LAUSR

Let B (H) be the C∗–algebra of all bounded linear operators on a Hilbert spaceH . As usual, we reserve m, M for scalars and I for the identity operator onH . A self adjoint operator A is said to be positive (written A ≥ 0) if 〈Ax, x〉 ≥ 0 for all x ∈ H , while it is said to be strictly positive (written A > 0) if A is positive and invertible. If A and B are self adjoint, we write B ≥ A in case B−A ≥ 0. The Gelfand map f (t) 7→ f (A) is an isometrical ∗–isomorphism between the C∗–algebra C (σ (A)) of continuous functions on the spectrum σ (A) of a self adjoint operator A and the C∗–algebra generated by A and the identity operator I. This is called the functional calculus of A. If f , 1 ∈ C (σ (A)), then f (t) ≥ 1 (t) (t ∈ σ (A)) implies f (A) ≥ 1 (A) (see [11, p. 3]). A linear map Φ : B (H)→ B (K ) is positive if Φ (A) ≥ 0 whenever A ≥ 0. It’s said to be unital if Φ (I) = I. For any strictly positive operator A,B ∈ B (H ) and 0 ≤ v ≤ 1, we write