Choi-Davis-Jensen Inequalities in Semifinite von Neumann Algebras

Let A and B be C-algebras, and let Φ be a linear map between A and B. It is said to be positive, if for all positive operators A ∈ A we have Φ(A) ≥ 0. If for all strictly positive operators A ∈ A (A > 0), it follows that Φ(A) is strictly positive, thenΦ is said to be strictly positive.Φ is called unital if Φ(1) = 1, where 1 denotes the unities of the algebras. Davis [1] and Choi [2] showed that ifΦ is a unital positive linearmap onB(H) and iff is an operator convex function on an interval I, then the so-called Choi-Davis-Jensen inequality


Introduction
Let A and B be * -algebras, and let Φ be a linear map between A and B. It is said to be positive, if for all positive operators ∈ A we have Φ( ) ≥ 0. If for all strictly positive operators ∈ A ( > 0), it follows that Φ( ) is strictly positive, then Φ is said to be strictly positive. Φ is called unital if Φ(1) = 1, where 1 denotes the unities of the algebras.
Davis [1] and Choi [2] showed that if Φ is a unital positive linear map on (H) and if is an operator convex function on an interval , then the so-called Choi-Davis-Jensen inequality holds for every self-adjoint operator on H whose spectrum is contained in , where (H) is the * -algebra of all bounded linear operators on Hilbert space H. Khosravi et al. [3] proved that (1) still holds for positive linear map Φ : A → (H) with 0 < Φ(1) ≤ . Antezana et al. [4] obtained the following type of Choi-Davis-Jensen inequality. Let A, B be unital *algebras, Φ : → a positive unital linear map, a convex function defined on an open interval , and ∈ A such that is self-adjoint and ( ) ⊂ . If B is a von Neumann algebra and is monotone, then (Φ( )) ⪯ Φ( ( )) (spectral preorder). One can find some related results to these topics in [5][6][7][8].

Preliminaries
We use standard notions from theory of noncommutative -spaces. Our main references are [9,10] (see also [9] for more historical references). Throughout the paper, let M be a semifinite von Neumann algebra acting on a Hilbert space H with a normal semifinite faithful trace . Let 0 (M) denote the topological * -algebra of measurable operators with respect to (M, ). The topology of 0 (M) is determined by the convergence in measure. The trace can be extended to the positive cone + 0 (M) of 0 (M): where ⊥ (| |) = ( ,∞) (| |) is the spectral projection of | | associated with the interval ( , ∞). The function → ( ) is called the distribution function of and ( ) is the generalized -number of . We will denote simply by ( ) and ( ) the functions → ( ) and → ( ), respectively. It is easy to check that both are decreasing and continuous from the right on (0, ∞). For further information we refer the reader to [11].
If , ∈ 0 (M), then we say that is submajorised by (in the sense of Hardy, Littlewood, and Polya) and write ≼ if and only if We remark that if M = M and is the standard trace, then and if , ∈ M, then ≼ is equivalent to For further information we refer the reader to [11][12][13].
Let , be self-adjoint elements of M; we say that spectrally dominates , denoted by ⪯ , if ( ,∞) ( ) is equivalent, in the sense of Murray-von Neumann, to a subprojection of ( ,∞) ( ) for every real number (see [6]). It is clear that if spectrally dominates , then is submajorised by .
Using the functional calculus and Corollary 1.2 in [13] observe that ∧ 1↑ , and so, by continuousness of Φ, it follows that Journal of Function Spaces 3 Using (vi) of Lemma 2.5 in [11] we obtain that that is, (10) holds.
(ii) The proof is similar to the proof of (i).
Proof. Let N be the von Neumann algebra: then Φ is a unital positive linear map from N into M. By Lemma 1, we obtain desired result.
Using Theorem 5.3 in [14] and Theorem 2 we obtain the following.
Proof. By Corollary 2.9 in [15] we have that ∑ =1 Φ is a tracepreserving positive contraction. Using Theorem 5.3 in [14], Lemma 3.1 in [16] (it is also holds for the semifinite case), and Theorem 2 we obtain the desired result.
Let M be von Neumann algebra of × complex matrices, and let 1 , 2 , . . . , be a family of mutually orthogonal projections in C such that ⊕ =1 = , where is unit matrix in M . Then the operation of taking to C( ) = ∑ =1 is called a pinching of . The pinching C : M → M is a trace-preserving positive map (see [17,18]).
Using the same arguments, we can prove (ii).