Extremality conditions for generalized channels

A generalized channel is a completely positive map that preserves trace on a given subspace. We find conditions under which a generalized channel with respect to a positively generated subspace J is an extreme point in the set of all such generalized channels. As a special case, this yields extremality conditions for quantum protocols. In particular, we obtain new extremality conditions for quantum 1-testers with 2 outcomes, which correspond to yes/no measurements on the set of quantum channels.