  We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.
