  Anar A. Dosiev in [Local operator spaces, unbounded operators and multinormed
$C^*$-algebras, J. Funct. Anal. 255 (2008), 1724-1760], obtained a
Stinespring's theorem for local completely positive maps (in short: local
CP-maps) on locally $C^{\ast}$-algebras. In this article a suitable notion of
minimality for this construction has been identified so as to ensure uniqueness
up to unitary equivalence for the associated representation. Using this a
Radon-Nikodym type theorem for local completely positive maps has been proved.
  Further, a Stinespring's theorem for unbounded operator valued local
completely positive maps on Hilbert modules over locally $C^{\ast}$-algebras
(also called as local CP-inducing maps) has been presented. Following a
construction of M. Joi\c{t}a, a Radon-Nikodym type theorem for local
CP-inducing maps has been shown. In both cases the Radon-Nikodym derivative
obtained is a positive contraction on some complex Hilbert space with an upward
filtered family of reducing subspaces.
