An Ergodic Dilation of Completely Positive Maps

Abstract

We shall prove the following Stinespring-type theorem: there exists a triple $(π,\mathcal{H},\mathbf{V})$ associated with an unital completely positive map $Φ:\mathfrak{A}\rightarrow \mathfrak{A}$ on C* algebra $\mathfrak{A}$ with unit, where $\mathcal{H}$ is a Hilbert space, $π:\mathfrak{A\rightarrow B}(\mathcal{H})$ is a faithful representation and $\mathbf{V}$ is a linear isometry on $\mathcal{H}$ such that $π(Φ(a)=\mathbf{V}^*π(a)\mathbf{V}$ for all $a$ belong to $\mathfrak{A}$. The Nagy dilation theorem, applied to isometry $\mathbf{V}$, allows to construct a dilation of ucp-map, $Φ$, in the sense of Arveson, that satisfies ergodic properties of a $Φ$-invariante state $ϕ$ on $\mathfrak{A}$, if $Φ$ admit a $ϕ$-adjoint.