Fixed points of dual quantum operations

Let phi(A) be a normal completely positive map on B(H) with Kraus operators {A(i)}(i=1)(infinity) . Denote M the subset of normal completely positive maps by M = {phi(A) : Sigma(infinity)(i=1) A(i) A(i)* <= 1 Sigma(infinity)(i=1) A(i) A(i)* <= 1 and phi A is normal). In this note, the relations between the fixed points of phi A and phi(+)(A) are investigated. We obtain that {B is an element of K(H): = phi(A) = (B) =B} = {B is an element of K(H): phi(+)(A)(B)=B), where K(H) is the set of all compact operators on H and phi(+)(A) is the dual of phi A is an element of M. In addition, we show that the map phi(A) -> phi(+)(A) is a bijection on M. (C) 2011 Elsevier Inc. All rights reserved.