- Open Access
Some convexity inequalities in noncommutative -spaces
Journal of Inequalities and Applications volume 2014, Article number: 385 (2014)
In this paper, we prove some convexity inequalities in noncommutative spaces generalizing the previous result of Hiai and Zhan. Moreover, we generalize a variational inequality for positive definite matrices due to Hansen to the case of noncommutative spaces.
MSC:47A30, 47L05, 47L50.
Let be a unitarily invariant norm on matrices. For matrices A, B, X in with A, B being positive semidefinite and X arbitrary, Bhatia and Davis  presented the matrix Cauchy-Schwarz inequality . In 2002, Hiai and Zhan  proved that under the same assumption for matrices A, B, and X, the function is convex on for each . Among other things, this convexity result interpolated the above matrix Cauchy-Schwarz inequality showed by Bhatia and Davis. Another result of interest is that, in 2014, Hansen , among other things, gave new and simplified proofs of the Carlen-Lieb theorem concerning concavity of certain trace functions by applying the theory of operator monotone functions. Let ℋ be a Hilbert space and denote the -algebra of all bounded linear operators on ℋ. Equipped with the usual adjoint as involution becomes a unital -algebra. We say a function is a unitary invariant norm (or symmetric norm) if it is a norm satisfying the invariance property for all x and all unitary operators u and v in .
In this paper, we consider the noncommutative spaces of τ-measurable operators affiliated with a semi-finite von Neumann algebra equipped with a normal faithful semi-finite trace τ. On one hand, we use the method of Hiai and Zhan, via the notion of generalized singular value studied by Fack and Kosaki , to prove some convexity inequalities in noncommutative spaces generalizing the previous result of Hiai and Zhan. On the other hand, by making use of the joint convexity and concavity of trace functions obtained by Bekjan  we generalize a variational inequality for positive definite matrices due to Hansen to the case of noncommutative spaces.
Throughout the paper, unless specified, we always denote by ℳ a semi-finite von Neumann algebra acting on the Hilbert space ℋ, with a normal faithful semi-finite trace τ. We denote the identity in ℳ by 1 and let denote the projection lattice of ℳ. A closed densely defined linear operator x in ℋ with domain is said to be affiliated with ℳ if for all unitary u which belong to the commutant of ℳ. If x is affiliated with ℳ, then x is said to be τ-measurable if for every there exists a projection such that and . The set of all τ-measurable operators will be denoted by , or simply . The set is a ∗-algebra with sum and product being the respective closures of the algebraic sum and product. A closed densely defined linear operator x admits a unique polar decomposition , where u is a partial isometry such that and (with ). We call and the left and right supports of x, respectively. Thus . Moreover, if x is self-adjoint, we let , the support of x.
Let be the positive part of ℳ. Set and let be the linear span of , we will often abbreviate and , respectively, as and . Let , the noncommutative -space is the completion of , where , . In addition, we put and denote by () the usual operator norm. It is well known that are Banach spaces under for and they have a lot of expected properties of classical -spaces (see  or ).
Let x be a τ-measurable operator and . The ‘t th singular number (or generalized s-number) of x’ is defined by
See  for basic properties and detailed information on the generalized s-numbers.
To achieve one of our main results, we state for easy reference the following fact, obtained from , which will be applied below.
Lemma 2.1 If , then the function is jointly concave in strictly positive operators .
3 Main results
Lemma 3.1 Let such that xy is a self-adjoint τ-measurable operator and let , then
Proof Notice that xy is a self-adjoint τ-measurable operator; then
Let f be an increasing function on satisfying and where is convex, then applying Lemma 2 of  we have
In particular, if , then
Taking the , by the usual Fatou lemma we get
Lemma 3.2 Let and . Let , then
Proof It suffices to prove that
Applying Lemma 2 of  together with the Hölder inequality we obtain
which implies the lemma. □
Theorem 3.3 Let be positive operators with and let , then for every real number , the function
is convex on the interval and attains its minimum at .
Proof (i) First we assume that τ is finite. By the density of in , we first consider the case . Since ϕ is continuous and symmetric with respect to , all the conclusions will follow after we show that
for . By (1) we have
Multiplying the above two inequalities we obtain
For the general case, namely, for any , there exist such that , are invertible and , in . Moreover, we have is convex for all and attains its minimum at . Applying Theorem 3.7 of  and using the method to prove Lemma 3.3 in , we get , in . Hence, we obtain , . Therefore, is convex on and attains its minimum at .
(ii) In the general case when τ is semi-finite, there exists an increasing family such that for every and such that converges to 1 in the strong operator topology (see  or ). Thus, is finite for each . Let , then . Write , , it follows from the case (i) that the function is convex on and attains its minimum at . In view of the fact that , in , by a similar computation we derive . Therefore, is convex on and attains its minimum at . □
An immediate consequence of Theorem 3.3 interpolates the inequality (1) as follows.
Corollary 3.4 Let x, y, z be τ-measurable operators as in Theorem 3.3. For every ,
holds for .
The following is another example of convex functions involving the noncommutative -norm.
Theorem 3.5 Let () be positive operators and let . Then, for every positive real number r, the function is convex on .
Proof The same proof as of Theorem 4 of Hiai and Zhan  works. □
If , that is to say, we consider the von Neumann algebra ℳ to be equipped with normal faithful finite unital trace, then we can also get the following noncommutative analog of the variational inequality and we refer the readers to Theorem 2.5 of  for more details of the Carlen-Lieb theorems concerning the concavity of certain trace functions.
Theorem 3.6 Let be positive operators and let . Then
for every such that z and are invertible. If , take
and we get the equality.
Proof The same proof as of Theorem 2.5 of Hansen  works. □
Bhatia R, Davis C: A Cauchy-Schwarz inequality for operators with applications. Linear Algebra Appl. 1995, 223/224: 119-129.
Hiai F, Zhan X: Inequalities involving unitarily invariant norms and operator monotone functions. Linear Algebra Appl. 2002, 341: 151-169. 10.1016/S0024-3795(01)00353-6
Hansen F: Trace functions with applications in quantum physics. J. Stat. Phys. 2014, 154: 807-818. 10.1007/s10955-013-0890-x
Fack T, Kosaki H: Generalized s -numbers of τ -measurable operators. Pac. J. Math. 1986, 123: 269-300. 10.2140/pjm.1986.123.269
Bekjan TN: On joint convexity of trace functions. Linear Algebra Appl. 2004, 390: 321-327.
Pisier G, Xu Q: Noncommutative -spaces. 2. Handbook of the Geometry of Banach Spaces 2003, 1459-1517.
Xu, Q: Noncommutative -spaces. Preprint
Muhammad M, Tleulessova A: Joint convexity and concavity of τ -measurable positive operator trace functions. J. Xinjiang Univ. Nat. Sci. 2011,28(4):421-425.
Fack T: Type and cotype inequalities for non commutative -spaces. J. Oper. Theory 1987, 17: 255-279.
Shao J: On Young and Heinz inequalities for τ -measurable operators. J. Math. Anal. Appl. 2013, 414: 814-821.
This work was partially supported by NSFC Grant No. 11371304 and NSFC Grant No. 11401507.
The authors declare that they have no competing interests.
The main idea of this paper was proposed by the corresponding author JJS. JJS and YZH prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
About this article
Cite this article
Shao, J., Han, Y. Some convexity inequalities in noncommutative -spaces. J Inequal Appl 2014, 385 (2014). https://doi.org/10.1186/1029-242X-2014-385
- von Neumann algebra
- positive operator
- noncommutative -space