# Choi-Davis-Jensen Inequalities in Semifinite von Neumann Algebras

Authors: Turdebek N. Bekjan; Kordan N. Ospanov; Asilbek Zulkhazhav
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove the Choi-Davis-Jensen type submajorization inequalities on semifinite von Neumann algebras for concave functions and convex functions.


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## Body

## 1. Introduction

LetA 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, 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 B(H) and if f is an operator convex function on an interval I, then the so-called Choi-Davis-Jensen inequality (1)fΦA≤ΦfAholds for every self-adjoint operator A on H whose spectrum is contained in I, where B(H) is the C∗-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→B(H) with 0<Φ(1)≤I. Antezana et al. [4] obtained the following type of Choi-Davis-Jensen inequality. Let A,B be unital C∗-algebras, Φ:A→B a positive unital linear map, f a convex function defined on an open interval I, and A∈A such that A is self-adjoint and σ(a)⊂I. If B is a von Neumann algebra and f is monotone, then f(Φ(A))⪯Φ(f(A)) (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 Φ1,…,Φn be strictly positive linear maps from a unital C∗-algebra A into a unital C∗-algebra B and let Φ=∑j=1n‍Φj be unital. If f is an operator convex function on an interval I, then for every self-adjoint operator A∈A with spectrum contained in I, (2)fΦA≤∑j=1nΦj11/2fΦj1-1/2ΦjAΦj1-1/2Φj11/2≤ΦfA.The purpose of this paper is to extend (2) for measurable operators and for convex functions. Let N, M be semifinite von Neumann algebras, Φ1,…,Φn positive linear continuous maps from L0(N) into L0(M) such that Φ1|N,…,Φn|N are positive linear maps from N into M, and Φ=∑j=1n‍Φj unital, and let xk∈L0(N)+ (k=1,2,…,n). We will prove (3)∑k=1nΦkfxk≼f∑k=1nΦkxkfor any concave function f:[0,∞)→[0,∞) and (4)g∑k=1nΦkxk≼∑k=1nΦkgxkfor any convex function g:[0,∞)→[0,∞) with g(0)=0, where “≼” mains submajorization.This paper is organized as follows. Section2 contains some preliminary definitions. In Section 3, we prove the main result and related results.


## 2. Preliminaries

We use standard notions from theory of noncommutativeLp-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 τ. Let L0(M) denote the topological ∗-algebra of measurable operators with respect to (M,τ). The topology of L0(M) is determined by the convergence in measure. The trace τ can be extended to the positive cone L0+(M) of L0(M): (5)τx=∫0∞λdτeλx,where x=∫0∞‍λdeλ(x) is the spectral decomposition of x.Forx∈L0(M) we define (6)λsx=τes⊥xs>0,μtx=inf⁡s>0:λsx≤tt>0,where es⊥(x)=es,∞(x) is the spectral projection of |x| associated with the interval (s,∞). The function s↦λs(x) is called thedistribution function of x and μt(x) is thegeneralizeds-number of x. We will denote simply by λ(x) and μ(x) the functions s↦λs(x) and t↦μt(x), 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].Ifx,y∈L0(M), then we say that y is submajorised by x (in the sense of Hardy, Littlewood, and Polya) and write y≼x if and only if (7)∫0tμsyds≤∫0tμsxds,∀t>0.We remark that ifM=Mn and τ is the standard trace, then (8)μtx=sjx,t∈j-1,j,j=1,2,…and if x,y∈M, then y≼x is equivalent to (9)∑j=1ksjy≤∑j=1ksjx,1≤k≤n.For further information we refer the reader to [11–13].Letx,y be self-adjoint elements of M; we say that x spectrally dominates y, denoted by y⪯x, if e(α,∞)(y) is equivalent, in the sense of Murray-von Neumann, to a subprojection of e(α,∞)(x) for every real number α (see [6]). It is clear that if x spectrally dominates y, then y is submajorised by x.


## 3. Main Results

Lemma 1.
LetN and M be semifinite von Neumann algebras. Let Φ be a positive linear continuous map from L0(N) into L0(M) such that the restriction of Φ on N is a unital positive linear map from N into M. (i)
Iff:[0,+∞)→[0,+∞) is a concave function, then (10)Φfx≼fΦx,∀x∈L0N+.(ii)
Ifg:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then (11)gΦx≼Φgx,∀x∈L0N+.Proof.
(i) We may assumef(0)=0. It implies f is nondecreasing. First assume that x∈N+. We use same method as in the proof of Theorem 2 [4] (see Remark  3.2 in [4]). Let a>0. If ξ∈et:ft≤a(Φ(x))(H)∩e(a,∞)(Φ(f(x)))(H) with ξ=1, then f(〈Φ(x)ξ,ξ〉)≤a and 〈Φ(f(x))ξ,ξ〉>a. On the other hand, using Jensen’s inequality for the state 〈Φ(·)ξ,ξ〉 and nondecreasing concave function f, we get 〈Φ(f(x))ξ,ξ〉≤f(〈Φ(x)ξ,ξ〉)≤a. Therefore et:ft≤a(Φ(x))(H)∩ea,∞(Φ(f(x)))(H)=0. Thus et:ft≤a(Φ(x))∧e(a,∞)(Φ(f(x)))=0 and (12)ea,∞Φfx=ea,∞Φfx-et:ft≤aΦx∧ea,∞Φfx~et:ft≤aΦx∨ea,∞Φfx-et:ft≤aΦx≤1-et:ft≤aΦx=et:ft>aΦx=ea,+∞fΦx;that is, Φ(f(x))⪯f(Φ(x)). Hence (10) holds.
Now letx∈L0(N)+. For each m=1,2,…, observe that x∧m1∈M, and so, using the first case, it follows that (13)Φfx∧m1≼fΦx∧m1.Using the functional calculus and Corollary 1.2 in [13] observe that (14)x∧m1↑mx,fx∧m1↑mfxand so, by continuousness of Φ, it follows that (15)Φx∧m1↑mΦx,Φfx∧m1↑mΦfx.Using (vi) of Lemma  2.5 in [11] we obtain that (16)∫0tμsfΦxds=∫0tfμsΦxds=limm→∞⁡∫0tfμsΦx∧m1ds=limm→∞⁡∫0tμsfΦx∧m1ds≥limm→∞⁡∫0tμsΦfx∧m1ds=∫0tμsΦfxds,∀t>0;that is, (10) holds.
(ii) The proof is similar to the proof of (i).Theorem 2.
LetN, M be semifinite von Neumann algebras, Φ1,…,Φn positive linear continuous maps from L0(N) into L0(M) such that the restriction of Φk on N is a positive linear map from N into M(k=1,2,…,n), and Φ=∑j=1n‍Φj unital. (i)
Iff:[0,+∞)→[0,+∞) is a concave function, then, for xk∈L0(N)+(k=1,2,…,n), (17)∑k=1nΦkfxk⪯f∑k=1nΦkxk.(ii)
Ifg:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then, for xk∈L0(N)+(k=1,2,…,n), (18)g∑k=1nΦkxk⪯∑k=1nΦkgxk.Proof.
LetNn be the von Neumann algebra: (19)Nn=x10⋯00x2⋯0⋱00⋯xn,xk∈N,k=1,2,…,non Hilbert space H⊕⋯⊕H. Define Φ:Nn→M by (20)Φx10⋯00x2⋯0⋱00⋯xn=∑k=1nΦkxk;then Φ is a unital positive linear map from Nn into M. By Lemma 1, we obtain desired result.Using Theorem  5.3 in [14] and Theorem 2 we obtain the following.Proposition 3.
LetN,M be semifinite von Neumann algebras, Φ1,…,Φn positive linear continuous maps from L0(N) into L0(M) such that the restriction of Φk on N is a positive linear map from N into M(k=1,2,…,n), and Φ=∑j=1n‍Φj unital.(i)
Iff:[0,+∞)→[0,+∞) is a concave function, then, for xk∈N+(k=1,2,…,n), (21)∑k=1nΦkfxk≼f∑k=1nΦkxk≼∑k=1nfΦkxk.(ii)
Ifg:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then, for xk∈N+(k=1,2,…,n), (22)∑k=1ngΦkxk≼g∑k=1nΦkxk≼∑k=1nΦkgxk.Proposition 4.
LetN, M be semifinite von Neumann algebras and Φ1,…,Φn positive linear continuous maps from L0(N) into L0(M) such that the restriction of Φk on N is a positive linear map from N into M(k=1,2,…,n). Suppose Φ=∑j=1n‍Φj is unital trace-preserving positive linear map from M into M. If g:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then (23)∑k=1ngΦkx≼g∑k=1nΦkx≼∑k=1nΦkgx≼gx,∀x∈M+.Proof.
By Corollary  2.9 in [15] we have that ∑j=1n‍Φj 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.
Letak∈M(k=1,2,…,n) and ∑k=1n‍ak∗ak=1.(i)
Iff:[0,+∞)→[0,+∞) is a concave function, then, for xk∈L0(M)+(k=1,2,…,n), (24)∑k=1nak∗fxkak≼f∑k=1nak∗xkak≼∑k=1nfak∗xkak.(ii)
Ifg:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then(25)∑k=1ngak∗xak≼g∑k=1nak∗xak≼∑k=1nak∗gxak≼gx,∀x∈L0M+.LetMn be von Neumann algebra of n×n complex matrices, and let P1,P2,…,Pr be a family of mutually orthogonal projections in Cn such that ⊕j=1rPj=I, where I is unit matrix in Mn. Then the operation of taking A to C(A)=∑j=1r‍PjAPj is called a pinching of A. The pinching C:Mn→Mn is a trace-preserving positive map (see [17, 18]).Proposition 6.
LetC:Mn→Mn be a pinching.(i)
Iff:[0,+∞)→[0,+∞) is a concave function, then (26)CfA≼fCA,∀A∈Mn+.(ii)
Ifg:[0,+∞)→[0,+∞) is a convex function with g(0)=0, then (27)gCA≼CgA≼gA,∀A∈Mn+.Letx=(xk)k≥1 be a sequence in L0(M). Define (28)diag⁡xk=x10⋯0⋯0x2⋯0⋯⋱00⋯xk⋯⋱.Proposition 7.
Letx=(xk)k≥1 be a sequence in L0(M).(i)
Iff:[0,+∞)→[0,+∞), where f(t) is a concave function, then (29)f∑k≥1xk21/2p≤∑k≥1fxkp1/p,∀0<p≤1.(ii)
Ifg:[0,+∞)→[0,+∞), where g(t) is a convex function with g(0)=0, then (30)∑k≥1gxkp1/p≤g∑k≥1xk21/2p,∀1≤p<∞.Proof.
(i) Since(31)x1∗20⋯0x2∗2⋯⋮⋮⋱=Cx1∗2x1x2∗⋯x2x1∗x2∗2⋯⋮⋮⋱=Cx10⋯x20⋯⋮⋮⋱x1∗x2∗⋯00⋯⋮⋮⋱and f(t)p is concave, by Lemma 1, we get (32)Cfpx1∗x2∗⋯00⋯⋮⋮⋱≼fx1∗p0⋯0fx2∗p⋯⋮⋮⋱.Hence (33)f∑k≥1xk21/2pp=τfpx10⋯x20⋯⋮⋮⋱=τfpx1∗x2∗⋯00⋯⋮⋮⋱=τCfpx1∗x2∗⋯00⋯⋮⋮⋱≤τfx1∗p0⋯0fx2∗p⋯⋮⋮⋱=∑k≥1fxkp.Using the same arguments, we can prove (ii).



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