Stability of Tri-Homomorphisms, Tri-Derivations, and Tri-Isomorphisms in C*-Ternary Algebras

Abstract

This work demonstrates the generalized Hyers–Ulam stability of tri-homomorphisms within $C^{\ast }$-ternary algebras. We investigate the stability of a tri-additive functional equation and apply these results to study tri-derivations and tri-isomorphisms between $C^{\ast }$-ternary algebras.

1. Introduction and Background

Ternary algebraic operations have been studied since the 19th century, notably by Cayley [1], who introduced the concept of cubic matrices. One of the simplest examples of such a nontrivial ternary operation is given by the following composition rule:

$${\{a,b,c\}}_{ijk}=\sum _{1\le l,m,n\le N}{a}_{nil}{b}_{ljm}{c}_{mkn}(i,j,k=1,2,\cdots ,N).$$

Definition 1.

If A is a Banach algebra, an involution is a map $a\mapsto a^{\ast }$ of A into A such that the following properties hold for a and b in A and $\lambda \in \mathbb{C}$:

(i) 

${\left(a^{\ast }\right)}^{\ast }=a$;

(ii) 

${\left(ab\right)}^{\ast }=b^{\ast }{a}^{\ast }$;

(iii) 

${(\lambda a+b)}^{\ast }=\overline{\lambda }{a}^{\ast }+b^{\ast }$.

A $C^{\ast }$-algebra is a Banach algebra A with an involution such that for every $a\in A$, $\parallel a^{\ast }{a\parallel =\parallel a\parallel }^2$. The identity $\parallel a^{\ast }{a\parallel =\parallel a\parallel }^2$ is called the $C^{\ast }$-identity.

The $C^{\ast }$-identity implies that $\parallel a^{\ast }\parallel =\parallel a\parallel$ for all $a\in A$. Classical examples include the algebra $B\left(H\right)$ of all bounded linear operators on a Hilbert space H, as well as any closed ∗-subalgebra of $B\left(H\right)$.

The notion of $C^{\ast }$-ternary rings together with ternary operations was first presented in [2].

Definition 2.

If A is a complex Banach space, a ternary product τ is a map $(x,y,z)\mapsto [x,y,z]$ of A into A such that the following properties hold:

(i) 

The mapping τ acts $\mathbb{C}$-linearly on the first and third variables.

(ii) 

On the second variable, τ is conjugate $\mathbb{C}$-linear.

(iii) 

τ is associative in the sense that, for every $x,y,z,w,v\in A$ and every $\lambda \in \mathbb{C}$,

$$[x,y,[z,w,v\left]\right]=[x,[w,z,y],v]=\left[\right[x,y,z],w,v].$$

A $C^{\ast }$-ternary algebra is a complex Banach space A, equipped with a ternary product such that $\parallel [x,y,z]\parallel \le \parallel x\parallel ·\parallel y\parallel ·\parallel z\parallel$ and $\parallel [x,x,x]\parallel ={\parallel x\parallel }^3$ for every $x,y,z\in A$.

Every left Hilbert $C^{\ast }$-module can be viewed as a $C^{\ast }$-ternary algebra with the ternary product $[x,y,z]:=⟨x,y⟩z$.

If a $C^{\ast }$-ternary algebra $(A,[·,·,·\left]\right)$ possesses an identity, i.e., an element $e\in A$ such that $x=[x,e,e]=[e,e,x]$ for all $x\in A$, then it is straightforward to check that A, equipped with the operations $x\circ y:=[x,e,y]$ and $x^{\ast }:=[e,x,e]$, becomes a unital $C^{\ast }$-algebra. Conversely, if $(A,\circ )$ is a unital $C^{\ast }$-algebra, then defining $[x,y,z]:=x\circ y^{\ast }\circ z$ turns A into a $C^{\ast }$-ternary algebra.

Definition 3

(see [3]). Let A and B denote $C^{\ast }$-ternary algebras. A $\mathbb{C}$-linear mapping $H:A\to B$ is called a $C^{\ast }$-ternary algebra homomorphism provided that

$$H\left(\right[x,y,z\left]\right)=\left[H\right(x),H(y),H(z\left)\right]$$

for every $x,y,z\in A$. If H is also bijective, then H is termed a $C^{\ast }$-ternary algebra isomorphism. In addition, a $\mathbb{C}$-linear mapping $\delta :A\to A$ is defined to be a $C^{\ast }$-ternary derivation if

$$\delta \left(\right[x,y,z\left]\right)=\left[\delta \right(x),y,z]+[x,\delta (y),z]+[x,y,\delta (z\left)\right]$$

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

Definition 4

(see [4,5]). Let A and B denote $C^{\ast }$-ternary algebras. We say that a $\mathbb{C}$-trilinear map $H:A^3\to B$ is a $C^{\ast }$-ternary algebra tri-homomorphism (equivalently, a $C^{\ast }$-ternary algebra 3-homomorphism) whenever

$$H([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3])=[H(x_1,x_2,x_3),H(y_1,y_2,y_3),H(z_1,z_2,z_3)]$$

for every $x_i,y_j,z_k\in A$ with $1\le i,j,k\le 3$. Similarly, a $\mathbb{C}$-trilinear mapping $\delta :A^3\to A$ is termed a $C^{\ast }$-ternary tri-derivation (equivalently, a $C^{\ast }$-ternary 3-derivation) provided that

$$\begin{array}{cc}\hfill D\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)=& \left[D(x_1,x_2,x_3),[y_1,y_2^{\ast },y_3],[z_1,z_2^{\ast },z_3]\right]\hfill \\ & +\left[[x_1,x_2^{\ast },x_3],D(y_1,y_2,y_3),[z_1,z_2^{\ast },z_3]\right]\hfill \\ & +\left[[x_1,x_2^{\ast },x_3],[y_1,y_2^{\ast },y_3],D(z_1,z_2,z_3)\right]\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$.

Since Ulam [6] first raised the stability problem of functional equations in 1940, numerous mathematicians have studied this topic extensively. One of these dealt with the stability of homomorphisms and may be formulated as follows:

Let G be a group and $G^{\prime }$ a metric group with metric $\rho (·,·)$. Given $ϵ>0$, is there a $\delta >0$ such that if $f:G\to G^{\prime }$ satisfies

$$\rho \left(f\right(xy),f(x\left)f\right(y\left)\right)<\delta$$

for every $x,y\in G$, then there exists a homomorphism $h:G\to G^{\prime }$ with

$$\rho \left(f\right(x),h(x\left)\right)<ϵ$$

for every $x\in G$?

In 1941, Hyers [7] gave the first partial solution to Ulam’s question for the case of approximate additive mappings under the assumption that $G_1$ and $G_2$ are Banach spaces. Then, Aoki [8] considered the stability problem with unbounded Cauchy differences. In 1978, Rassias [9] generalized the theorem of Hyers [7] by considering the stability problem with unbounded Cauchy differences.

In 2011, the authors [4] studied the stability of the functional equation:

$$f(x_1+x_2,y_1+y_2,z_1+z_2)=\sum _{1\le i,j,k\le 2}f(x_i,y_j,z_k).$$

(1)

As clarified in the Abstract and Introduction, our 2011 paper dealt exclusively with the generalized Hyers–Ulam stability of $C^{\ast }$-ternary algebra tri-homomorphisms, whereas the present work extends these results to include tri-derivations and tri-isomorphisms in $C^{\ast }$-ternary algebras.

Let $l,m$ and n be integers larger than 1. Suppose X and Y are real or complex linear spaces. For a mapping $f:X^3\to Y$, we examine the following functional equation:

$$f\left(\sum _{i=1}^l{x}_i,\sum _{j=1}^m{y}_j,\sum _{k=1}^n{z}_k\right)=\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i,y_j,z_k).$$

(2)

For convenience, let $l,m$ and n be integers greater than 2. In fact, throughout this paper, one can obtain similar results when $l,m,n>1$.

One can verify that a mapping $f:X^3\to Y$ satisfies the Equation (2) if and only if the mapping f is tri-additive, as described below.

If $f:X^3\to Y$ is tri-additive, then

$$\begin{array}{cc}\hfill f\left(\sum _{i=1}^l{x}_i,\sum _{j=1}^m{y}_j,\sum _{k=1}^n{z}_k\right)& =\sum _{i=1}^{l}f\left(x_i,\sum _{j=1}^m{y}_j,\sum _{k=1}^n{z}_k\right)\hfill \\ & =\sum _{i=1}^l\sum _{j=1}^{m}f\left(x_i,y_j,\sum _{k=1}^n{z}_k\right)=\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i,y_j,z_k)\hfill \end{array}$$

for every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in X$. Conversely, assume that f satisfies (2). By choosing $x_1=\cdots =x_l=y_1=\cdots =y_m=z_1=\cdots =z_n=0$ in (2), we get $f(0,0,0)=0$. Letting $x_1=x$ and $x_2=\cdots =x_l=y_1=\cdots =y_m=z_1=\cdots =z_n=0$ in (2), we have $f(x,0,0)=0$ for every $x\in X$. If we put $x_1=x$, $x_2=y$, $y_1=z$, $z_1=w$, $x_3=\cdots =x_l=y_2=\cdots =y_m=z_2=\cdots =z_n=0$ in (2), we get

$$f(x+y,z,w)=f(x,z,w)+f(y,z,w)$$

for every $x,y,z,w\in X$. By setting $y_1=y$ and $x_1=\cdots =x_l=y_2=\cdots =y_m=z_1=\cdots =z_n=0$ in (2), we obtain $f(0,y,0)=0$ for all $y\in X$. If we take $x_1=x$, $y_1=y$, $y_2=z$, $z_1=w$, and $x_2=\cdots =x_l=y_3=\cdots =y_m=z_2=\cdots =z_n=0$ in (2), we see that

$$f(x,y+z,w)=f(x,y,w)+f(x,z,w)$$

for every $x,y,z,w\in X$. Taking $z_1=z$ and $x_1=\cdots =x_l=y_1=\cdots =y_m=z_2=\cdots =z_n=0$ in (2), we obtain $f(0,0,z)=0$ for every $z\in X$. Putting $x_1=x$, $y_1=y$, $z_1=z$, $z_2=w$, and $x_2=\cdots =x_l=y_2=\cdots =y_m=z_3=\cdots =z_n=0$ in (2), we see that

$$f(x,y,z+w)=f(x,y,z)+f(x,y,w)$$

for every $x,y,z,w\in X$. Hence a mapping $f:X^3\to Y$ is tri-additive if and only if it satisfies (2).

2. Stability of Tri-Homomorphisms in ${\mathit{C}}^{\ast }$-Ternary Algebras

Lemma 1

([4]). Let V and W be $\mathbb{C}$-linear spaces, and let $f:V^3\to W$ be a tri-additive mapping such that $f(\lambda x,\mu y,\nu z)=\lambda \mu \nu f(x,y,z)$ for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1:=\{\alpha \in \mathbb{C}:|\alpha |=1\}$ and every $x,y,z\in V$, then f is $\mathbb{C}$-trilinear.

Lemma 2.

Let V and W be two $\mathbb{C}$-linear spaces. And let $f:V^3\to W$ be a mapping such that

$$f\left(\lambda \sum _{i=1}^l{x}_i,\mu \sum _{j=1}^m{y}_j,\nu \sum _{k=1}^n{z}_k\right)=\lambda \mu \nu \sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i,y_j,z_k)$$

(3)

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in V$. Then the mapping f is $\mathbb{C}$-trilinear.

Proof. 

Putting $\lambda =\mu =\nu =1$, we see that f satisfies (2). Thus f is tri-additive. If we take $x_1=x$, $x_2=\cdots =x_l=0$, $y_1=y$, $y_2=\cdots =y_m=0$, $z_1=z$ and $z_2=\cdots =z_n=0$ in the given functional equation, we obtain $f(\lambda x,\mu y,\nu z)=\lambda \mu \nu f(x,y,z)$ for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x,y,z\in V$. Hence, by Lemma 1, the mapping f is $\mathbb{C}$-trilinear. □

From now on, assume that A is a $C^{\ast }$-ternary algebra with norm ${\parallel ·\parallel }_A$ and that B is a $C^{\ast }$-ternary algebra with norm ${\parallel ·\parallel }_B$. For a given mapping $f:A^3\to B$, we define

$$\begin{array}{cc}\hfill D_{\lambda ,\mu ,\nu }f(x_1,\cdots ,& x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n)\hfill \\ & :=f\left(\lambda \sum _{i=1}^l{x}_i,\mu \sum _{j=1}^m{y}_j,\nu \sum _{k=1}^n{z}_k\right)-\lambda \mu \nu \sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}f(x_i,y_j,z_k)\hfill \end{array}$$

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in A$.

We proved the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{\ast }$-ternary algebras.

Theorem 1.

Let $p,\theta >0$ be real numbers with $p<1$. Consider a mapping $f:A^3\to B$ such that

$$\begin{array}{cc}\hfill \parallel D_{\lambda ,\mu ,\nu }f(x_1,\cdots ,x_l,& y_1,\cdots ,y_m,z_1,\cdots ,z_n){\parallel }_B\hfill \\ & \le \theta \left(\sum _{i=1}^l\parallel x_i{\parallel }_A^p+\sum _{j=1}^m\parallel y_j{\parallel }_A^p+\sum _{k=1}^n{\parallel z_k\parallel }_A^p\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \parallel f([x_1,y_1,z_1],[x_2,y_2,& z_2],[x_3,y_3,z_3])-[f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3)]{\parallel }_B\hfill \\ & \le \theta \left(\sum _{i=1}^3\parallel x_i{\parallel }_A^p+\sum _{j=1}^3\parallel y_j{\parallel }_A^p+\sum _{k=1}^3{\parallel z_k\parallel }_A^p\right)\hfill \end{array}$$

(5)

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in A$. Then there is a unique $C^{\ast }$-ternary algebra tri-homomorphism $H:A^3\to B$ such that

$${\parallel f(x,y,z)-H(x,y,z)\parallel }_B\le \theta \left({\displaystyle \frac{l}{lmn-l^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{m}{lmn-m^p}}{\parallel y\parallel }_A^p+{\displaystyle \frac{n}{lmn-n^p}}{\parallel z\parallel }_A^p\right)$$

(6)

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

Proof. 

Taking $\lambda =\mu =\nu =1$, $x_1=\cdots =x_l=x$, $y_1=\cdots =y_m=y$, and $z_1,\cdots ,z_n=z$ in (4), we gain

$${\parallel f(lx,my,nz)-lmnf(x,y,z)\parallel }_B\le \theta \left({l\parallel x\parallel }_A^p+{m\parallel y\parallel }_A^p+n{\parallel z\parallel }_A^p\right)$$

(7)

for every $x,y,z\in A$. By substituting x with $l^{j}x$, y with $m^{j}y$, and z with $n^{j}z$, and then dividing the above inequality by ${\left(lmn\right)}^{j+1}$, we obtain

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(lmn\right)}^j}}f\left(l^{j}x,m^{j}y,n^{j}z\right)& -{\displaystyle \frac{1}{{\left(lmn\right)}^{j+1}}}f\left(l^{j+1}x,m^{j+1}y,n^{j+1}z\right){\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{{\left(lmn\right)}^{j+1}}}\left(l^{1+jp}{\parallel x\parallel }_A^p+m^{1+jp}{\parallel y\parallel }_A^p+m^{1+jp}{\parallel z\parallel }_A^p\right)\hfill \end{array}$$

for every $x,y,z\in A$ and each $j=0,1,2,\cdots$. Given integers $j,k(0\le j<k)$, we obtain that

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(lmn\right)}^j}}f\left(l^{j}x,m^{j}y,n^{j}z\right)& -{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f\left(l^{k}x,m^{k}y,n^{k}z\right){\parallel }_B\hfill \\ & \le \sum _{i=j}^{k-1}{\displaystyle \frac{\theta }{{\left(lmn\right)}^{i+1}}}\left(l^{1+ip}{\parallel x\parallel }_A^p+m^{1+ip}{\parallel y\parallel }_A^p+n^{1+ip}{\parallel z\parallel }_A^p\right)\hfill \end{array}$$

(8)

for every $x,y,z\in A$. From the above inequality, the sequence $\left\{{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f(l^{d}x,m^{d}y,n^{d}z)\right\}$ is a Cauchy sequence for any $x,y,z\in A$. Since B is complete, this sequence $\left\{{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f(l^{d}x,m^{d}y,n^{d}z)\right\}$ converges for every $x,y,z\in A$. We thus define $H:A^3\to B$ by

$$H(x,y,z):=\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f\left(l^{d}x,m^{d}y,n^{d}z\right)$$

for every $x,y,z\in A$. By (4), we have

$$\begin{array}{cc}& {\parallel D_{\lambda ,\mu ,\nu }H(x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n)\parallel }_B\hfill \\ & ={\parallel H\left(\lambda \sum _{i=1}^l{x}_i,\mu \sum _{j=1}^m{y}_j,\nu \sum _{k=1}^n{z}_k\right)-\lambda \mu \nu \sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}H(x_i,y_j,z_k)\parallel }_B\hfill \\ & =\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}\parallel f\left(l^d\lambda \sum _{i=1}^l{x}_i,m^d\mu \sum _{j=1}^m{y}_j,n^d\nu \sum _{k=1}^n{z}_k\right)\hfill \\ & -\lambda \mu \nu \sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^n{f(l^d{x}_i,m^d{y}_j,n^d{z}_k)\parallel }_B\hfill \\ & \le \underset{d\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(lmn\right)}^d}}\left(\sum _{i=1}^l{\parallel l^d{x}_i\parallel }_A^p+\sum _{j=1}^m{\parallel m^d{y}_j\parallel }_A^p+\sum _{k=1}^n{\parallel n^d{z}_k\parallel }_A^p\right)\hfill \\ & =\theta \underset{d\to \infty }{\lim }\left[{\left({\displaystyle \frac{l^p}{lmn}}\right)}^d\sum _{i=1}^l{\parallel x_i\parallel }_A^p+{\left({\displaystyle \frac{m^p}{lmn}}\right)}^d\sum _{j=1}^m{\parallel y_j\parallel }_A^p+{\left({\displaystyle \frac{n^p}{lmn}}\right)}^d\sum _{k=1}^n{\parallel z_k\parallel }_A^p\right]=0\hfill \end{array}$$

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in A$. According to Lemma 2, the mapping H is $\mathbb{C}$-trilinear. If we set $j=0$ and taking $k\to \infty$ in (8), we derive the inequality (6).

It follows from (5) that

$$\begin{array}{cc}& \parallel H([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3])\hfill \\ & -[H(x_1,x_2,x_3),H(y_1,y_2,y_3),H(z_1,z_2,z_3)]{\parallel }_B\hfill \\ & =\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}\parallel f(l^d[x_1,y_1,z_1],m^d[x_2,y_2,z_2],n^d[x_3,y_3,z_3])\hfill \\ & -[f(l^d{x}_1,m^d{x}_2,n^d{x}_3),f(l^d{y}_1,m^d{y}_2,n^d{y}_3),f(l^d{z}_1,m^d{z}_2,n^d{z}_3)]{\parallel }_B\hfill \\ & \le \underset{d\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(lmn\right)}^d}}\left(l^{dp}\sum _{i=1}^3{\parallel x_i\parallel }_A^p+m^{dp}\sum _{j=1}^3{\parallel y_j\parallel }_A^p+n^{dp}\sum _{k=1}^3{\parallel z_k\parallel }_A^p\right)=0\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$. Thus we have

$$H\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)=\left[H(x_1,x_2,x_3),H(y_1,y_2,y_3),H(z_1,z_2,z_3)\right]$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$.

Now, suppose that $T:A^3\to B$ is another $C^{\ast }$-ternary algebra tri-homomorphism which also satisfies (6). Then we obtain

$$\begin{array}{cc}& \parallel H(x,y,z)-T(x,y,z){\parallel }_B\hfill \\ \hfill =& {\displaystyle \frac{1}{{\left(lmn\right)}^d}}\parallel H\left(l^{d}x,m^{d}y,n^{d}z\right)-T\left(l^{d}x,m^{d}y,n^{d}z\right){\parallel }_B\hfill \\ \hfill \le & {\displaystyle \frac{1}{{\left(lmn\right)}^d}}[{\parallel H\left(l^{d}x,m^{d}y,n^{d}z\right)-f\left(l^{d}x,m^{d}y,n^{d}z\right)\parallel }_B\hfill \\ & +{\parallel f\left(l^{d}x,m^{d}y,n^{d}z\right)-T\left(l^{d}x,m^{d}y,n^{d}z\right)\parallel }_B]\hfill \\ \hfill \le & {\displaystyle \frac{2\theta }{{\left(lmn\right)}^d}}\left({\displaystyle \frac{l^{1+dp}}{lmn-l^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{m^{1+dp}}{lmn-m^p}}{\parallel y\parallel }_A^p+{\displaystyle \frac{n^{1+dp}}{lmn-n^p}}{\parallel z\parallel }_A^p\right),\hfill \end{array}$$

which approaches to zero as $d\to \infty$ for every $x,y,z\in A$. It follows that $H(x,y,z)=T(x,y,z)$ for every $x,y,z\in A$, and hence H is unique. Thus, H constitutes the unique $C^{\ast }$-ternary algebra tri-homomorphism fulfilling (6). □

Next, we establish the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{\ast }$-ternary algebras, provided that $p>1+\max \{{\log }_{l}mn,{\log }_{m}ln,{\log }_{n}lm\}$.

Theorem 2.

Let θ be a positive real number and p a real number greater than $1+\max \{{\log }_{l}mn,{\log }_{m}ln,{\log }_{n}lm\}$, and suppose that $f:A^3\to B$ is a mapping satisfying (4) and (5). Then there exists a unique $C^{\ast }$-ternary algebra tri-homomorphism $H:A^3\to B$ such that

$${\parallel f(x,y,z)-H(x,y,z)\parallel }_B\le \theta \left({\displaystyle \frac{l}{l^p-lmn}}{\parallel x\parallel }_A^p+{\displaystyle \frac{m}{m^p-lmn}}{\parallel y\parallel }_A^p+{\displaystyle \frac{n}{n^p-lmn}}{\parallel z\parallel }_A^p\right)$$

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

Proof. 

Using the same reasoning as in the proof of Theorem 1, one obtains the inequality (7). It follows from (7) that

$${\parallel f(x,y,z)-lmnf\left({\displaystyle \frac{x}{l}},{\displaystyle \frac{y}{m}},{\displaystyle \frac{z}{n}}\right)\parallel }_B\le \theta \left(l^{1-p}{\parallel x\parallel }_A^p+m^{1-p}{\parallel y\parallel }_A^p+n^{1-p}{\parallel z\parallel }_A^p\right)$$

for every $x,y,z\in A$. So

$$\begin{array}{cc}& \parallel {\left(lmn\right)}^{j}f\left({\displaystyle \frac{x}{l^j}},{\displaystyle \frac{y}{m^j}},{\displaystyle \frac{y}{n^j}}\right)-{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{y}{n^k}}\right){\parallel }_B\hfill \\ \hfill \le & \sum _{i=j}^{k-1}\parallel {\left(lmn\right)}^{i}f\left({\displaystyle \frac{x}{l^i}},{\displaystyle \frac{y}{m^i}},{\displaystyle \frac{y}{n^i}}\right)-{\left(lmn\right)}^{i+1}f\left({\displaystyle \frac{x}{l^{i+1}}},{\displaystyle \frac{y}{m^{i+1}}},{\displaystyle \frac{z}{n^{i+1}}}\right){\parallel }_B\hfill \\ \hfill \le & \theta \sum _{i=j}^{k-1}{\left(lmn\right)}^i\left(l^{1-p(i+1)}{\parallel x\parallel }_A^p+m^{1-p(i+1)}{\parallel y\parallel }_A^p+n^{1-p(i+1)}{\parallel z\parallel }_A^p\right)\hfill \end{array}$$

(9)

for every $x,y,z\in A$ and every nonnegative integers $j,k$ with $j<k$. By the above inequality, the sequence $\left\{{\left(lmn\right)}^{k}f({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{z}{n^k}})\right\}$ is a Cauchy sequence for any $x,y,z\in A$. Since B is complete, the sequence $\left\{{\left(lmn\right)}^{k}f({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{z}{n^k}})\right\}$ converges for every $x,y,z\in A$. Hence we may define the mapping $H:A^3\to B$ by

$$H(x,y,z):=\underset{k\to \infty }{\lim }{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{y}{n^k}}\right)$$

for every $x,y,z\in A$. Furthermore, by taking $j=0$ and letting $k\to \infty$ in (9), we obtain the required inequality relating f and H.

The remainder of the proof proceeds in the same manner as in Theorem 1. □

We thus establish the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{\ast }$-ternary algebras whenever $p<\frac{1}{\max \{l,m,n\}}$.

Theorem 3.

Let $p,\theta >0$ be real numbers with $p<{\displaystyle \frac{1}{\max \{l,m,n\}}}$, and consider a mapping $f:A^3\to B$ such that

$$\begin{array}{cc}& \parallel D_{\lambda ,\mu ,\nu }f(x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n){\parallel }_B\hfill \\ & \le \theta {\left(\parallel x_1{\parallel }_A\cdots \parallel x_l{\parallel }_A\parallel y_1{\parallel }_A\cdots \parallel y_m{\parallel }_A\parallel z_1{\parallel }_A\cdots {\parallel z_n\parallel }_A\right)}^p,\hfill \end{array}$$

(10)

$$\begin{array}{cc}& \parallel f([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3])-[f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3)]{\parallel }_B\hfill \\ & \le \theta {\left({\parallel x_1\parallel }_A{\parallel x_2\parallel }_A{\parallel x_3\parallel }_A{\parallel y_1\parallel }_A{\parallel y_2\parallel }_A{\parallel y_3\parallel }_A{\parallel z_1\parallel }_A{\parallel z_2\parallel }_A{\parallel z_3\parallel }_A\right)}^p\hfill \end{array}$$

(11)

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in A$. Then there exists a unique $C^{\ast }$-ternary algebra tri-homomorphism $H:A^3\to B$ satisfying

$${\parallel f(x,y,z)-H(x,y,z)\parallel }_B\le {\displaystyle \frac{\theta }{lmn-{\left(l^l{m}^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

(12)

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

Proof. 

If we set $\lambda =\mu =\nu =1$, $x_1=\cdots =x_l=x$, $y_1=\cdots =y_m=y$ and $z_1,\cdots ,z_n=z$ in (10), we gain

$$\parallel f(lx,my,nz)-lmnf(x,y,z){\parallel }_B\le \theta {\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

(13)

for every $x,y,z\in A$. Substituting $x=l^{j}x$, $y=m^{j}y$, and $z=n^{j}z$ into the above inequality and dividing by ${\left(lmn\right)}^{j+1}$, we obtain

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(lmn\right)}^j}}f\left(l^{j}x,m^{j}y,n^{j}z\right)& -{\displaystyle \frac{1}{{\left(lmn\right)}^{j+1}}}f\left(l^{j+1}x,m^{j+1}y,n^{j+1}z\right){\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{lmn}}{\left(l^{lp-1}{m}^{mp-1}{n}^{np-1}\right)}^j{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p\hfill \end{array}$$

for every $x,y,z\in A$ and each $j=0,1,2,\cdots$. If $j,k$ are integers with $0\le j<k$, then we obtain

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{1}{{\left(lmn\right)}^j}}f\left(l^{j}x,m^{j}y,n^{j}z\right)& -{\displaystyle \frac{1}{{\left(lmn\right)}^k}}{f\left(l^{k}x,m^{k}y,n^{k}z\right)\parallel }_B\hfill \\ & \le \sum _{i=j}^{k-1}{\displaystyle \frac{\theta }{lmn}}{\left(l^{lp-1}{m}^{mp-1}{n}^{np-1}\right)}^i{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p\hfill \end{array}$$

(14)

for every $x,y,z\in A$. From the above inequality, the sequence $\left\{{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f(l^{d}x,m^{d}y,n^{d}z)\right\}$ is a Cauchy sequence for any $x,y,z\in A$. Since B is complete, the sequence $\left\{{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f(l^{d}x,m^{d}y,n^{d}z)\right\}$ converges for every $x,y,z\in A$. Define $H:A^3\to B$ by

$$H(x,y,z):=\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}f\left(l^{d}x,m^{d}y,n^{d}z\right)$$

for every $x,y,z\in A$. From (10), we gain

$$\begin{array}{cc}& \parallel D_{\lambda ,\mu ,\nu }H(x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n){\parallel }_B\hfill \\ & =\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}\parallel f\left(l^d\lambda \sum _{i=1}^l{x}_i,m^d\mu \sum _{j=1}^m{y}_j,n^d\nu \sum _{k=1}^n{z}_k\right)\hfill \\ & -\lambda \mu \nu \sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^n{f(l^d{x}_i,m^d{y}_j,n^d{z}_k)\parallel }_B\hfill \\ & \le \theta {\left(\parallel x_1{\parallel }_A\cdots \parallel x_l{\parallel }_A\parallel y_1{\parallel }_A\cdots \parallel y_m{\parallel }_A\parallel z_1{\parallel }_A\cdots {\parallel z_n\parallel }_A\right)}^p\underset{d\to \infty }{\lim }{\left(l^{lp-1}{m}^{mp-1}{n}^{np-1}\right)}^d\hfill \\ & =0\hfill \end{array}$$

for every $\lambda ,\mu ,\nu \in {\mathbb{T}}^1$ and every $x_1,\cdots ,x_l,y_1,\cdots ,y_m,z_1,\cdots ,z_n\in A$. According to Lemma 2, the map H is $\mathbb{C}$-trilinear. By taking $j=0$ and passing to the limit $k\to \infty$ in (14), the inequality (12) is derived. Consequently, (11) implies that

$$\begin{array}{cc}& \parallel H\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)\hfill \\ & -\left[H(x_1,x_2,x_3),H(y_1,y_2,y_3),H(z_1,z_2,z_3)\right]{\parallel }_B\hfill \\ & =\underset{d\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^d}}\parallel f\left(l^d[x_1,y_1,z_1],m^d[x_2,y_2,z_2],n^d[x_3,y_3,z_3]\right)\hfill \\ & -\left[f(l^d{x}_1,m^d{x}_2,n^d{x}_3),f(l^d{y}_1,m^d{y}_2,n^d{y}_3),f(l^d{z}_1,m^d{z}_2,n^d{z}_3)\right]{\parallel }_B\hfill \\ & \le \theta {\left({\parallel x_1\parallel }_A{\parallel x_2\parallel }_A{\parallel x_3\parallel }_A{\parallel y_1\parallel }_A{\parallel y_2\parallel }_A{\parallel y_3\parallel }_A{\parallel z_1\parallel }_A{\parallel z_2\parallel }_A{\parallel z_3\parallel }_A\right)}^p\underset{d\to \infty }{\lim }{\left(lmn\right)}^{(3p-1)d}\hfill \\ & =0\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$. Thus we obtain

$$H\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)=\left[H(x_1,x_2,x_3),H(y_1,y_2,y_3),H(z_1,z_2,z_3)\right]$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$.

Consider another $C^{\ast }$-ternary algebra tri-homomorphism $T:A^3\to B$ satisfying (12). Then we obtain

$$\begin{array}{cc}& {\parallel H(x,y,z)-T(x,y,z)\parallel }_B\hfill \\ & ={\displaystyle \frac{1}{{\left(lmn\right)}^d}}{\parallel H(l^{d}x,m^{d}y,n^{d}z)-T(l^{d}x,m^{d}y,n^{d}z)\parallel }_B\hfill \\ & \le {\displaystyle \frac{2\theta {\left(l^{lp-1}{m}^{mp-1}{n}^{np-1}\right)}^d}{lmn-{\left(l^l{m}^m{n}^n\right)}^p}}{(\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n{)}^p,\hfill \end{array}$$

which converges to zero as $d\to \infty$ for every $x,y,z\in A$. Therefore $H=T$, showing that H is uniquely determined. Thus H is the only $C^{\ast }$-ternary algebra tri-homomorphism that satisfies (12). □

Next, we proved the generalized Hyers–Ulam stability of the functional Equation (3) in $C^{\ast }$-ternary algebras for $p>{\displaystyle \frac{\log lmn}{\log l^l{m}^m{n}^n}}$.

Theorem 4.

Let $p,\theta$ be a positive real numbers with $p>{\displaystyle \frac{\log lmn}{\log l^l{m}^m{n}^n}}$, and let $f:A^3\to B$ be a mapping satisfying (10) and (11). Then there is a unique $C^{\ast }$-ternary algebra tri-homomorphism $H:A^3\to B$ for which

$${\parallel f(x,y,z)-H(x,y,z)\parallel }_B\le {\displaystyle \frac{\theta }{{\left(l^l{m}^m{n}^n\right)}^p-lmn}}{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

holds for every $x,y,z\in A$.

Proof. 

It follows directly from (13) that

$${\parallel f(x,y,z)-lmnf\left({\displaystyle \frac{x}{l}},{\displaystyle \frac{y}{m}},{\displaystyle \frac{z}{n}}\right)\parallel }_B\le {\displaystyle \frac{\theta }{{\left(l^l{m}^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

for every $x,y,z\in A$. For integers $j,k$ with $0\le j<k$, we then obtain that

$$\begin{array}{cc}\hfill \parallel {\left(lmn\right)}^{j}f({\displaystyle \frac{x}{l^j}},{\displaystyle \frac{y}{m^j}}& ,{\displaystyle \frac{z}{n^j}})-{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{z}{n^k}}\right){\parallel }_B\hfill \\ & \le \sum _{i=j}^{k-1}\parallel {\left(lmn\right)}^{i}f\left({\displaystyle \frac{x}{l^i}},{\displaystyle \frac{y}{m^i}},{\displaystyle \frac{z}{n^i}}\right)-{\left(lmn\right)}^{i+1}{f\left({\displaystyle \frac{x}{l^{i+1}}},{\displaystyle \frac{y}{m^{i+1}}},{\displaystyle \frac{z}{n^{i+1}}}\right)\parallel }_B\hfill \\ & \le {\displaystyle \frac{\theta }{{\left(l^l{m}^m{n}^n\right)}^p}}\sum _{i=j}^{k-1}{\left(l^{1-lp}{m}^{1-mp}{n}^{1-np}\right)}^i{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p\hfill \end{array}$$

(15)

for every $x,y,z\in A$. From the above inequality, the sequence

$$\left\{{\left(lmn\right)}^{k}f\left(\frac{x}{l^k},\frac{y}{m^k},\frac{z}{n^k}\right)\right\}$$

forms a Cauchy sequence for every $x,y,z\in A$. Since B is complete, this sequence converges for every such $x,y,z$. We thus define $H:A^3\to B$ by

$$H(x,y,z):=\underset{k\to \infty }{\lim }{\left(lmn\right)}^{k}f\left(\frac{x}{l^k},\frac{y}{m^k},\frac{z}{n^k}\right),x,y,z\in A.$$

If we put $j=0$ and let $k\to \infty$ in (15), we arrive at the required inequality between f and H. The rest of the argument proceeds in the same way as in Theorem 3. □

3. Stability of Tri-Derivations and Tri-Isomorphisms in ${\mathit{C}}^{\ast }$-Ternary Algebras

We proved the stability of tri-derivations and tri-isomorphisms in $C^{\ast }$-ternary algebras. In particular, we proved the existence and uniqueness of such mappings, showing that they approximate given mappings within prescribed bounds.

Theorem 5.

Let $p,\theta$ be positive real numbers with $p<1$, and let $f:A^3\to A$ be a mapping satisfying (4) such that

$$\begin{array}{cc}\hfill \parallel f\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)& -\left[f(x_1,x_2,x_3),[y_1,y_2^{\ast },y_3],[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],f(y_1,y_2,y_3),[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],[y_1,y_2^{\ast },y_3],f(z_1,z_2,z_3)\right]{\parallel }_A\hfill \\ & \le \theta \sum _{k=1}^3\left({\parallel x_k\parallel }_A^p+{\parallel y_k\parallel }_A^p+{\parallel z_k\parallel }_A^p\right)\hfill \end{array}$$

(16)

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$. If f satisfies

$$\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f(l^{k}x,m^{k}y,n^{k}z)=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f(l^{3k}x,m^{3k}y,n^{3k}z)$$

(17)

for every $x,y,z\in A$ and $f(0,0,0)=0$, then exists a unique $C^{\ast }$-ternary tri-derivation $\delta :A^3\to A$ such that

$${\parallel f(x,y,z)-\delta (x,y,z)\parallel }_A\le \theta \left({\displaystyle \frac{l}{lmn-l^p}}{\parallel x\parallel }_A^p+{\displaystyle \frac{m}{lmn-m^p}}{\parallel y\parallel }_A^p+{\displaystyle \frac{n}{lmn-n^p}}{\parallel z\parallel }_A^p\right)$$

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

Proof. 

By the proof of Theorem 1, the sequence $\left\{{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f(l^{k}x,m^{k}y,n^{k}z)\right\}$ is a Cauchy sequence for any $x,y,z\in A$. Because A is complete, this sequence converges for every $x,y,z\in A$. Hence, we define the mapping $\delta :A^3\to A$ by

$$\delta (x,y,z):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f\left(l^{k}x,m^{k}y,n^{k}z\right)$$

for every $x,y,z\in A$. Using the same argument as in the proof of Theorem 1, we derive the desired inequality for f and $\delta$, and conclude that $\delta$ is $\mathbb{C}$-trilinear.

From (16) and (17), it follows that

$$\begin{array}{cc}& \parallel \delta \left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)-\left[\delta (x_1,x_2,x_3),[y_1,y_2^{\ast },y_3],[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],\delta (y_1,y_2,y_3),[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],[y_1,y_2^{\ast },y_3],\delta (z_1,z_2,z_3)\right]{\parallel }_A\hfill \\ & =\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^k}}\parallel f\left(l^{3k}[x_1,y_1,z_1],m^{3k}[x_2,y_2,z_2],n^{3k}[x_3,y_3,z_3]\right)\hfill \\ & -\left[f(l^k{x}_1,m^k{x}_2,n^k{x}_3),[l^k{y}_1,m^k{y}_2^{\ast },n^k{y}_3],[l^k{z}_1,m^k{z}_2^{\ast },n^k{z}_3]\right]\hfill \\ & -\left[[l^k{x}_1,m^k{x}_2^{\ast },n^k{x}_3],f(l^k{y}_1,m^k{y}_2,n^k{y}_3),[l^k{z}_1,m^k{z}_2^{\ast },n^k{z}_3]\right]\hfill \\ & -\left[[l^k{x}_1,m^k{x}_2^{\ast },n^k{x}_3],[l^k{y}_1,m^k{y}_2^{\ast },n^k{y}_3],f(l^k{z}_1,m^k{z}_2,n^k{z}_3)\right]{\parallel }_A\hfill \\ & \le \underset{k\to \infty }{\lim }{\displaystyle \frac{\theta }{{\left(lmn\right)}^k}}[l^{kp}\left({\parallel x_1\parallel }_A^p+{\parallel y_1\parallel }_A^p+{\parallel z_1\parallel }_A^p\right)+m^{kp}\left({\parallel x_2\parallel }_A^p+{\parallel y_2\parallel }_A^p+{\parallel z_2\parallel }_A^p\right)\hfill \\ & +n^{kp}\left({\parallel x_3\parallel }_A^p+{\parallel y_3\parallel }_A^p+{\parallel z_3\parallel }_A^p\right)]\hfill \\ & =0\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$. So

$$\begin{array}{cc}\hfill \delta \left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)=& \left[\delta (x_1,x_2,x_3),[y_1,y_2^{\ast },y_3],[z_1,z_2^{\ast },z_3]\right]\hfill \\ & +\left[[x_1,x_2^{\ast },x_3],\delta (y_1,y_2,y_3),[z_1,z_2^{\ast },z_3]\right]\hfill \\ & +\left[[x_1,x_2^{\ast },x_3],[y_1,y_2^{\ast },y_3],\delta (z_1,z_2,z_3)\right]\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$.

By an argument similar to that in the proof of Theorem 1, the mapping $\delta :A^3\to A$ is the unique $C^{\ast }$-ternary tri-derivation satisfying the required inequality. □

In Theorem 5, for the case that $p>{\displaystyle \frac{1}{3}}$ and

$$\underset{k\to \infty }{\lim }{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{z}{n^k}}\right)=\underset{k\to \infty }{\lim }{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^{3k}}},{\displaystyle \frac{y}{m^{3k}}},{\displaystyle \frac{z}{n^{3k}}}\right)$$

(18)

for every $x,y,z\in A$, we derive a similar result.

Theorem 6.

Let $p,\theta$ denote positive real numbers. Consider a mapping $f:A^3\to A$ which satisfies (10) such that

$$\begin{array}{cc}& \parallel f\left([x_1,y_1,z_1],[x_2,y_2,z_2],[x_3,y_3,z_3]\right)-\left[f(x_1,x_2,x_3),[y_1,y_2^{\ast },y_3],[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],f(y_1,y_2,y_3),[z_1,z_2^{\ast },z_3]\right]\hfill \\ & -\left[[x_1,x_2^{\ast },x_3],[y_1,y_2^{\ast },y_3],f(z_1,z_2,z_3)\right]{\parallel }_A\hfill \\ & \le \theta {\left({\parallel x_1\parallel }_A{\parallel y_1\parallel }_A{\parallel z_1\parallel }_A{\parallel x_2\parallel }_A{\parallel y_2\parallel }_A{\parallel z_2\parallel }_A{\parallel x_3\parallel }_A{\parallel y_3\parallel }_A{\parallel z_3\parallel }_A\right)}^p\hfill \end{array}$$

for every $x_i,y_j,z_k\in A$$(1\le i,j,k\le 3)$.

For $p<{\displaystyle \frac{1}{\max \{l,m,n\}}}$, if f satisfies (17), then there exists only one $C^{\ast }$-ternary tri-derivation $\delta :A^3\to A$ such that

$${\parallel f(x,y,z)-\delta (x,y,z)\parallel }_A\le {\displaystyle \frac{\theta }{lmn-{\left(l^l{m}^m{n}^n\right)}^p}}{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

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

For $p>{\displaystyle \frac{1}{3}}$, if f satisfies (18), then there exists only one $C^{\ast }$-ternary tri-derivation $\delta :A^3\to A$ such that

$${\parallel f(x,y,z)-\delta (x,y,z)\parallel }_A\le {\displaystyle \frac{\theta }{{\left(l^l{m}^m{n}^n\right)}^p-lmn}}{\left({\parallel x\parallel }_A^l{\parallel y\parallel }_A^m{\parallel z\parallel }_A^n\right)}^p$$

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

Proof. 

Let $p<{\displaystyle \frac{1}{\max \{l,m,n\}}}$. Analogous to the proof of Theorem 3, we define the mapping $\delta :A^3\to A$ by

$$\delta (x,y,z):=\underset{k\to \infty }{\lim }{\displaystyle \frac{1}{{\left(lmn\right)}^k}}f(l^{k}x,m^{k}y,n^{k}z)$$

for every $x,y,z\in A$, which satisfies the required inequality for f and $\delta$. The remainder of the proof follows in the same manner as that of Theorem 5.

If $p>{\displaystyle \frac{1}{3}}$. Analogous to the proof of Theorem 4, the mapping $\delta :A^3\to A$ can be defined by

$$\delta (x,y,z):=\underset{k\to \infty }{\lim }{\left(lmn\right)}^{k}f\left({\displaystyle \frac{x}{l^k}},{\displaystyle \frac{y}{m^k}},{\displaystyle \frac{z}{n^k}}\right)$$

for every $x,y,z\in A$, thereby satisfying the required inequality for f and $\delta$. The remainder of the proof proceeds in the same way as the proofs of Theorems 4 and 5, as well as the case $p<{\displaystyle \frac{1}{\max \{l,m,n\}}}$. □

4. Conclusions

In this work, we carried out a refined investigation of Hyers–-Ulam stability for functional equations in three-dimensional $C^{\ast }$-ternary algebras. By employing a generalized norm inequality method, we established the existence and uniqueness of tri-homomorphic and tri-isomorphic approximations. This study provides a refined analysis of the Hyers–-Ulam stability of functional equations within three-dimensional $C^{\ast }$-ternary algebras. Using a generalized norm inequality framework, we confirmed both the existence and uniqueness of tri-homomorphic and tri-isomorphic approximations. In this paper, we presented an enhanced analysis of Hyers–Ulam stability for functional equations in three-dimensional $C^{\ast }$-ternary algebras. Our approach, based on generalized norm inequalities, guarantees the existence and uniqueness of tri-homomorphic and tri-isomorphic approximations.

Since $C^{\ast }$-ternary algebras are closely related to Hilbert $C^{\ast }$-modules, the stability results have potential applications in operator algebra theory and quantum information theory, where ternary operations naturally appear.