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.

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 |𝑎+𝑏|²⩽𝑝|𝑎|²+𝑞|𝑏|² (1) for complex numbers 𝑎,𝑏 and real numbers 𝑝,𝑞>1 such that 1/𝑝+1/𝑞=1. The equality in (1) occurs if and only if 𝑝𝑎=𝑞𝑏. 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 ∗-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 2-by-2 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 𝑎 in a Banach ∗-algebra is called self-adjoint if 𝑎^∗=𝑎. An element 𝑎 which has real spectrum, that is, 𝜎(𝑎)⊆ℝ, 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 𝐶^∗-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, 𝒜 denotes a hermitian Banach ∗-algebra.

Every hermitian Banach ∗-algebra is equipped with a natural order structure as follows. Given self-adjoint elements 𝑎,𝑏∈𝒜, the relation 𝑎⩽𝑏 means that 𝑏−𝑎 is self-adjoint and 𝜎(𝑏−𝑎)⊆[0,∞). Then the relation “⩽” forms a partial order on the real vector space of self-adjoint elements in 𝒜. The set of 𝑎∈𝒜 such that 𝑎⩾0 forms a positive cone in 𝒜 (see [16, Lemma 41.4]), that is, if 𝑎,𝑏∈𝒜 are such that 𝑎,𝑏⩾0, then 𝛼𝑎+𝛽𝑏⩾0 for any 𝛼,𝛽∈[0,∞).

The Shirali-Ford Theorem [17, Theorem 1] assures that 𝑎^∗𝑎⩾0 for any 𝑎∈𝒜. Then the absolute value of 𝑎∈𝒜 is defined to be (𝑎^∗𝑎)^1/2. By the spectral mapping theorem, 𝜎(|𝑎|)⊆[0,∞) and hence |𝑎|⩾0 for every 𝑎∈𝒜. Note that |𝑎|=0 if and only if 𝑎=0 by [18, Lemma 3].

Denote by 𝐻₂(ℂ) the hermitian Banach ∗-subalgebra of 2-by-2 hermitian matrices of the hermitian Banach ∗-algebra 𝑀₂(ℂ) of 2-by-2 complex matrices. The natural order structure on 𝐻₂(ℂ) is called the Löwner partial order: the relation 𝐴⩽𝐵 in 𝐻₂(ℂ) means that 𝐵−𝐴 is a positive semidefinite matrix, that is, a hermitian matrix with nonnegative eigenvalues.

Theorem 1. Let 𝑎,𝑏∈𝒜. The map Ψ∶𝑀₂(ℂ)→𝒜 given by |𝑎|𝛼𝛽𝛾𝜇↦𝛼²+𝛽𝑎^∗𝑏+𝛾𝑏^∗𝑎+𝜇|𝑏|², (2) for 𝛼,𝛽,𝛾,𝜇∈ℂ, is ℂ-linear. The map Φ∶𝐻₂(ℂ)→𝒜 given by 𝛼𝛽|𝑎|𝛽𝛾↦𝛼²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝑎+𝛾|𝑏|², (3) for 𝛼,𝛾∈ℝ and 𝛽∈ℂ, is ℝ-linear, positive, and order-preserving. The positivity of Φ means 𝐴⩾0 in 𝐻₂(ℂ) implies Φ(𝐴)⩾0 in 𝒜. The order preserving of Φ means 𝐴⩾𝐵 in 𝐻₂(ℂ) implies Φ(𝐴)⩾Φ(𝐵) in 𝒜.

Proof. The ℂ-linearity of Ψ and the ℝ-linearity of Φ are clear. For the positivity of Φ, consider 𝛼,𝛾⩾0 and 𝛽∈ℂ such that 𝛼𝛾⩾|𝛽|². If 𝛽=0, we are done. If 𝛽≠0, then 𝛼>0 and 𝛾>0. Set 𝜆=𝛼𝛾−|𝛽|². Since 𝛼=(𝜆+|𝛽|²)/𝛾, it follows that 𝛼|𝑎|²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝑎+𝛾|𝑏|²=𝜆+𝛽²𝛾|𝑎|²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝑎+𝛾|𝑏|²=𝜆𝛾|𝑎|²+𝛽√𝛾𝑎+√𝛾𝑏²⩾0. (4) So, Φ is positive. Now, if 𝐴⩽𝐵 in 𝐻₂(ℂ), it follows from the linearity and positivity of Φ that 𝐵−𝐴⩾0 which implies Φ(𝐵)−Φ(𝐴)=Φ(𝐵−𝐴)⩾Φ(0)=0, that is, Φ(𝐴)⩽Φ(𝐵).

The most useful form of Φ in this paper is Φ|𝛼|²𝛼𝛼𝛽𝛽𝛽²=𝛼𝑎+𝛽𝑏². (5) The next theorem is very useful to get a necessity and sufficiency condition for the equality case in the later discussions.

Theorem 2. Let 𝑎,𝑏 be nonzero elements in 𝒜. For each 𝛼,𝛾⩾0 and 𝛽∈ℂ such that 𝛼𝛾⩾|𝛽|², the equation 𝛼|𝑎|²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝑎+𝛾|𝑏|²=0 (6) holds if and only if either 𝛼=𝛽=𝛾=0 or 𝛼𝛾=|𝛽|²≠0 and 𝛼𝑎+𝛽𝑏=0. In particular, the restriction of Φ in the previous theorem to the 2-by-2 positive semidefinite matrices satisfies Φ⁻¹({0})=𝛼𝛽𝛽𝛽²𝛼∪.∶𝛼𝑎+𝛽𝑏=0,𝛼>0,𝛽∈ℂ0000 (7)

Proof. If either 𝛼=𝛽=𝛾=0 or 𝛼𝛾=𝛽² and 𝛼𝑎+𝛽𝑏=0 holds, then (6) holds. Suppose now that (6) holds for 𝛼,𝛾⩾0 and 𝛽∈ℂ such that 𝛼𝛾⩾|𝛽|². If 𝛼=0, then 𝛽=0 and 0=𝛾|𝑏|², that is, 𝛾=0. Consider the case 𝛼>0. If 𝛾=0, we get 𝛽=0 and then 𝑎=0, a contradiction. Now for 𝛾>0, let 𝜆=𝛼𝛾−|𝛽|². We have 𝜆⩾0 and 𝛾=(𝜆+|𝛽|²)/𝛼. Hence 0=𝛼|𝑎|²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝑎+𝛾|𝑏|²=𝛼|𝑎|²+𝛽𝑎^∗𝑏+𝛽𝑏^∗𝛽𝑎+𝜆+²𝛼|𝑏|²=√𝛽𝛼𝑎+√𝛼𝑏²+𝜆𝛼|𝑏|². (8) This forces |√𝛼𝑎+(𝛽/√𝛼)𝑏|=(𝜆/𝛼)|𝑏|=0 and hence 𝛼𝑎+𝛽𝑏=0 and 𝛼𝛾=|𝛽|².

3. Applications

The ℝ-linearity and order preserving of the map Φ and the ℂ-linearity of Ψ 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 𝒜 be a hermitian Banach ∗-algebra.

Corollary 3. Let 𝑎,𝑏∈𝒜 and 𝑛∈ℕ. If 𝜔 is an 𝑛th -root of unity (i.e., 𝜔∈ℂ⧵{1} and 𝜔^𝑛=1), then |𝑎|²+|𝑏|²=1𝑛^𝑛_𝑘=1𝑎+𝜔^𝑘𝑏². (9) In particular, the usual parallelogram law holds: 2|𝑎|²+2|𝑏|²=|𝑎−𝑏|²+|𝑎+𝑏|². (10)

Proof. Since 𝜔 is an 𝑛th-root of unity, 𝜔^𝑛−1+𝜔^𝑛−2+⋯+1=0 and |𝜔|=1. Then a computation shows that 1𝜔𝜔|𝜔|²+1𝜔²𝜔²|𝜔|⁴+⋯+1𝜔^𝑛𝜔^𝑛|𝜔|^2𝑛=𝑛𝜔+𝜔²+⋯+𝜔^𝑛𝜔+𝜔²+⋯+𝜔^𝑛|𝜔|²+|𝜔|⁴+⋯+|𝜔|^2𝑛=.𝑛00𝑛 (11) By linearity of Ψ or Φ, we get (9). To get a usual parallelogram law, take 𝑛=2 and 𝜔=−1.

Corollary 4. Let 𝑎,𝑏∈𝒜. Then for any 𝑡∈ℝ⧵{0}, |𝑎+𝑏|²+1𝑡|𝑡𝑎−𝑏|²=(1+𝑡)|𝑎|²+11+𝑡|𝑏|². (12) In particular, for 𝑝,𝑞∈ℝ such that 𝑝,𝑞≠1,𝑝𝑞>0 and 1/𝑝+1/𝑞=1, |𝑎+𝑏|²+𝑝𝑞𝑎−𝑞𝑝𝑏²=𝑝|𝑎|²+𝑞|𝑏|². (13)

Proof. Equation (12) is done by applying Ψ or Φ to a matrix identity +11111𝑡𝑡²=1−𝑡−𝑡11+𝑡001+𝑡. (14) To get (13), choose 𝑡=𝑝/𝑞>0 which implies 1+𝑡=𝑝 and 1+1/𝑡=𝑞 from the condition 1/𝑝+1/𝑞=1.

The identities (12) and (13) become the parallelogram law when 𝑡=1 and 𝑝=𝑞=2, 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 0⩽𝛼⩽1, |𝛼𝑎+(1−𝛼)𝑏|²+𝛼(1−𝛼)|𝑎−𝑏|²=𝛼|𝑎|²+(1−𝛼)|𝑏|². (15)

Corollary 5. Let 𝑎,𝑏∈𝒜 and 𝑛∈ℕ. If 𝜔 is an 𝑛th -root of unity, then 𝑏^∗1𝑎=𝑛^𝑛_𝑘=1𝜔^𝑘𝑎+𝜔^𝑘𝑏². (16) In particular, the usual polarization identity holds when we denote ⟨𝑎,𝑏⟩=𝑏^∗𝑎 (e.g., in ℂ^𝑛): 1⟨𝑎,𝑏⟩=4⁴_𝑘=1𝑖^𝑘𝑎+𝑖^𝑘𝑏². (17)

Proof. Since 𝜔 is an 𝑛th-root of unity, 𝜔^𝑛−1+𝜔^𝑛−2+⋯+1=0 and |𝜔|=1. Note that 𝜔² is also an 𝑛th-root of unity and hence 1+𝜔²+𝜔⁴+⋯+𝜔^2𝑛−2=0. Then a computation shows that 𝜔1𝜔𝜔|𝜔|²+𝜔²1𝜔²𝜔²|𝜔|⁴+⋯+𝜔^𝑛1𝜔^𝑛𝜔^𝑛|𝜔|^2𝑛=𝜔+𝜔²+⋯+𝜔^𝑛𝜔²+𝜔⁴+⋯+𝜔^2𝑛𝜔𝜔+𝜔²𝜔²+⋯+𝜔^𝑛𝜔^𝑛𝜔|𝜔|²+𝜔²|𝜔|⁴+⋯+𝜔^𝑛|𝜔|^2𝑛=𝜔1+𝜔+⋯+𝜔^𝑛−1𝜔²1+𝜔²+𝜔⁴+⋯+𝜔^2𝑛−2|𝜔|²+|𝜔|⁴+⋯+|𝜔|^2𝑛𝜔1+𝜔+⋯+𝜔^𝑛−1=.00𝑛0 (18) By ℂ-linearity of Ψ, we get (16). To get a usual polarization identity, take 𝑛=4 and 𝜔=𝑖.

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 𝑎,𝑏∈𝒜 and 𝑝,𝑞∈ℝ such that 1/𝑝+1/𝑞=1. (i)Bohr’s inequality: if 𝑝,𝑞>1, then |𝑎+𝑏|²⩽𝑝|𝑎|²+𝑞|𝑏|² (19) with equality if and only if 𝑝𝑎=𝑞𝑏 (i.e., 𝑏=(𝑝−1)𝑎 or 𝑎=(𝑞−1)𝑏). (ii)Reverse Bohr’s inequality: if 𝑝,𝑞<1, then |𝑎+𝑏|²⩾𝑝|𝑎|²+𝑞|𝑏|² (20) with equality if and only if 𝑝𝑎=𝑞𝑏.

Proof. Let 𝑝,𝑞>1 be such that 1/𝑝+1/𝑞=1. Then (𝑝−1)(𝑞−1)=1 which implies ⩽1111𝑝00𝑞. (21) Since Φ is order preserving, we obtain the Bohr’s inequality (19). By Theorem 2, (𝑝−1)|𝑎|²−𝑎^∗𝑏−𝑏^∗𝑎+(𝑞−1)|𝑏|²=0 if and only if (𝑝−1)𝑎−𝑏=0, that is, 𝑏=(𝑝−1)𝑎 (note that 𝑝,𝑞≠1). Part (ii) is similarly proven.

Corollary 7. Let 𝑎,𝑏∈𝒜 and 𝑝,𝑞 real numbers such that 1/𝑝+1/𝑞=1. (i)If 𝑝<1, then |𝑎+𝑏|²+𝑝−1𝑎−𝑏²⩾𝑝|𝑎|²+𝑞|𝑏|²,|𝑎+𝑏|²+𝑎−𝑞−1𝑏²⩾𝑝|𝑎|²+𝑞|𝑏|², (22) with equality if and only if 𝑏=(𝑝−1)𝑎. (ii)If 1<𝑝⩽2, then |𝑎+𝑏|²+𝑝−1𝑎−𝑏²⩽𝑝|𝑎|²+𝑞|𝑏|²,|𝑎+𝑏|²+𝑎−𝑞−1𝑏²⩾𝑝|𝑎|²+𝑞|𝑏|², (23)(24) with equality if and only if 𝑝=𝑞=2 or 𝑏=(𝑝−1)𝑎. (iii)If 𝑝>2, then |𝑎+𝑏|²+𝑝−1𝑎−𝑏²⩾𝑝|𝑎|²+𝑞|𝑏|²,|𝑎+𝑏|²+𝑎−𝑞−1𝑏²⩽𝑝|𝑎|²+𝑞|𝑏|², (25) with equality if and only if 𝑏=(𝑝−1)𝑎.

Proof. Assume 1<𝑝⩽2. The inequality (23) is obtained by applying the order-preserving Φ to a matrix ordering +1111𝑝−1²=𝑝1−𝑝1−𝑝1₂⩽.−2𝑝+22−𝑝2−𝑝2𝑝00𝑞 (26) By Theorem 2, (𝑝²−2𝑝+2−𝑝)|𝑎|²+(2−𝑝)𝑎^∗𝑏+(2−𝑝)𝑏^∗𝑎+(2−𝑞)|𝑏|²=0 holds if and only if either 𝑝²−3𝑝−2=𝑝−2=𝑞−2=0 or (𝑝−1)(𝑝−2)𝑎+(2−𝑝)𝑏=0 and (𝑝²−3𝑝−2)(2−𝑞)=(2−𝑝)². The first case is 𝑝=𝑞=2. The latter case is 𝑏=(𝑝−1)𝑎 since (𝑝²−3𝑝−2)(2−𝑞)=(2−𝑝)² always holds from the fact that 𝑝+𝑞=𝑝𝑞.
Now, a matrix ordering +111111−𝑞1−𝑞𝑞−1²=22−𝑞2−𝑞𝑞²⩾−2𝑞+2𝑝00𝑞 (27) yields (24) via the map Φ. 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,𝑏∈𝒜,(a)if 0<𝑡⩽1, then |𝑎+𝑏|²+|𝑡𝑎−𝑏|²⩽(1+𝑡)|𝑎|²+(1+1/𝑡)|𝑏|² with equality if and only if 𝑡=1 or 𝑏=𝑡𝑎, (b)if 𝑡<0 or 𝑡⩾1, then |𝑎+𝑏|²+|𝑡𝑎−𝑏|²⩾(1+𝑡)|𝑎|²+(1+1/𝑡)|𝑏|² with equality if and only if 𝑡=1 or 𝑏=𝑡𝑎.

Proof. Assume 0<𝑡⩽1. The order-preserving Φ brings a matrix inequality +11𝑡11²⩽1−𝑡−𝑡11+𝑡001+𝑡 (28) to the desired inequality in 𝒜. This inequality becomes an equality if and only if (𝑡−1)𝑡|𝑎|²−𝑎^∗𝑏−𝑏^∗1𝑎+𝑡|𝑏|²=0 (29) which is equivalent to the condition 𝑡=1 or 𝑡𝑎−𝑏=0 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 𝑎,𝑏∈𝒜 and 𝑠,𝑡∈ℂ, 𝑝,𝑞∈ℝ⧵{0} such that |𝑠|²/𝑝+|𝑡|²/𝑞⩽1. (a)Generalized Bohr’s inequality: if |𝑠|²⩽𝑝 and |𝑡|²⩽𝑞, then |𝑠𝑎+𝑡𝑏|²⩽𝑝|𝑎|²+𝑞|𝑏|². (30)(b)Generalized reverse Bohr’s inequality: if |𝑠|²⩾𝑝 and |𝑡|²⩾𝑞, then |𝑠𝑎+𝑡𝑏|²⩾𝑝|𝑎|²+𝑞|𝑏|². (31) In both cases, equality holds if and only if one of the following occurs∶(i)𝑎=𝑏=0,(ii)𝑎=0,𝑏≠0, and 𝑞=|𝑡|², (iii)𝑎≠0,𝑏=0, and 𝑝=|𝑠|², (iv)𝑎≠0,𝑏≠0,(𝑝−|𝑠|²)𝑎=𝑠𝑡𝑏, and |𝑠𝑡|=(𝑝−|𝑠|²)(𝑞−|𝑡|²) (equivalently, 𝑎≠0,𝑏≠0,𝑠𝑡𝑎=(𝑞−|𝑡|²)𝑏 and |𝑠𝑡|=(𝑝−|𝑠|²)(𝑞−|𝑡|²)).

Proof. The proofs of (a) and (b) are similar. For (a) assume |𝑠|²⩽𝑝 and |𝑡|²⩽𝑞. The conditions |𝑠|²−𝑝⩽0,|𝑡|²−𝑞⩽0 and (𝑝−|𝑠|²)(𝑞−|𝑡|²)=|𝑠𝑡|² imply |𝑠|²𝑠𝑠𝑡𝑡|𝑡|²⩽𝑝00𝑞. (32) So, by passing through the order-preserving Φ, we obtain (30). When 𝑎≠0 and 𝑏≠0, the equality holds by Theorem 2 if and only if either 𝑝−|𝑠|²=𝑠𝑡=𝑞−|𝑡|²=0 or (𝑝−|𝑠|²)𝑎=𝑠𝑡𝑏 and (𝑝−|𝑠|²)(𝑞−|𝑡|²)=|𝑠𝑡|². The first case is impossible. The latter case is equivalent to 𝑠𝑡𝑎=(𝑞−|𝑡|²)𝑏 and |𝑠𝑡|=(𝑝−𝑠|²)(𝑞−𝑡|²)). For 𝑎=0 and 𝑏≠0, it forces 𝑞=|𝑡|². For 𝑎≠0 and 𝑏=0, it forces 𝑝=|𝑠|².

Corollary 10. For any 𝑎,𝑏∈𝒜, we have ±𝑎^∗𝑏+𝑏^∗𝑎⩽12|𝑎±𝑏|²⩽|𝑎|²+|𝑏|² (33) and each equality holds if and only if 𝑎=±𝑏.

Proof. The inequality (33) follows via applying Φ to a matrix inequality ⩽0±11±102⩽1±1±111001. (34) Each equality holds if and only if |𝑎|²∓𝑎^∗𝑏∓𝑏^∗𝑎+|𝑏|²=0. By Theorem 2, it holds if and only if 𝑎=±𝑏.

In particular, for any 𝑎,𝑏∈𝒜 and 𝑡∈ℂ⧵{0}±𝑎^∗𝑏+𝑏^∗𝑎⩽121𝑡𝑎±𝑡𝑏²⩽|𝑡|²|𝑎|²+1|𝑡|²|𝑏|², (35) with each equality holds if and only if 𝑏=±𝑡²𝑎. The inequalities of this form for operators appear in [15, Theorem 3].

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