Homomorphism-derivation functional inequalities in C^*-algebras

1.   Introduction and preliminaries

Let $A$ be a complex Banach algebra. A $\mathbb{C}$-linear mapping $g: A\to A$ is a derivation if $g: A\to A$ satisfies

for all $x, y\in A$, and a $\mathbb{C}$-linear mapping $h: A\to A$ is a homomorphism if $h: A\to A$ satisfies

for all $x, y\in A$,

A $C^*$-ternary algebra is a complex Banach space $A$, equipped with a ternary product $(x, y, z) \mapsto [x, y, z]$ of $A^3$ into $A$, which is $\mathbb C$-linear in the outer variables, conjugate $\mathbb C$-linear in the middle variable, and associative in the sense that $[x, y, [z, w, v]] = [x, [w, z, y], v] = [[x, y, z], w, v]$, and satisfies $\|[x, y, z]\| \le \|x\| \cdot \|y\| \cdot \|z\|$ and $\|[x, x, x]\| = \|x\|^3$ (see ^[18]).

Let $A$ be a $C^*$-ternary algebra. A $\mathbb{C}$-linear mapping $g: A\to A$ is a $C^*$-ternary derivation if $g: A\to A$ satisfies

for all $x, y, z\in A$, and a $\mathbb{C}$-linear mapping $h: A\to A$ is a $C^*$-ternary homomorphism if $h: A\to A$ satisfies

for all $x, y, z\in A$.

In 1940, Ulam ^[16] raised a question the stability problem for functional equations, and Hyers ^[8], Aoki ^[1], Rassias ^[14] and Găvruta ^[7] have given positive answers for additive additive functional equations in Banach spaces. Park ^[11,12] introduced new additive functional inequalities and proved the Hyers-Ulam stability of the additive functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations in various spaces have been extensively investigated by a number of authors (see ^{[2,3,4,5,6,13,15,17]}).

This paper is organized as follows: In Sections 2 and 3, we solve the additive-additive $(s, t)$-functional inequality $\rm(0.1)$. Furthermore, we investigate homomorphisms and derivations on compex Banach algebras and unital $C^*$-algebras associated to the additive-additive $(s, t)$-functional inequality $\rm(0.1)$ and the following functional inequality

In Section 4, we solve the additive-additive $(s, t)$-functional inequality $\rm(0.2)$. Furthermore, we investigate $C^*$-ternary homomorphisms and $C^*$-ternary derivations on $C^*$-ternary algebras associated to the additive-additive $(s, t)$-functional inequality $\rm(0.2)$ and the following functional inequality

Throughout this paper, assume that $A$ is a complex Banach algebra and that $B$ is a $C^*$-ternary algebra and that $s$ and $t$ are fixed nonzero complex numbers with $|s| < 1$ and $|t| < 1$.

2.   Additive-additive $(s, t)$-functional inequality $\rm(0.1)$ in Banach algebras

In this section, we solve and investigate the additive-additive $(s, t)$-functional inequality (0.1) in complex Banach algebras.

Lemma 2.1. If mappings $g, h : A \rightarrow A$ satisfy $g(0) = h(0) = 0$ and

for all $x, y \in A$, then the mappings $g, h : A \rightarrow A$ are additive.

Proof. Letting $x = y$ in $\rm(2.1)$, we get

for all $x \in A$.

Letting $y = 0$ in $\rm(2.1)$, we get

and so

for all $x \in A$. Thus

for all $x \in A$. So $h(2x) = 2h(x)$ and $g(2x) = 2g(x)$ for all $x\in A$, since $|st| < 1$. It follows from (2.1) that

for all $x, y \in A$. Thus $g\left(x+y\right) -g(x) -g(y) = 0$ and $h\left(x+y\right) - h(x) - h(y) = 0$ for all $x\in A$, since $|s| < 1$ and $|t| < 1$. So the mappings $g, h: A \to A$ are additive.

Lemma 2.2. [10,Theorem 2.1] Let $f: A \to A$ be a mapping such that

for all $\lambda \in { {\mathbb{T}}^1 } : = \{ \xi \in {\mathbb{C}} : | \xi | = 1\}$ and all $a, b\in A$. Then the mapping $f: A \to A$ is $\mathbb{C}$-linear.

Now, we investigate homomorphisms and derivations in complex Banach algebras associated to the additive-additive $(s, t)$-functional inequality (2.1).

Theorem 2.3. Let $\varphi: A^2 \to [0, \infty)$ be a function such that

for all $x, y\in A$. Let $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$ and

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y \in A$. If $g, h: A\to A$ satisfy

for all $x, y\in A$, then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. Let $\lambda = 1$ in (2.3). By Lemma 2.1, the mappings $g, h: A\to A$ are additive.

It follows from (2.3) that

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y \in A$. Since $|s| < 1$ and $|t| < 1$,

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y \in A$. Thus by Lemma 2.2, the mappings $g, h: A\to A$ are $\mathbb{C}$-linear.

It follows from (2.4) and the additivity of $g, h$ that

which tends to zero as $n\to \infty$, by (2.2). So

for all $x, y\in A$. Hence the mapping $g: A\to A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Theorem 2.4. Let $\varphi : A^2\to [0, \infty)$ be a function and $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(2.3)$, $\rm(2.4)$ and

for all $x, y \in A$. Then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. By the same reasoning as in the proof of Theorem 2.3, one can show that the mappings $g, h: A\to A$ are $\mathbb{C}$-linear.

It follows from (2.4) and the additivity of $g, h$ that

which tends to zero as $n\to \infty$, by (2.5). So

for all $x, y\in A$. Hence the mapping $g: A\to A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Next, we investigate homomorphisms and derivations in complex Banach algebras associated to the additive-additive $(s, t)$-functional inequality (2.1) by using a similar method to the fixed point alternative.

Theorem 2.5. Let $\varphi: A^2 \to [0, \infty)$ be a function such that there exists an $L < 1$ with

for all $x, y \in A$. Let $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(2.3)$ and $\rm(2.4)$. Then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. It follows from (2.6) that

for all $x, y \in A$. By Theorem 2.3, the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Theorem 2.6. Let $\varphi: A^2 \to [0, \infty)$ be a function such that there exists an $L < 1$ with

for all $x, y\in A$. Let $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(2.3)$ and $\rm(2.4)$. Then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. It follows from (2.7) that

for all $x, y \in A$. By Theorem 2.4, the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Remark 2.7. In (2.4), the pair $(g, h)$ of a derivation $g$ and a homomorphism $h$ can be replaced by the pair of a derivation and a derivation or the pair of a homomorphism and a homomorphism.

3.   Additive-additive $(s, t)$-functional inequality $\rm(0.1)$ in $C^*$-algebras

In this section, we study homorphisms and derivations in unital $C^*$-algebras associated to the additive-additive $(s, t)$-functional inequality (2.1) by using a similar method to the fixed point alternative. Throughout this scetion, assume that $A$ is a unital $C^*$-algebra with unitary group $U(A)$.

Theorem 3.1. Let $\varphi: A^2 \to [0, \infty)$ be a function satisfying $\rm(2.2)$. Let $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$ and $\rm(2.3)$. If $g, h: A\to A$ satisfy

for all $u, v\in U(A)$, then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. Let $\lambda = 1$ in (2.3). By Lemma 2.1, the mappings $g, h: A\to A$ are additive.

It follows from (2.3) that

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y \in A$. Since $|s| < 1$ and $|t| < 1$,

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y \in A$. Thus by Lemma 2.2, the mappings $g, h: A\to A$ are $\mathbb{C}$-linear.

Since $D$ is $\mathbb {C}$-linear in the first variable and each $y \in A$ is a finite linear combination of unitary elements (see ^[9]), i.e., $y = \sum_{i = 1}^k \lambda_i v_i \ (\lambda_i \in {\mathbb{C}}, \; v_i \in U(A))$.

for all $x, y \in A$.

Similarly, let $x = \sum_{j = 1}^m \lambda_j u_j \ (\lambda_j \in {\mathbb{C}}, \; u_j \in U(A))$. Then

for all $x, y \in A$. So

for all $x, y\in A$. Hence the $\mathbb{C}$-linear mapping $g: A\to A$ is a derivation.

Since $D$ is $\mathbb {C}$-linear in the first variable and each $y \in A$ is a finite linear combination of unitary elements (see ^[9]), $y = \sum_{i = 1}^k \lambda_i v_i\ (\lambda_i \in {\mathbb{C}}, \; v_i \in U(A))$,

for all $x, y \in A$.

Similarly, let $x = \sum_{j = 1}^m \lambda_j u_j \ (\lambda_j \in {\mathbb{C}}, \; u_j \in U(A))$. Then

for all $x, y \in A$. So

for all $x, y\in A$. Hence the $\mathbb{C}$-linear mapping $g: A\to A$ is a homomorphism.

Theorem 3.2. Let $\varphi : A^2\to [0, \infty)$ be a function and $g, h : A \rightarrow A$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(2.3)$, $\rm(3.1)$ and $\rm(2.5)$. Then the mapping $g : A \rightarrow A$ is a derivation and the mapping $h: A\to A$ is a homomorphism.

Proof. The proof is similar to the proofs of Theorems 2.4 and 3.1.

Remark 3.3. Using the fixed point method given in Theorems 2.5 and 2.6, one can obtain the same results as in Theorems 3.1 and 3.2.

4.   Additive-additive $(s, t)$-functional inequality $\rm(0.2)$ in $C^*$-ternary algebras

In this section, we solve and investigate the additive-additive $(s, t)$-functional inequality (0.2) in $C^*$-ternary algebras.

Lemma 4.1. If mappings $g, h : B \rightarrow B$ satisfy $g(0) = h(0) = 0$ and

for all $x, y, z \in B$, then the mappings $g, h : B \rightarrow B$ are additive.

Proof. Letting $x = y = z$ in $\rm(4.1)$, we get

for all $x \in B$.

Letting $y = z = 0$ in $\rm(4.1)$, we get

and so

for all $x \in B$. Thus

for all $x \in B$. So $h(3x) = 3h(x)$ and $g(3x) = 3g(x)$ for all $x\in B$, since $|st| < 1$. It follows from (4.1) that

for all $x, y, z \in B$. Thus

for all $x, y, z\in A$, since $|s| < 1$ and $|t| < 1$. So the mappings $g, h: B \to B$ are additive.

Now, we investigate $C^*$-ternary homomorphisms and $C^*$-ternary derivations in $C^*$-ternary algebras associated to the additive-additive $(s, t)$-functional inequality (4.1).

Theorem 4.2. Let $\varphi: B^3 \to [0, \infty)$ be a function such that

for all $x, y, z\in B$. Let $g, h : B \rightarrow B$ be mappings satisfying $g(0) = h(0) = 0$ and

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y, z \in B$. If $g, h: B\to B$ satisfy

for all $x, y, z\in B$, then the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Proof. Let $\lambda = 1$ in (4.3). By Lemma 4.1, the mappings $g, h: B\to B$ are additive.

It follows from (4.3) that

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y, z \in B$. Since $|s| < 1$ and $|t| < 1$,

for all $\lambda \in {{\mathbb{T}}^1}$ and all $x, y, z \in B$. Thus by Lemma 2.2, the mappings $g, h: B\to B$ are $\mathbb{C}$-linear.

It follows from (4.4) and the additivity of $g, h$ that

which tends to zero as $n\to \infty$, by (4.2). So

for all $x, y, z\in B$. Hence the mapping $g: B \to B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Theorem 4.3. Let $\varphi : B^3\to [0, \infty)$ be a function and $g, h : B \rightarrow B$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(4.3)$, $\rm(4.4)$ and

for all $x, y, z \in B$. Then the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Proof. By the same reasoning as in the proof of Theorem 4.2, one can show that the mappings $g, h: B\to B$ are $\mathbb{C}$-linear.

It follows from (4.4) and the additivity of $g, h$ that

which tends to zero as $n\to \infty$, by (4.5). So

for all $x, y, z\in B$. Hence the mapping $g: B\to B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Next, we investigate $C^*$-ternary homomorphisms and $C^*$-ternary derivations in $C^*$-ternary algebras associated to the additive-additive $(s, t)$-functional inequality (4.1) by using a similar method to the fixed point alternative.

Theorem 4.4. Let $\varphi: B^3 \to [0, \infty)$ be a function such that there exists an $L < 1$ with

for all $x, y, z \in B$. Let $g, h : B \rightarrow B$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(4.3)$ and $\rm(4.4)$. Then the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Proof. It follows from (4.6) that

for all $x, y, z \in B$. By Theorem 4.2, the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Theorem 4.5. Let $\varphi: B^3 \to [0, \infty)$ be a function such that there exists an $L < 1$ with

for all $x, y, z\in B$. Let $g, h : B \rightarrow B$ be mappings satisfying $g(0) = h(0) = 0$, $\rm(4.3)$ and $\rm(4.4)$. Then the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Proof. It follows from (4.7) that

for all $x, y, z \in B$. By Theorem 4.3, the mapping $g : B \rightarrow B$ is a $C^*$-ternary derivation and the mapping $h: B\to B$ is a $C^*$-ternary homomorphism.

Remark 4.6. In (4.4), the pair $(g, h)$ of a $C^*$-ternary derivation $g$ and a $C^*$-ternary homomorphism $h$ can be replaced by the pair of a $C^*$-ternary derivation and a $C^*$-ternary derivation or the pair of a $C^*$-ternary homomorphism and a $C^*$-ternary homomorphism.

5.   Conclusions

We have introduced the following additive-additive $(s, t)$-functional inequality (0.1) and have investigated homomorphisms and derivations in complex Banach algebras and unital $C^*$-algebra, associated to the additive-additive $(s, t)$-functional inequality (0.1) and the functional inequality (1.1). Moreover, we have introduced the following additive-additive $(s, t)$-functional inequality (0.2) and have investigated $C^*$-ternary derivations and $C^*$-ternary homomorphisms in $C^*$-ternary algebras, associated to the additive-additive $(s, t)$-functional inequality (0.2) and the functional inequality (1.2).

Conflict of interests

The authors declare no conflicts of interest in this paper.