k Positivity and Schwarz inequality for two linear maps

For a given A ∈ M n , we define the linear maps Ψ A , - and Φ A , + from M n into M n by Ψ A , - ( X ) : = t r ( A X ) I - X and Φ A , + ( X ) : = t r ( A X ) I + X t , where M n is the n × n matrix algebra and X t is the transpose of X . First, we show that Ψ A , - is a k positive map if and only if A ≥ I and ‖ A - 1 ‖ ( k ) ≤ 1 . Then Φ A , + is a k ( k ≥ 2 ) positive map if and only if A ≥ 0 with λ i λ j ≥ 1 for all i ≠ j = 1 , 2 , … n , where λ 1 , λ 2 , … λ n are eigenvalues of A . At last, we consider k positivity of above maps on B ( H ) (the von Neumann algebra of all bounded linear operators on the separable complex Hilbert space H ) .