In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below.
Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is extreme in $CP[\mathcal{M}_n,\mathcal{M}_m]$, iff $\phi$ has an expression $\phi(x)=\sum_iV_i^*xV_i$ for all $x\in\mathcal{M}_n$ and $\lbrace V_i^*V_j\rbrace_{i,j}$ is a linearly independent set.
(I am assuming here unitality of the map. Otherwise, $\phi(I)=K\in\mathcal{M}_m$ for some fixed positive operator $K$.)
I have two questions.
1> Is there any extension of this theorem in any arbitrary $C^*$ algebra? (I mean extremal property, of course).
2> Even in finite dimension case, what can be the analogous result for (unital) completely bounded maps. Is it meaningful to ask this question for completely bounded maps?
I searched for the answers; but perhaps I am not giving the correct string; or did not understand some obvious points. Advanced thanks for any help.