Fixed points of completely positive maps and their dual maps

Abstract Let $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ A ⊂ B ( H ) be a row contraction and $\Phi _{\mathcal {A}}$ Φ determined by {A}$ completely positive map on ${\mathcal . In this paper, we mainly consider fixed points of its dual {A}}^{\dagger}$ † It is given that {A}}(X)\leq X $ X ≤ (or {A}}(X)\geq ≥ ) implies {A}}(X)= X$ = {A}}^{\dagger}(X)= when $X\in {\mathcal ∈ compact operator. Some necessary conditions are given.