Extensions of Kadison’s inequality on positive linear maps

Let U be a unital C∗-algebra, B(H) the algebra of all bounded linear operators on a Hilbert space H, and P[U,B(H)] the set of all positive linear maps from U to B(H). The well-known Kadison’s inequality on unital positive linear maps is said that, if Φ∈P[U,B(H)] and Φ is unital, then Φ(A2)≥Φ(A)2 for each Hermitian A. This paper is to consider the extensions of Kadison’s inequality, some inequalities for unital Φ∈P[U,B(H)] are obtained which generalize Furuta’s result, and a complement to a result of Bourin and Ricard is provided.