Journal of Function Spaces
Volume 2015 (2015), Article ID 208923, 5 pages
http://dx.doi.org/10.1155/2015/208923

Research Article

Choi-Davis-Jensen Inequalities in Semifinite von Neumann Algebras

Turdebek N. Bekjan,¹ Kordan N. Ospanov,² and Asilbek Zulkhazhav²

¹College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
²L.N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan

Received 4 July 2015; Accepted 19 August 2015

Academic Editor: Quanhua Xu

Copyright © 2015 Turdebek N. Bekjan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

References

C. Davis, “A Schwarz inequality for convex operator functions,” Proceedings of the American Mathematical Society, vol. 8, pp. 42–44, 1957.

M. D. Choi, “A Schwarz inequality for positive linear maps on C ∗-algebras,” Illinois Journal of Mathematics, vol. 18, pp. 565–574, 1974.

M. Khosravi, J. S. Aujla, S. S. Dragomir, and M. S. Moslehian, “Refinements of Choi-Davis-Jensen's inequality,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 127–133, 2011.

J. Antezana, P. Massey, and D. Stojanoff, “Jensen's inequality for spectral order and submajorization,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 297–307, 2007.

T. Ando, Topics on Operator Inequalities, Hokkaido University, Sapporo, Japan, 1978.

L. G. Brown and H. Kosaki, “Jensen's inequality in semi-finite von Neumann algebras,” Journal of Operator Theory, vol. 23, no. 1, pp. 3–19, 1990.

F. Hansen and G. K. Pedersen, “Jensen's operator inequality,” The Bulletin of the London Mathematical Society, vol. 35, no. 4, pp. 553–564, 2003.

F. Hansen and G. K. Pedersen, “Jensen’s inequality for operators and Lowner’s theorem,” Mathematische Annalen, vol. 258, no. 3, pp. 229–241, 1982.

G. Pisier and Q. Xu, “Noncommutative L p-spaces,” in Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517, 2003.

Q. Xu, Noncommutative L p-Spaces and Martingale Inequalities, 2007.

T. Fack and H. Kosaki, “Generalized -numbers of -measure operators,” Pacific Journal of Mathematics, vol. 123, no. 2, pp. 269–300, 1986.

P. G. Dodds, T. K. Dodds, and B. de Pagter, “Noncommutative Banach function spaces,” Mathematische Zeitschrift, vol. 201, no. 4, pp. 583–597, 1989.

P. G. Dodds, T. K. Dodds, and B. de Pager, “Noncommutative Köthe duality,” Transactions of the American Mathematical Society, vol. 339, pp. 717–750, 1993.

P. G. Dodds and F. A. Sukochev, “Submajorisation inequalities for convex and concave functions of sums of measurable operators,” Positivity., vol. 13, no. 1, pp. 107–124, 2009.

V. Paulsen, Completely Bounded Maps and Operator Algebras, vol. 78 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2002.

T. N. Bekjan and M. Raikhan, “An Hadamard-type inequality,” Linear Algebra and its Applications, vol. 443, pp. 228–234, 2014.

R. Bhatia, M. D. Choi, and C. Davis, “Comparing a matrix to its off-diagonal part,” Operator Theory: Advances and Applications, vol. 40, pp. 151–163, 1989.

J.-C. Bourin and M. Uchiyama, “A matrix subadditivity inequality for and ,” Linear Algebra and its Applications, vol. 423, no. 2-3, pp. 512–518, 2007.