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Journal of Mathematics
Volume 2013 (2013), Article ID 328510, 5 pages
http://dx.doi.org/10.1155/2013/328510
Research Article

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

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

Received 30 August 2012; Accepted 17 October 2012

Academic Editor: Tepper L. Gill

Copyright © 2013 Pattrawut Chansangiam. 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 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.

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 for complex numbers and real numbers such that . The equality in (1) occurs if and only if . 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 [24]. The context of matrices is given in [5]. Bohr’s inequality and related results were generalized to operator algebras in [615].

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 -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 -algebra. The image of this map for each -by- 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 -algebras.

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

An element in a Banach -algebra is called self-adjoint if . An element which has real spectrum, that is, , is said to be hermitian. A Banach -algebra is called hermitian if each self-adjoint element is hermitian. The class of hermitian Banach -algebras includes any -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, denotes a hermitian Banach -algebra.

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

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

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

Theorem 1. Let . The map given by for , is -linear. The map given by for and , is -linear, positive, and order-preserving. The positivity of means in implies in . The order preserving of means in implies in .

Proof. The -linearity of and the -linearity of are clear. For the positivity of , consider and such that . If , we are done. If , then and . Set . Since , it follows that So, is positive. Now, if in , it follows from the linearity and positivity of that which implies , that is, .

The most useful form of in this paper is The next theorem is very useful to get a necessity and sufficiency condition for the equality case in the later discussions.

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

Proof. If either or and holds, then (6) holds. Suppose now that (6) holds for and such that . If , then and , that is, . Consider the case . If , we get and then , a contradiction. Now for , let . We have and . Hence This forces and hence and .

3. Applications

The -linearity and order preserving of the map and the -linearity of 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 be a hermitian Banach -algebra.

Corollary 3. Let and . If is an th -root of unity (i.e., and ), then In particular, the usual parallelogram law holds:

Proof. Since is an th-root of unity, and . Then a computation shows that By linearity of or , we get (9). To get a usual parallelogram law, take and .

Corollary 4. Let . Then for any , In particular, for such that and ,

Proof. Equation (12) is done by applying or to a matrix identity To get (13), choose which implies and from the condition .

The identities (12) and (13) become the parallelogram law when and , 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 ,

Corollary 5. Let and . If is an th -root of unity, then In particular, the usual polarization identity holds when we denote (e.g., in ):

Proof. Since is an th-root of unity, and . Note that is also an th-root of unity and hence . Then a computation shows that By -linearity of , we get (16). To get a usual polarization identity, take and .

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 and such that . (i)Bohr’s inequality: if , then with equality if and only if (i.e., or ). (ii)Reverse Bohr’s inequality: if , then with equality if and only if .

Proof. Let be such that . Then which implies Since is order preserving, we obtain the Bohr’s inequality (19). By Theorem 2, if and only if , that is, (note that ). Part (ii) is similarly proven.

Corollary 7. Let and real numbers such that . (i)If , then with equality if and only if . (ii)If , then with equality if and only if or . (iii)If , then with equality if and only if .

Proof. Assume . The inequality (23) is obtained by applying the order-preserving to a matrix ordering By Theorem 2, holds if and only if either or and . The first case is . The latter case is since always holds from the fact that .
Now, a matrix ordering yields (24) via the map . 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 ,(a)if , then with equality if and only if or , (b)if or , then with equality if and only if or .

Proof. Assume . The order-preserving brings a matrix inequality to the desired inequality in . This inequality becomes an equality if and only if which is equivalent to the condition or 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 -algebras.

Corollary 9. Let and , such that . (a)Generalized Bohr’s inequality: if and , then (b)Generalized reverse Bohr’s inequality: if and , then In both cases, equality holds if and only if one of the following occurs(i),(ii), and , (iii), and , (iv), and (equivalently, ,, and ).

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

Corollary 10. For any , we have and each equality holds if and only if .

Proof. The inequality (33) follows via applying to a matrix inequality Each equality holds if and only if . By Theorem 2, it holds if and only if .

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

We finally comment that the order-preserving can generate many absolute-value identities and inequalities from any 2-by-2 matrix inequalities.

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