Dosiev (2008) 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. Joiţa, 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.

Dosiev（2008）为局部上的局部完全正图（简称：局部CP图）获得了Stinespring定理。 $C^{\ast }$-代数。在本文中，已经确定了此构造的最小化的适当概念，以确保唯一性，直到相关表示的统一等价。已经证明使用Radon-Nikodym型定理可以求解局部完全正图。

此外，一个无穷算子的Stinespring定理对Hilbert模块上的局部完全正图进行局部估值 $C^{\ast }$-代数（也称为局部CP诱导图）已被提出。在构造了乔亚（M.Joiţa）之后，显示了局部CP诱导图的Radon-Nikodym型定理。在这两种情况下，所获得的Radon-Nikodym导数都是在某些复杂的希尔伯特空间上的正收缩，并且具有向上过滤的还原子空间族。