On Some Dilation Problem

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 $C^\ast$-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 $\phi$ from $\mathscr{A}$ to the $C^\ast$-algebra $\mathscr{L}(\mathscr{H})$ of all bounded linear operators acting on a Hilbert space $\mathscr{H}$ is of the form $\phi(\cdot) = S^\ast\pi(\cdot)S$, where $\pi$ is a $\ast$-representation of $\mathscr{A}$ on a Hilbert space $\mathscr{K}$ and $S$ is a bounded linear operator from $\mathscr{H}$ to $\mathscr{K}$. The aim of this article is to consider some dilation problem for completely positive maps defined on an abstract Hilbert $C^\ast$-module and taking value in a Hilbert $C^\ast$-module of linear continuous operators from a Hilbert space $\mathscr{H}$ to a Hilbert space $\mathscr{K}$. We prove an analogue of Stinespring theorem for these maps and show that any two minimal Stinespring representations are unitarily equivalent.

Keywords: Completely positive maps, $\ast$-representation, locally compact group, Stinespring representation, Hilbert $C^\ast$-module, locally $C^\ast$-algebra.