An Ergodic Dilation of Completely Positive Maps

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