Abstract:
The study of completely positive linear maps is motivated by
applications of the theory of completely positive linear maps to
quantum information theory, where operator valued completely
positive linear maps on $C^*$-algebras are used as a mathematical
model for quantum operations and quantum probability. Stinespring in
the first part of 20 century showed that a completely positive
linear map $\varphi$ from $\mathscr{A}$ to the $C^*$-algebra
$\mathscr{L}(\mathcal{H})$ of all bounded linear operators acting on
a Hilbert space $\mathcal{H}$ is of the form
$\varphi(\cdot)=S^*\pi(\cdot)S$, where $\pi$ is a $*$-representation
of $\mathscr{A}$ on a Hilbert space $\mathcal{K}$ and $S$ is a
bounded linear operator from $\mathcal{H}$ to $\mathcal{K}$. The aim
of this article is to consider some dilation problem for
completely positive maps defined on an abstract Hilbert $C^*$-module
and taking value in a Hilbert $C^*$-module of linear continuous
operators from a Hilbert space $H$ to a Hilbert space $K$. We prove
an analogue of Stinespring theorem for these maps and show that any two
minimal Stinespring representations are unitarily equivalent.
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