and Operator Algebras" (corollary 2.9), as a corollary of the Russo-Dye theorem. Have I perhaps misquoted the result in the first paragraph?
I'm also less certain that I've seen this similar result stated before: if $\psi : A\to B^*$ is (completely) positive, then $\|\psi\| = \lim_\lambda\|\psi(e_\lambda)\|$. Positivity in $B^*$ is defined as you might expect, giving us an order structure, and the order structure on $M_n(B^*)$ is obtained by the canonical identification of $M_n(B^*)$ with $M_n(B)^*$. I was hoping to find a proof of the result in the paragraph above and see what modifications needed to be made to prove this, but now I can't find either.
 A: A reference is Lance's book "Hilbert $C^*$-modules", lemma 5.3, p46. However, here is my proof which I believe is more straightforward than the one you will encounter in this reference.
Theorem: Let $\varphi: A \to B$ be a c.p. map between $C^*$-algebras and let $\{u_i\}_{i \in I}$ be an approximate unit for $A$. Then $\|\varphi\| = \lim_i \|\varphi(u_i)\|.$ In particular $\|\varphi\| = \|\varphi(1_A)\|$ if $A$ is unital.
Proof: By rescaling, we may assume $\|\varphi\| = 1$. The net $\{\| \varphi(u_i)\|\}_i$ is increasing and hence converges to its supremum in $\mathbb{R}$. Thus $L:=\lim_i \|\varphi(u_i)\|$ exists and clearly $L \leq \|\varphi\|=1$. If $a\in A$ with $\| a \| \leq 1$, then $u_ia^* a u_i \leq \|a^*a \| u_i u_i \leq u_i$. Applying the Schwarz-inequality,
$$\varphi(au_i)^*  \varphi(au_i) \leq \varphi((au_i)^*(au_i)) = \varphi(u_i a^*a u_i) \leq \varphi(u_i).$$
Taking norms, we obtain $\|\varphi(au_i)\|^2 \leq \|\varphi(u_i)\| \leq L$. Letting $i \to \infty$, we see that $\|\varphi(a)\|^2 \leq L$ so since $a$ was an arbitrary element in the unit ball of $A$ we get $1=\|\varphi\|^2 \leq L$. This ends the proof.
