  For a class of linear maps on a von Neumann factor, we associate two objects,
bounded operators and trace class operators, both of which play the roles of
Choi matrices. Each of them is positive if and only if the original map on the
factor is completely positive. They are also useful to characterize positivity
of maps as well as complete positivity. It turns out that such correspondences
are possible for every normal completely bounded map if and only if the factor
is of type I. As an application, we provide criteria for Schmidt numbers of
normal positive functionals in terms of Choi matrices of $k$-positive maps, in
infinite dimensional cases. We also define the notion of $k$-superpositive
maps, which turns out to be equivalent to the property of $k$-partially
entanglement breaking.
