  In this paper we give a simple sequence of necessary and sufficient finite
dimensional conditions for a positive map between certain subspaces of bounded
linear operators on separable Hilbert spaces to be completely positive. These
criterions are natural generalization of Choi's characterization for completely
positive maps between pairs of linear operators on finite dimensional Hilbert
spaces. We apply our conditions to a completely positive map between two trace
class operators on separable Hilbert spaces. A map $\mu$ is called a quantum
channel, if it is trace preserving, and $\mu$ is called a quantum subchannel if
it decreases the trace of a positive operator. We give simple neccesary and
sufficient condtions for $\mu$ to be a quantum subchannel. We show that $\mu$
is a quantum subchannel if and only if it has Hellwig-Kraus representation. The
last result extends the classical results of Kraus and the recent result of
Holevo for characterization of a quantum channel.
