Fixed points of completely positive maps and its dual maps *

Let A ⊂ B(H) be a row contraction and Φ_A determined by A be a completely positive map on B(H). In this paper, we mainly consider fixed points of Φ_A and its dual map {Φ_A}^ † . It is given that Φ_A(X) ≤ X (or Φ_A(X) ≥ X) implies Φ_A(X) = X and {Φ_A}^ † (X) = X when X ∈ B(H) is a compact operator. Some necessary conditions of Φ_A(X) = X and {Φ_A}^ † (X) = X are given. MR (2010) Subject Classification 47A05; 47A62