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 characterise them extending both the characterisation of operator
multipliers from
[16] and that of local Schur
multipliers from
[27]. We provide a description of the positive local operator
multipliers in terms of approximation by elements of canonical positive cones.