Fixed points of dual quantum operations

Let ϕ A be a normal completely positive map on B ( H ) with Kraus operators { A i } i = 1 ∞ . Denote M the subset of normal completely positive maps by M = { ϕ A : ∑ i = 1 ∞ A i A i ⁎ ⩽ I , ∑ i = 1 ∞ A i ⁎ A i ⩽ I and ϕ A is normal } . In this note, the relations between the fixed points of ϕ A and ϕ A † are investigated. We obtain that { B ∈ K ( H ) : ϕ A ( B ) = B } = { B ∈ K ( H ) : ϕ A † ( B ) = B } , where K ( H ) is the set of all compact operators on H and ϕ A † is the dual of ϕ A ∈ M . In addition, we show that the map ϕ A → ϕ A † is a bijection on M .