无界算子值局部完全正映射的Stinespring定理及其应用
Stinespring's Theorem for Unbounded Operator valued Local completely positive maps and Its Applications
摘要：
阿纳尔河Dosiev在[Local算子空间，unbounded operators and multinormed $C^*$ -algebras，J. Funct. Anal. 255（2008），1724-1760]，得到了局部$C^{\ast}$ -代数上局部完全正映射（简称：局部CP-映射）的Stinespring定理。在这篇文章中，已经确定了一个合适的概念，这种结构的极小，以确保唯一性，以酉等价的相关表示。利用这一定理证明了局部完全正映射的Radon-Nikodym型定理。进一步，给出了局部$C^{\ast}$ -代数上Hilbert模上的无界算子值局部完全正映射（又称局部CP诱导映射）的Stinespring定理.根据M. Joienza，证明了局部CP诱导映射的Radon-Nikodym型定理。在这两种情况下，得到的Radon-Nikodym导数是复希尔伯特空间上的正压缩，其上有一个向上滤子族.
英文摘要：
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. 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.
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