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