Abstract
We prove the Choi-Davis-Jensen type submajorization inequalities on semifinite von Neumann algebras for concave functions and convex functions.
1. Introduction
Let and be -algebras, and let be a linear map between and . It is said to be positive, if for all positive operators we have . If for all strictly positive operators (), it follows that is strictly positive, then is said to be strictly positive. is called unital if , where 1 denotes the unities of the algebras.
Davis [1] and Choi [2] showed that if is a unital positive linear map on 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 whose spectrum is contained in , where is the -algebra of all bounded linear operators on Hilbert space . Khosravi et al. [3] proved that (1) still holds for positive linear map with . Antezana et al. [4] obtained the following type of Choi-Davis-Jensen inequality. Let be unital -algebras, a positive unital linear map, a convex function defined on an open interval , and such that is self-adjoint and . If is a von Neumann algebra and is monotone, then (spectral preorder). One can find some related results to these topics in [5–8].
In [3], the authors proved the following refinement of the Choi-Davis-Jensen inequality: let be strictly positive linear maps from a unital -algebra into a unital -algebra and let be unital. If is an operator convex function on an interval , then for every self-adjoint operator with spectrum contained in ,
The purpose of this paper is to extend (2) for measurable operators and for convex functions. Let , be semifinite von Neumann algebras, positive linear continuous maps from into such that are positive linear maps from into , and unital, and let (). We will prove for any concave function and for any convex function with , where “” mains submajorization.
This paper is organized as follows. Section 2 contains some preliminary definitions. In Section 3, we prove the main result and related results.
2. 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 Our main references are [9, 10] (see also [9] for more historical references).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 be a semifinite von Neumann algebra acting on a Hilbert space with a normal semifinite faithful trace . Let denote the topological -algebra of measurable operators with respect to . The topology of is determined by the convergence in measure. The trace can be extended to the positive cone of : where is the spectral decomposition of .
For we define 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 . For further information we refer the reader to [11].
If , 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 and is the standard trace, then and if , then is equivalent to For further information we refer the reader to [11–13].
Let be self-adjoint elements of ; 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 .
3. Main Results
Lemma 1. Let and be semifinite von Neumann algebras. Let be a positive linear continuous map from into such that the restriction of on is a unital positive linear map from into . (i)If is a concave function, then (ii)If is a convex function with , then
Proof. (i) We may assume . It implies is nondecreasing. First assume that . We use same method as in the proof of Theorem 2 [4] (see Remark 3.2 in [4]). Let . If with , then and . On the other hand, using Jensen’s inequality for the state and nondecreasing concave function , we get . Therefore . Thus and that is, . Hence (10) holds.
Now let . For each , observe that , and so, using the first case, it follows that Using the functional calculus and Corollary in [13] observe that and so, by continuousness of , it follows that 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).
Theorem 2. Let , be semifinite von Neumann algebras, positive linear continuous maps from into such that the restriction of on is a positive linear map from into , and unital. (i)If is a concave function, then, for , (ii)If is a convex function with , then, for ,
Proof. Let be the von Neumann algebra: on Hilbert space . Define by then is a unital positive linear map from into . By Lemma 1, we obtain desired result.
Using Theorem 5.3 in [14] and Theorem 2 we obtain the following.
Proposition 3. Let be semifinite von Neumann algebras, positive linear continuous maps from into such that the restriction of on is a positive linear map from into , and unital.(i)If is a concave function, then, for , (ii)If is a convex function with , then, for ,
Proposition 4. Let , be semifinite von Neumann algebras and positive linear continuous maps from into such that the restriction of on is a positive linear map from into . Suppose is unital trace-preserving positive linear map from into . If is a convex function with , then
Proof. By Corollary 2.9 in [15] we have that is a trace-preserving 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.
Corollary 5. Let and .(i)If is a concave function, then, for , (ii)If is a convex function with , then
Let be von Neumann algebra of complex matrices, and let be a family of mutually orthogonal projections in such that , where is unit matrix in . Then the operation of taking to is called a pinching of . The pinching is a trace-preserving positive map (see [17, 18]).
Proposition 6. Let be a pinching.(i)If is a concave function, then (ii)If is a convex function with , then
Let be a sequence in . Define
Proposition 7. Let be a sequence in .(i)If , where is a concave function, then (ii)If , where is a convex function with , then
Proof. (i) Since and is concave, by Lemma 1, we get Hence Using the same arguments, we can prove (ii).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Turdebek N. Bekjan is partially supported by NSFC Grant no. 11371304. Kordan N. Ospanov is partially supported by Project 3606/GF4 of Science Committee of Ministry of Education and Science of the Republic of Kazakhstan.