# Extension of Choi's theorem on extreme completely positive maps

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.

The only characterization I know of the extreme points of the CP maps on a C$^*$-algebra is Theorem 1.4.6 in Arveson's "Subalgebras of C$^*$-Algebras".
Arveson proves that if $\phi:A\subset B(H)\to B$ is CP with $\phi(I)=K$, and if $\phi=V^*\pi V$ is a Stinespring decomposition of $\phi$, then $\phi$ is extremal if and only if $\overline {VH}$ is faithful for $\pi(B)'$.
Here, faithful means that for every $x\in\pi(B)'$, $pxp=0\ \iff\ x=0$, where $p$ is the projection onto $\overline{VH}$.
• I'm no expert myself, but I did ask a colleague who is a C$^*$-convexity expert, and his answer was "Arveson I". The way I see it, there is no "Kraus formula" in a general C$^*$-algebra, so Stinespring is the best you have. In any case, you might want to ask this at Math Overflow. Oct 15 '12 at 9:43