Extensions of Kadison’s inequality on positive linear maps

Abstract

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 ( A 2 ) ≥ 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.