An Order-Preserving Linear Map from Matrices to Banach *-Algebras and Applications

Abstract

We establish a theorem for which a number of absolute-value identities and inequalities in the framework of Banach * -algebras can be generated. To do this job, we construct an order-preserving linear map from the vector space of 2-by-2 hermitian matrices to a hermitian Banach * -algebras. This map can convert any suitable matrix ordering to a number of identities and inequalities in Banach * -algebras. Hence, we obtain a number of analogues of the well-known results in a framework of hermitian Banach * -algebras.

Academic Editor:Tepper L. Gill

Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received 30 August 2012; Accepted 17 October 2012

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.

1. Introduction

There are many works involving absolute-value identities and inequalities. The famous such identities are the parallelogram law and the polarization identity. It is well known that these identities hold in the context of Euclidean spaces, normed linear spaces, and inner product spaces. On the other hand for inequalities, Bohr [1] established the classical Bohr's inequality which asserts that [figure omitted; refer to PDF] for complex numbers [figure omitted; refer to PDF] and real numbers [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . The equality in (1) occurs if and only if [figure omitted; refer to PDF] . Then a number of extensions and variations of absolute-value inequalities concerning Bohr's inequality were developed in various contexts by many authors. The results for complex numbers are obtained in [2-4]. The context of matrices is given in [5]. Bohr's inequality and related results were generalized to operator algebras in [6-15].

The first version of operator Bohr's inequality was established via direct computations by Hirzallah [12]. Later, Hirzallah's results were extended by the same technique in [6, 10, 14]. Zhang [15] used operator identities and inequalities for approaching operator inequalities related to (1). Recently, the idea of matrix ordering for discussing operator absolute-value inequalities appears in [8, 9, 11]. However, the results mentioned above has been proved in separate ways.

This paper consists of two main purposes. The first goal is to construct a theorem for which a number of absolute-value identities and inequalities can be generated from it. The second is to extend identities and inequalities involving absolute values to an abstract framework of Banach [figure omitted; refer to PDF] -algebras. In Section 2, after recalling some terminology, we propose an order-preserving linear map from the set of 2-by-2 hermitian matrices to a hermitian Banach [figure omitted; refer to PDF] -algebra. The image of this map for each [figure omitted; refer to PDF] -by- [figure omitted; refer to PDF] hermitian matrix is in the form involving absolute values. In Section 3, we apply this order preserving to elementary matrix identities and inequalities to get a number of identities and inequalities about absolute values of elements in hermitian Banach [figure omitted; refer to PDF] -algebras.

2. An Order-Preserving Linear Map from Matrices to Banach *-Algebras

An element [figure omitted; refer to PDF] in a Banach [figure omitted; refer to PDF] -algebra is called self-adjoint if [figure omitted; refer to PDF] . An element [figure omitted; refer to PDF] which has real spectrum, that is, [figure omitted; refer to PDF] , is said to be hermitian . A Banach [figure omitted; refer to PDF] -algebra is called hermitian if each self-adjoint element is hermitian. The class of hermitian Banach [figure omitted; refer to PDF] -algebras includes any [figure omitted; refer to PDF] -algebra, any group algebra of an abelian group, any group algebra of a compact group, and any measure algebra of discrete group. Throughout this paper, [figure omitted; refer to PDF] denotes a hermitian Banach [figure omitted; refer to PDF] -algebra.

Every hermitian Banach [figure omitted; refer to PDF] -algebra is equipped with a natural order structure as follows. Given self-adjoint elements [figure omitted; refer to PDF] , the relation [figure omitted; refer to PDF] means that [figure omitted; refer to PDF] is self-adjoint and [figure omitted; refer to PDF] . Then the relation " [figure omitted; refer to PDF] " forms a partial order on the real vector space of self-adjoint elements in [figure omitted; refer to PDF] . The set of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] forms a positive cone in [figure omitted; refer to PDF] (see [16, Lemma 41.4]), that is, if [figure omitted; refer to PDF] are such that [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] .

The Shirali-Ford Theorem [17, Theorem 1] assures that [figure omitted; refer to PDF] for any [figure omitted; refer to PDF] . Then the absolute value of [figure omitted; refer to PDF] is defined to be [figure omitted; refer to PDF] . By the spectral mapping theorem, [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] by [18, Lemma 3].

Denote by [figure omitted; refer to PDF] the hermitian Banach [figure omitted; refer to PDF] -subalgebra of 2-by-2 hermitian matrices of the hermitian Banach [figure omitted; refer to PDF] -algebra [figure omitted; refer to PDF] of 2-by-2 complex matrices. The natural order structure on [figure omitted; refer to PDF] is called the Löwner partial order: the relation [figure omitted; refer to PDF] in [figure omitted; refer to PDF] means that [figure omitted; refer to PDF] is a positive semidefinite matrix, that is, a hermitian matrix with nonnegative eigenvalues.

Theorem 1.

Let [figure omitted; refer to PDF] . The map [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , is [figure omitted; refer to PDF] -linear. The map [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , is [figure omitted; refer to PDF] -linear, positive, and order-preserving. The positivity of [figure omitted; refer to PDF] means [figure omitted; refer to PDF] in [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] in [figure omitted; refer to PDF] . The order preserving of [figure omitted; refer to PDF] means [figure omitted; refer to PDF] in [figure omitted; refer to PDF] implies [figure omitted; refer to PDF] in [figure omitted; refer to PDF] .

Proof.

The [figure omitted; refer to PDF] -linearity of [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] -linearity of [figure omitted; refer to PDF] are clear. For the positivity of [figure omitted; refer to PDF] , consider [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , we are done. If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Set [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] , it follows that [figure omitted; refer to PDF] So, [figure omitted; refer to PDF] is positive. Now, if [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , it follows from the linearity and positivity of [figure omitted; refer to PDF] that [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] .

The most useful form of [figure omitted; refer to PDF] in this paper is [figure omitted; refer to PDF] The next theorem is very useful to get a necessity and sufficiency condition for the equality case in the later discussions.

Theorem 2.

Let [figure omitted; refer to PDF] be nonzero elements in [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , the equation [figure omitted; refer to PDF] holds if and only if either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In particular, the restriction of [figure omitted; refer to PDF] in the previous theorem to the 2-by-2 positive semidefinite matrices satisfies [figure omitted; refer to PDF]

Proof.

If either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] and [figure omitted; refer to PDF] holds, then (6) holds. Suppose now that (6) holds for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] . Consider the case [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] and then [figure omitted; refer to PDF] , a contradiction. Now for [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] . We have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] This forces [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

3. Applications

The [figure omitted; refer to PDF] -linearity and order preserving of the map [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] -linearity of [figure omitted; refer to PDF] are used in order to obtain a number of identities and inequalities from suitable matrix identities and inequalities. We give some applications of Theorem 1 as follows:

(1) parallelogram law and its generalizations,

(2) polarization identity and its generalizations,

(3) Bohr's inequality and its reverse,

(4) generalizations of Bohr's inequality,

(5) related absolute-value identities and inequalities.

Let [figure omitted; refer to PDF] be a hermitian Banach [figure omitted; refer to PDF] -algebra.

Corollary 3.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] th -root of unity (i.e., [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ), then [figure omitted; refer to PDF] In particular, the usual parallelogram law holds: [figure omitted; refer to PDF]

Proof.

Since [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] th-root of unity, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then a computation shows that [figure omitted; refer to PDF] By linearity of [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , we get (9). To get a usual parallelogram law, take [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Corollary 4.

Let [figure omitted; refer to PDF] . Then for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF] In particular, for [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

Equation (12) is done by applying [figure omitted; refer to PDF] or [figure omitted; refer to PDF] to a matrix identity [figure omitted; refer to PDF] To get (13), choose [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] and [figure omitted; refer to PDF] from the condition [figure omitted; refer to PDF] .

The identities (12) and (13) become the parallelogram law when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. The identities (12) and (13) for Hilbert space operators are provided in [11, Theorem 4.1] and [15, Theorem 2], respectively. The identity (13) can be stated equivalently that for any [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Corollary 5.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] th -root of unity, then [figure omitted; refer to PDF] In particular, the usual polarization identity holds when we denote [figure omitted; refer to PDF] (e.g., in [figure omitted; refer to PDF] ): [figure omitted; refer to PDF]

Proof.

Since [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] th -root of unity, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF] is also an [figure omitted; refer to PDF] th-root of unity and hence [figure omitted; refer to PDF] . Then a computation shows that [figure omitted; refer to PDF] By [figure omitted; refer to PDF] -linearity of [figure omitted; refer to PDF] , we get (16). To get a usual polarization identity, take [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Next, consider Bohr's inequality and its extensions. The operator version of Bohr's inequality is first proved in [12, Corallary 1]. The following result gives a generalization of Bohr's inequality and also includes its reverse.

Corollary 6.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

(i) Bohr's inequality: if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] (i.e., [figure omitted; refer to PDF] or [figure omitted; refer to PDF] ).

(ii) Reverse Bohr's inequality: if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] .

Proof.

Let [figure omitted; refer to PDF] be such that [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is order preserving, we obtain the Bohr's inequality (19). By Theorem 2, [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] (note that [figure omitted; refer to PDF] ). Part (ii) is similarly proven.

Corollary 7.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] real numbers such that [figure omitted; refer to PDF] .

(i) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] .

(ii) If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

(iii): If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] .

Proof.

Assume [figure omitted; refer to PDF] . The inequality (23) is obtained by applying the order-preserving [figure omitted; refer to PDF] to a matrix ordering [figure omitted; refer to PDF] By Theorem 2, [figure omitted; refer to PDF] holds if and only if either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The first case is [figure omitted; refer to PDF] . The latter case is [figure omitted; refer to PDF] since [figure omitted; refer to PDF] always holds from the fact that [figure omitted; refer to PDF] .

Now, a matrix ordering [figure omitted; refer to PDF] yields (24) via the map [figure omitted; refer to PDF] . The proofs of others results are similar to that one.

The analogue results for the case of operators on a Hilbert space are obtained in [10] (cf. Theorem 2, Theorem 1 and Corollary 1 in [10], resp.). The next result generalizes [11, Theorem 3.2].

Corollary 8.

Let [figure omitted; refer to PDF] ,

(a) if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] ,

(b) if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] with equality if and only if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] .

Proof.

Assume [figure omitted; refer to PDF] . The order-preserving [figure omitted; refer to PDF] brings a matrix inequality [figure omitted; refer to PDF] to the desired inequality in [figure omitted; refer to PDF] . This inequality becomes an equality if and only if [figure omitted; refer to PDF] which is equivalent to the condition [figure omitted; refer to PDF] or [figure omitted; refer to PDF] by Theorem 2. The proof of (b) is similar to that one.

In 2009, generalized Bohr's inequality and its reverse for operators acting on a Hilbert space are done in [9, Theorem 9]. The following results give analogues results in the framework of Banach [figure omitted; refer to PDF] -algebras.

Corollary 9.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

(a) Generalized Bohr's inequality: if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

(b) Generalized reverse Bohr's inequality: if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] In both cases, equality holds if and only if one of the following occurs [figure omitted; refer to PDF]

(i) [figure omitted; refer to PDF] ,

(ii) [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ,

(iii): [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ,

(iv) [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] (equivalently, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ).

Proof.

The proofs of (a) and (b) are similar. For (a) assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The conditions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] imply [figure omitted; refer to PDF] So, by passing through the order-preserving [figure omitted; refer to PDF] , we obtain (30). When [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the equality holds by Theorem 2 if and only if either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The first case is impossible. The latter case is equivalent to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it forces [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , it forces [figure omitted; refer to PDF] .

Corollary 10.

For any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and each equality holds if and only if [figure omitted; refer to PDF] .

Proof.

The inequality (33) follows via applying [figure omitted; refer to PDF] to a matrix inequality [figure omitted; refer to PDF] Each equality holds if and only if [figure omitted; refer to PDF] . By Theorem 2, it holds if and only if [figure omitted; refer to PDF] .

In particular, for any [figure omitted; refer to PDF] and [figure omitted; refer to PDF] [figure omitted; refer to PDF] with each equality holds if and only if [figure omitted; refer to PDF] . The inequalities of this form for operators appear in [15, Theorem 3].

We finally comment that the order-preserving [figure omitted; refer to PDF] can generate many absolute-value identities and inequalities from any 2-by-2 matrix inequalities.

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