Generalized ($\alpha,\beta, \gamma$)-derivations on Lie $C^*$-algebras

1.   Introduction and preliminaries

The derivation theory of Lie algebras play a key role in Lie theory. In particular, Physically motivated relations between two Lie algebras have been extensively discussed ^[27]. The problems for the structures and characteristics of $(\alpha, \beta, \gamma)$-derivations of Lie algebras have been extensively investigated by a range of scholars, as for this, many scholars have made useful researches (see ^[22,28,37]). The authors set up the structure and properties of $(\alpha, \beta, \gamma)$-derivations of Lie algebras.

In this work, The definition of a Lie $C^*$-algebra come from ^[29,30,34]). In ^[28], the definition of $(\alpha, \beta, \gamma)$-derivation can be found.

1940, the stability problem of group homomorphisms was raised by Ulam ^[38]. In 1941, Hyers ^[20] answers this question with a qualified yes to the question of Ulam for additive groups in Banach spaces. Hyers' theorem was generalized by Aoki ^[2], Rassias ^[35] and Gǎvruta ^[17] for linear mappings. In recent years, a lot of experts and scholars have studied in this area and made many achievements (see ^{[1,3,6,7,9,12,23,24,25,33,39,40]}).

Gilányi ^[18] and ^[36] considered the functional inequality

then $f$ satisfies the Jordan-von Neumann functional equation

respectively. The Hyers-Ulam stability of the above functional inequality is discussed by Fechner ^[16] and Gilányi ^[19]. Park ^[31,32] gave the definition of additive $\rho$-functional inequalities and discussed the Hyers-Ulam stability of the additive $\rho$-functional inequalities in different spaces.

To obtain a Jordan and von Neumann type characterization theorem for the quasi-inner-product spaces, Drygas ^[11] considered the functional equation

which solution is called a Drygas mapping. The general solution of the above functional equation was given by Ebanks, Kannappan and Sahoo ^[13] as

here $A$ is an additive mapping and $Q$ is a quadratic mapping.

In this work, we consider the stability of $(\alpha, \beta, \gamma)$-derivations on Lie $C^*$-algebras by the general Drygas functional equation

the coefficients $a, b$ is complex number, the proof of stability of the (1.3) is difference in ^[13]. The additive mapping $A$ and quadratic mapping $Q$ is constructed by the function relations, this method is called “directed method”. In the (1.3), $a, b$ action will cause difficulties for the stability of functional inequalities. We can overcome the influence of $a, b$, the stability of $(\alpha, \beta, \gamma)$-derivations using the fixed method. The beautiful examples about $(\alpha, \beta, \gamma)$-derivations can be found in ^[41].

The Hyers-Ulam stability analysis on $C^*$-algebras about functional equations have been discussed by fixed point theorem (see^{[5,8,14,15,21]}).

Next, the concept of the “generalized complete metric space” is introduced following Luxemburg ^[26].

Definition 1.1. Let $X$ be an abstract (nonempty) set, the elements of which are denoted by $x, y, \cdots$ and assume that on the Cartesian product $X\times X$ a distance function $d(x, y)(0\leq d(x, y)\leq \infty)$ is defined, satisfying the following conditions

$(1)\; d(x, y) = 0$ if and only if $x = y$,

$(2)\; d(x, y) = d(y, x)$(symmetry),

$(3)\; d(x, y)\leq d(x, z)+d(z, y)$(triangle inequality),

$(4)$ every $d$-Cauchy sequence in $X$ is $d$-convergent, i.e. $\lim_{n, m\rightarrow \infty}d(x_n, x_m) = 0$ for a sequence $x_n\in X(n = 1, 2, \cdots)$ implies the existence of an element $x\in X$ with $\lim_{n\rightarrow \infty}d(x, x_n) = 0$, ($x$ is unique).

By the concept, every two points in $X$ may be have the infinite distance. The space is called a generalized complete metric space.

We recall fixed point theorem that plays an key role to prove the stability of derivation.

Theorem 1.2. ^[4,10] Let $(X, d)$ be a complete generalized metric space and $J:X\rightarrow X$ be a strictly contractive mapping with Lipschitz constant $L < 1$. Then for any $x\in X$, either

for all nonnegative integers $n$ or there exists a positive integer $n_0$ such that

(1) $d(J^nx, J^{n+1}x) < \infty$, for all $n\geq n_0$;

(2) the sequence $\{J^nx\}$ converges to a fixed point $y^*$ of $J$;

(3) $y^*$ is the unique fixed point of $J$ in the set $Y = \{y\in X| d(J^{n_0}x, y) < \infty\}$;

(4) $d(y, y^*)\leq \frac{1}{1-L}d(y, Jy)$ for all $y\in Y$.

Now, using some thoughts from (^[4,10,15]) we discuss the stability for $(\alpha, \beta, \gamma)$-derivations and Lie $C^*$-algebra homomorphisms on Lie $C^*$-algebras related to (1.3) via the above fixed point theorem.

2.   The stability of $(\alpha, \beta, \gamma)$-derivations

Now, suppose that $s$ is complex fixed point and $\mathcal{A}$ is a Lie $C^*$-algebra with norm $\|\cdot\|$. The following lemma is necessary to prove our main theorems.

Lemma 2.1. ^[30] Suppose $X$ and $Y$ are linear spaces, $f:X \rightarrow Y$ is an additive map satisfying $f(\mu x) = \mu f(x)$, $\forall x\in X$ and $\mu \in T^1: = \{\lambda \in {\mathbb{C}}\; : \; |\lambda| = 1\}$. Then $f$ is $\mathbb{C}$-linear.

Lemma 2.2. Assume $f: \mathcal{A}\rightarrow \mathcal{A}$ is a map satisfying

$\forall x, y\in \mathcal{A}$, $|s|\leq |1-2b|\leq 1$. Then $f$ is additive.

Proof. If $x = y = 0$ in (2.1), then $f(0) = 0$. If $x = \frac{b}{a}y$ in (2.1) with $b\neq 0$, one obtain $f(-y) = -f(y)$.

Next, we discuss that $f$ is additive. Since $f(-y) = -f(y)$ in (2.1),

for $\forall x, y\in \mathcal{A}$. So $f$ is additive.

Theorem 2.3. If there are a mapping $\phi :\mathcal{A}^2 \rightarrow [0, \infty)$

and a mapping $\psi : \mathcal{A}^2 \rightarrow [0, \infty)$ with a constant $0 < L < 1$

Let $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfy

$\forall x, y\in \mathcal{A}, \mu \in T^1$, some $\alpha, \beta, \gamma, a, b$ and $|s|\leq |1-2b|\leq 1$. Then we can find a unique $(\alpha, \beta, \gamma)$-derivation $\delta : \mathcal{A}\rightarrow \mathcal{A}$ satisfies (1.3) and

Proof. Suppose $\Omega$ is a set of all mappings from $\mathcal{A}$ into $\mathcal{A}$, on $\Omega$, a generalized metric is introduced,

Then $(\Omega, d)$ becomes a generalized complete metric space. One define a map $T:\Omega \rightarrow \Omega$ by

Let $g, h \in \Omega$ with $d(g, h)\leq C$, here $C \in (0, \infty)$ is an arbitrary constant. Then we obtain $\|g(x)-h(x)\|\leq C\phi\left(\frac{x}{a}, 0\right)$,

i.e. $d(Tg-Th)\leq L d(g, h), \forall g, h \in \Omega$. Therefore, $T$ is a strictly contractive self-mapping on $\Omega$ associated with the Lipschitz constant $L$.

If $x = y = 0$ in (2.4), $f(0) = 0$.

If $y = 0$ and $\mu = 1$ in (2.4), then

Thus

for $\forall x\in \mathcal{A}$. Then we have $d(Tf, f)\leq \frac{1}{2(1-|s|)}$. By Theorem 1.2, there is a unique fixed point of $T$, map $\delta$, in the set $\Omega_1 = \{g\in \Omega : d(f, g) < \infty\}$,

since $\lim_{n\rightarrow \infty}d(T^n f, \delta) = 0$. Again by Theorem 1.2,

Then (2.6) holds.

By (2.4) and (2.7) and the property of $\phi$,

That is, $\delta$ is additive by Lemma 2.2. Next, letting $y = 0$, we get $2\delta (a\mu x) = \mu \delta(2ax)$ and so the map $\delta$ is $\mathbb{C}$-linear. Therefore, by the property of $\psi$, (2.5) and (2.7), then

for $\forall x, y\in\mathcal{A}$, some $\alpha, \beta$ and $\gamma\in \mathbb{C}$. Thus

for some $\alpha, \beta$ and $\gamma\in \mathbb{C}$. Hence $\delta$ is an unique derivation satisfying (2.6).

Corollary 2.4. If $r, k$ and $\theta$ belong to real numbers, $0 < r < 1, 0 < k < 2$ and $\theta \geq 0$. Let the map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfy

for $\forall x, y\in \mathcal{A}, \mu \in T^1$ and $|s|\leq |1-2b|\leq 1$. Then we can find a unique $(\alpha, \beta, \gamma)$-derivation $\delta : \mathcal{A}\rightarrow \mathcal{A}$,

for $\forall x\in \mathcal{A}$.

Proof. Let $\phi(x, y) = \theta(\|x\|^r+\|y\|^r), \psi(x, y) = \theta(\|x\|^k+\|y\|^k)$ and $L = 2^{r-1}$ in Theorem 2.3, the desired result is obtained.

Theorem 2.5. If there exists a map $\psi : \mathcal{A}^2 \rightarrow [0, \infty)$ satisfying (2.3). Let a map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfy

for $\forall x, y\in \mathcal{A}, \mu \in T^1$ and $|s|\leq |1-2b|\leq 1$. Thus the map $f:\mathcal{A}\rightarrow \mathcal{A}$ is a $(\alpha, \beta, \gamma)$-derivation.

Proof. Let $\mu = 1$ in (2.8), the map $f$ is additive by Lemma 2.2. Let $y = 0$ in (2.8), we get

for $\forall x\in \mathcal{A}$, $\mu \in T^1$. So $f(\mu x) = \mu f(x), \forall x\in \mathcal{A}$ and $\mu \in T^1$. The map $f$ is $\mathbb{C}$-linear by Lemma 2.1. On account of $f$ is additive, by (2.9),

for $\forall x, y\in \mathcal{A}$. Thus

Corollary 2.6. If $k$ and $\theta$ belong to real numbers with $0 < k < 2$ and $\theta\geq 0$. Assume a map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfies

for $\forall x, y\in \mathcal{A}, \mu \in T^1$ and $|s|\leq |1-2b|\leq 1$. Then the map $f$ is a $(\alpha, \beta, \gamma)$-derivation.

Lemma 2.7. If $f: \mathcal{A}\rightarrow \mathcal{A}$ is a map satisfying

for $\forall x, y\in \mathcal{A}$, $|s|\geq |1-2b|\geq 1$. Then $f$ is additive.

Proof. Using the same technique with the Lemma 2.2, we can show that the Lemma 2.7.

Theorem 2.8. Assume the map $\phi :\mathcal{A}^2 \rightarrow [0, \infty)$ satisfies (2.2) and a map $\psi : \mathcal{A}^2 \rightarrow [0, \infty)$ satisfies (2.3). Let the map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfy

for $\forall x, y\in \mathcal{A}, \mu \in T^1$, some $\alpha, \beta, \gamma, a, b$, and $|s|\geq |1-2b|\geq 1$. Then we can find a unique derivation $\delta$ satisfying (1.3), and

for $\forall x\in \mathcal{A}$.

Proof. In a similar vein of Theorem 2.3, the theorem can be proved.

Corollary 2.9. Suppose $r, k, \theta\in \mathbb{R}$ and $0 < r < 1, 0 < k < 2$, $\theta \geq 0$, let the map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfy

for $\forall x, y\in \mathcal{A}, \mu \in T^1$, some $\alpha, \beta, \gamma, a, b$, and $|s|\geq |1-2b|\geq 1$. Then there is only one $(\alpha, \beta, \gamma)$-derivation $\delta : \mathcal{A}\rightarrow \mathcal{A}$ satisfying

for $\forall x\in \mathcal{A}$.

Proof. In Theorem 2.8, let $\phi(x, y) = \theta(\|x\|^r+\|y\|^r), \psi(x, y) = \theta(\|x\|^k+\|y\|^k)$, $\forall x, y\in \mathcal{A}$ and $L = 2^{r-1}$, then the Corollary is proved.

Theorem 2.10. If the map $\psi : \mathcal{A}^2 \rightarrow [0, \infty)$ satisfies (2.3). The map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfies

for $\forall x, y\in \mathcal{A}, \mu \in T^1$, some $\alpha, \beta, \gamma, a, b$, and $|s|\geq |1-2b|\geq 1$. Then the map $f:\mathcal{A}\rightarrow \mathcal{A}$ is a $(\alpha, \beta, \gamma)$-derivation.

Corollary 2.11. If $k, \theta \in \mathbb{R}$, $0 < k < 2$, $\theta\geq 0$, assume the map $f: \mathcal{A}\rightarrow \mathcal{A}$ satisfies

for $\forall x, y\in \mathcal{A}, \mu \in T^1$, $|s|\geq |1-2b|\geq 1$. Then the map $f:\mathcal{A}\rightarrow \mathcal{A}$ is a $(\alpha, \beta, \gamma)$-derivation.

3.   Conclusions

In this work, the general Drygas functional equation is introduced, the Hyers-Ulam stability of ($\alpha, \beta, \gamma$)-derivations on Lie $C^*$-algebras is discussed by general Drygas functional inequality with the participation of coefficient $a$ and $b$.

Acknowledgment

This work was supported by National Natural Science Foundation of China (No. 11761074), the Projection of the Department of Science and Technology of JiLin Province and the Education Department of Jilin Province (No. 20170101052JC) and the scientific research project of Guangzhou College of Technology and Business in 2020 (No. KA202032).

Conflict of interest

The authors of this paper declare that they have no conflict of interest.