1 Introduction

Ternary algebraic operations were considered in the 19th century by several mathematicians, such as Cayley [1], who introduced the notion of cubic matrix which, in turn, was generalized by Kapranov et al. [2]. The simplest example of such a non-trivial ternary operation is given by the following composition rule:

$$\{a,b,c\}_{ijk}=\sum_{1\leq l,m,n\leq N}a_{nil}b_{ljm}c_{mkn} $$

for each \(i,j,k=1,2,\ldots,N\).

Ternary structures and their generalization, the so-called n-ary structures, raise certain hopes in view of their applications in physics. Some significant applications are as follows (see [3, 4]):

  1. (1)

    The algebra of nonions generated by two matrices

    $$ \left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right ),\qquad \left ( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & 1 & 0 \\ 0 & 0 & \omega \\ \omega^{2} & 0 & 0 \end{array} \right ), $$

    where \(\omega=e^{\frac{2\pi i}{3}}\), was introduced by Sylvester as a ternary analog of Hamilton’s quaternions (see [5]).

  2. (2)

    The quark model inspired a particular brand of ternary algebraic systems. The so-called Nambu mechanics is based on such structures (see [6]).

2 Random \(C^{*}\)-ternary algebra

In the section, we adopt the usual terminology, notations and conventions of the theory of random \(C^{*}\)-ternary algebra.

Throughout this paper, \(\Delta^{+}\) is the space of distribution functions, that is, the space of all mappings \(F:{\mathbf{R}} \cup\{-\infty,\infty\} \to [0,1]\) such that F is left-continuous and non-decreasing on R, \(F(0)=0\), and \(F(+\infty)=1\). \(D^{+}\) is a subset of \(\Delta^{+}\) consisting of all functions \(F \in\Delta^{+}\) for which \(l^{-}F(+\infty)=1\), where \(l^{-}f(x)\) denotes the left limit of the function f at the point x, that is, \(l^{-}f(x)=\lim_{t \to x^{-}}f(t)\). The space \(\Delta^{+}\) is partially ordered by the usual point-wise ordering of functions, i.e., \(F \leq G\) if and only if \(F(t) \leq G(t)\) for all t in R. For example an element for \(\Delta^{+}\) is the distribution function \(\varepsilon_{a}\) given by \(\varepsilon_{a}(t)= 0\), if \(t\leq a\) and 1 if \(t>a\).

The maximal element for \(\Delta^{+}\) in this order is the distribution function \(\varepsilon_{0}\) (see [7–9]).

Definition 2.1

([8])

A mapping \(T:[0,1] \times[0,1]\to[0,1]\) is called a continuous triangular norm (briefly, a continuous t-norm) if T satisfies the following conditions:

  1. (a)

    T is commutative and associative;

  2. (b)

    T is continuous;

  3. (c)

    \(T(a,1)=a\) for all \(a\in[0,1]\);

  4. (d)

    \(T(a,b)\leq T(c,d)\) whenever \(a\leq c\) and \(b\leq d\) for all \(a,b,c,d\in[0,1]\).

Typical examples of continuous t-norms are \(T_{P}(a,b)=ab\), \(T_{M} (a,b)=\min(a,b)\), and \(T_{L}(a,b)=\max(a+b-1,0)\) (the Lukasiewicz t-norm).

Definition 2.2

([9])

A random normed space (briefly, RN-space) is a triple \((X,\mu,T)\), where X is a vector space, T is a continuous t-norm and μ is a mapping from X into \(D^{+}\) such that the following conditions hold:

  1. (RN1)

    \(\mu_{x}(t)=\varepsilon_{0}(t)\) for all \(t>0\) if and only if \(x=0\);

  2. (RN2)

    \(\mu_{\alpha x}(t)=\mu_{x}(\frac{t}{|\alpha|})\) for all \(x\in X\), \(\alpha\neq0\);

  3. (RN3)

    \(\mu_{x+y}(t+s)\geq T(\mu_{x}(t),\mu_{y}(s))\) for all \(x,y\in X\) and \(t,s \geq0\).

Every normed space \((X,\|\cdot\|)\) defines a random normed space \((X,\mu,T_{M})\), where

$$\mu_{x}(t)=\frac{t}{t+\|x\|} $$

for all \(t>0\), and \(T_{M}\) is the minimum t-norm. This space is called the induced random normed space.

Definition 2.3

([10])

A random normed algebra \((X,\mu,T,T')\) is a random normed space \((X,\mu,T)\) with algebraic structure such that

  1. (RN4)

    \(\mu_{xy}(ts)\geq T'(\mu_{x}(t), \mu_{y}(s))\) for all \(x,y\in X\) and \(t,s> 0\), in which \(T'\) is a continuous t-norm.

Every normed algebra \((X,\|\cdot\|)\) defines a random normed algebra \((X,\mu,T_{M},T_{P})\), where

$$\mu_{x}(t)=\frac{t}{t+\|x\|} $$

for all \(t>0\) if and only if

$$\|xy\|\le\|x\|\|y\| + s\|y\| + t\|x\| $$

for all \(x, y \in X\) and \(t,s > 0\). This space is called the induced random normed algebra. For more properties and examples of the theory of random normed spaces, we refer to [11–27].

Definition 2.4

Let \(({\mathcal{U}} ,\mu,T,T')\) be a random Banach algebra. Then an involution on \(\mathcal{U}\) is a mapping \(u\to u^{*}\) from \(\mathcal{U}\) into \(\mathcal{U}\) which satisfies the following conditions:

  1. (1)

    \(u^{**}=u\) for \(u\in\mathcal{U}\);

  2. (2)

    \((\alpha u+ \beta v)^{*}=\overline{\alpha} u^{*} + \overline{\beta} v^{*}\);

  3. (3)

    \((uv)^{*}=v^{*}u^{*}\) for \(u,v\in\mathcal{U}\).

If, in addition, \(\mu_{u^{*}u}(ts)=T'(\mu_{u}(t),\mu_{u}(s))\) for all \(u\in\mathcal{U}\) and \(t,s>0\), then \(\mathcal{U}\) is a random \(C^{*}\)-algebra.

Following the terminology of [28], a non-empty set G with a ternary operation \([\cdot, \cdot, \cdot] : G \times G \times G \rightarrow G\) is called a ternary groupoid and is denoted by \((G, [\cdot, \cdot, \cdot])\). The ternary groupoid \((G, [\cdot, \cdot, \cdot])\) is called commutative if \([x_{1}, x_{2}, x_{3}] = [x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}]\) for all \(x_{1}, x_{2}, x_{3} \in G\) and all permutations σ of \(\{1, 2, 3 \}\).

If a binary operation ∘ is defined on G such that \([x, y, z] = (x\circ y) \circ z\) for all \(x, y, z\in G\), then we say that \([\cdot, \cdot, \cdot]\) is derived from ∘. We say that \((G, [\cdot, \cdot, \cdot])\) is a ternary semigroup if the operation \([\cdot, \cdot, \cdot]\) is associative, i.e., if

$$\bigl[[x, y, z], u, v\bigr] = \bigl[x, [y, z, u], v\bigr] = \bigl[x, y, [z, u, v]\bigr] $$

for all \(x, y, z, u, v \in G\) (see [29]).

A random \(C^{*}\)-ternary algebra is a random complex Banach space A, equipped with a ternary product \((x, y, z) \mapsto[x, y, z]\) of \(A^{3}\) into A, which are C-linear in the outer variables, conjugate C-linear in the middle variable, associative in the sense that

$$\bigl[x, y, [z, w, v]\bigr] = \bigl[x, [w, z, y], v\bigr] = \bigl[[x, y, z], w,v \bigr], $$

and satisfies

$$\mu_{[x, y, z]}(tsp) \ge T\bigl(\mu_{x}(t),\mu_{y}(s), \mu_{z}(p)\bigr) $$

and

$$\mu_{[x, x, x]}\bigl(t^{3}\bigr) \ge T\bigl(\mu_{x}(t), \mu_{x}(t),\mu_{x}(t)\bigr) $$

(see [28, 30]).

Every random left Hilbert \(C^{*}\)-module is a random \(C^{*}\)-ternary algebra via the ternary product \([x, y, z] := \langle x, y \rangle z\).

If a random \(C^{*}\)-ternary algebra \((A, [\cdot, \cdot, \cdot] )\) has the 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 routine to verify that A, endowed with \(x\circ y : = [x, e, y]\) and \(x^{*}:=[e, x, e]\), is a unital \(C^{*}\)-algebra. Conversely, if \((A, \circ)\) is a unital \(C^{*}\)-algebra, then \([x, y, z] : = x \circ y^{*} \circ z\) makes A into a \(C^{*}\)-ternary algebra.

A C-linear mapping \(H : A \rightarrow B\) is called a \(C^{*}\)-ternary algebra homomorphism if

$$H\bigl([x, y, z]\bigr) = \bigl[H(x), H(y), H(z)\bigr] $$

for all \(x, y, z \in A\). If, in addition, the mapping H is bijective, then the mapping \(H : A \rightarrow B\) is called a \(C^{*}\)-ternary algebra isomorphism. A C-linear mapping \(\delta: A \rightarrow A\) is called a \(C^{*}\)-ternary algebra derivation if

$$\delta\bigl([x, y, z]\bigr) = \bigl[\delta(x), y, z\bigr] + \bigl[x, \delta(y), z\bigr] + \bigl[x, y, \delta(z)\bigr] $$

for all \(x, y, z \in A\) (see [28, 31]).

There are some applications, although still hypothetical, in the fractional quantum Hall effect, the non-standard statistics, supersymmetric theory, and the Yang-Baxter equation (cf. [5, 32, 33]).

Throughout this paper, assume that p, d are non-negative integers with \(p+d \ge3\) and A, B are random \(C^{*}\)-ternary algebras.

Definition 2.5

Let \((X,\mu,T)\) be an RN-space.

  1. (1)

    A sequence \(\{x_{n}\}\) in X is said to be convergent to x in X if, for any \(\epsilon>0\) and \(\lambda>0\), there exists a positive integer N such that \(\mu_{x_{n}-x}(\epsilon)>1-\lambda\) whenever \(n\geq N\).

  2. (2)

    A sequence \(\{x_{n}\}\) in X is called a Cauchy sequence if, for any \(\epsilon>0\) and \(\lambda>0\), there exists a positive integer N such that

    $$\mu_{x_{m}-x_{n}}(\epsilon)>1-\lambda $$

    whenever \(n \geq m \geq N\).

  3. (3)

    An RN-space \((X,\mu,T)\) is said to be complete if every Cauchy sequence in X is convergent to a point in X.

3 Approximation of random \(C^{*}\)-ternary algebras homomorphisms

In this section, we approximate random \(C^{*}\)-ternary algebras homomorphisms of a Cauchy-Jensen additive mapping (see also [34–45]).

For a given mapping \(f: A \to B\), we define

$$\begin{aligned}& C_{\mu}f(x_{1},\ldots,x_{p},y_{1}, \ldots,y_{d}) \\& \quad := 2f \Biggl(\frac{\sum_{j=1}^{p}\mu x_{j}}{2}+\sum_{j=1}^{d} \mu y_{j} \Biggr)-\sum_{j=1}^{p} \mu f(x_{j})-2\sum_{j=1}^{d}\mu f(y_{j}) \end{aligned}$$

for all \(\mu\in{{\mathbf{T}}}^{1}:=\{ \lambda\in\mathbf{C}: |\lambda|=1 \}\) and \(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}\in A\).

One can easily show that a mapping \(f:A \rightarrow B\) satisfies

$$C_{\mu}f(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d}) =0 $$

for all \(\mu\in{\mathbf{T}}^{1}\) and \(x_{1},\ldots,x_{p},y_{1},\ldots ,y_{d}\in A\) if and only if

$$f(\mu x+\lambda y)=\mu f(x)+\lambda f(y) $$

for all \(\mu, \lambda\in {\mathbf{T}}^{1}\) and \(x, y \in A\).

We use the following lemma in this paper.

Lemma 3.1

([46])

Let \(f : A \rightarrow B\) be an additive mapping such that \(f(\mu x) = \mu f(x)\) for all \(x\in A\) and \(\mu\in{\mathbf{T}}^{1}\). Then the mapping f is C-linear.

Theorem 3.2

Let r, s, and θ be non-negative real numbers such that \(0< r\neq1\), \(0< s\neq3\). Let \(\varphi: A^{p+d}\to D^{+}\) (\(d\geq2\)) and \(\psi:A^{3}\to D^{+}\) such that

$$ \varphi_{a(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d})}(t)=\varphi_{ x_{1},\ldots ,x_{p},y_{1},\ldots,y_{d} } \biggl( \frac{t}{a^{r}} \biggr) $$
(1)

and

$$ \psi_{a(x,y,z)}(t)=\psi_{ x,y,z } \biggl( \frac{t}{a^{s}} \biggr) $$
(2)

for all \(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d},x,y,z\in A\) and \(a\in\mathbf{C}\). Suppose that \(f : A \rightarrow B\) is a mapping with \(f(0)=0\), satisfying

$$ \mu_{C_{\mu}f(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d})}(t) \ge\varphi_{x_{1},\ldots,x_{p},y_{1},\ldots,y_{d}}(t) $$
(3)

and

$$ \mu_{f([x,y,z])-[f(x),f(y),f(z)]}(t)\ge\psi_{x,y,z}(t) $$
(4)

for all \(\mu\in\mathbf{T}^{1}\), \(x_{1},\ldots,x_{p},y_{1},\ldots,y_{d},x,y,z \in A\), and \(t>0\). Then there exists a unique \(C^{*}\)-ternary algebra homomorphism \(H : A \rightarrow B\) such that

(5)

for all \(x \in A\) and \(t>0\).

Proof

We prove the theorem when \(0< r<1\) and \(0< s<3\). Similarly, one can prove the theorem for other cases. Letting \(\mu=1\), \(x_{1}=\cdots=x_{p}=0\), and \(y_{1}=\cdots=y_{d}=x\) in (3), we get

(6)

for all \(x\in A\) and \(t>0\). If we replace x by \(d^{n} x\) in (6), we get

for all \(x\in A\), all non-negative integers n and \(t>0\). Therefore,

(7)

for all \(x\in A\), non-negative integers \(n,m\) and \(t>0\). From this, it follows that the sequence \(\{\frac{1}{d^{n}} f(d^{n} x)\}\) is a Cauchy sequence for all \(x \in A\). Since B is complete, the sequence \(\{\frac{1}{d^{n}} f(d^{n} x)\}\) converges. Thus one can define the mapping \(H : A \rightarrow B\) by

$$H(x) : = \lim_{n\to\infty} \frac{1}{d^{n}} f\bigl(d^{n} x \bigr) $$

for all \(x \in A\). Moreover, letting \(m =0\) and passing to the limit \(n \to\infty\) in (7), we get (5). It follows from (3) that

$$\begin{aligned}& \mu_{2H (\frac{\sum_{j=1}^{p} \mu x_{j}}{2}+\sum_{j=1}^{d} \mu y_{j} ) - \sum_{j=1}^{p} \mu H(x_{j})-2 \sum_{j=1}^{d} \mu H(y_{j}) }(t) \\& \quad = \lim_{n\to\infty} \mu_{ \frac{1}{d^{n}} ( 2 f (d^{n}\frac{\sum_{j=1}^{p} \mu x_{j}}{2}+d^{n}\sum_{j=1}^{d} \mu y_{j} ) - \sum_{j=1}^{p} \mu f(d^{n} x_{j})-2 \sum_{j=1}^{d} \mu f(d^{n}y_{j}) )}(t) \\& \quad \ge \lim_{n\to\infty} \varphi_{d^{n}(x_{1},\ldots,x_{p},y_{1},\ldots ,y_{d})} \bigl( {d^{n}} t \bigr) \\& \quad \ge \lim_{n\to\infty} \varphi_{ x_{1},\ldots,x_{p},y_{1},\ldots,y_{d} } \biggl( \frac{d^{n}}{d^{nr}}t \biggr) \\& \quad =1 \end{aligned}$$

for all \(\mu\in\mathbf{T}^{1}\), \(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d} \in A \), and \(t>0\). Hence we have

$$2H \Biggl(\frac{\sum_{j=1}^{p} \mu x_{j}}{2}+\sum_{j=1}^{d} \mu y_{j} \Biggr) = \sum_{j=1}^{p} \mu H(x_{j})+2 \sum_{j=1}^{d} \mu H(y_{j}) $$

for all \(\mu\in\mathbf{T}^{1}\) and \(x_{1}, \ldots, x_{p}, y_{1}, \ldots, y_{d} \in A\) and so

$$H(\lambda x+\mu y)=\lambda H(x)+\mu H(y) $$

for all \(\lambda, \mu\in\mathbf{T}^{1}\) and \(x, y \in A\). Therefore, by Lemma 3.1, the mapping \(H : A \rightarrow B\) is C-linear. It follows from (4) that

$$\begin{aligned}& \mu_{H([x, y, z])- [H(x), H(y), H(z)]}(t) \\& \quad = \lim_{n\to\infty}\mu_{\frac{1}{d^{3n}} (f ([d^{n} x, d^{n} y, d^{n} z] ) - [f(d^{n} x), f(d^{n} y), f(d^{n} z) ] )}(t) \\& \quad = \lim_{n\to \infty}\mu_{ (f ([d^{n} x, d^{n} y, d^{n} z] ) - [f(d^{n} x), f(d^{n} y), f(d^{n} z) ] )}\bigl({d^{3n}}t \bigr) \\& \quad \ge \lim_{n\to\infty}\psi_{d^{n} x, d^{n} y, d^{n} z}\bigl({d^{3n}}t \bigr) \\& \quad \ge \lim_{n\to\infty}\psi_{x,y,z} \biggl( \frac{d^{3n}}{d^{ns}} \biggr)=1 \end{aligned}$$

for all \(x, y, z \in A\) and \(t>0\) and so

$$H\bigl([x, y, z]\bigr) = \bigl[H(x), H(y), H(z)\bigr] $$

for all \(x, y, z \in A\).

Now, let \(T : A \rightarrow B\) be another generalized Cauchy-Jensen additive mapping satisfying (5). Then we have

for all \(x \in A\) and \(t>0\). So we can conclude that \(H(x)=T(x)\) for all \(x \in A\). This proves the uniqueness of H. Thus the mapping \(H : A\rightarrow B\) is a unique \(C^{*}\)-ternary algebra homomorphism satisfying (5). This completes the proof. □

Theorem 3.3

Let \(r<1\), \(s<2\), θ be non-negative real numbers and let \(f : A \rightarrow B\) be a mapping satisfying (1), (2), (3) and (4). If there exist a real number \(\lambda>1\) (\(0<\lambda<1\)) and an element \(x_{0}\in A\) such that

$$\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f\bigl(\lambda^{n} x_{0}\bigr) = e'\qquad \biggl(\lim_{n\rightarrow\infty} \lambda^{n} f\biggl(\frac{x_{0}}{\lambda^{n}}\biggr) = e' \biggr), $$

then the mapping \(f : A \rightarrow B\) is a \(C^{*}\)-ternary algebra homomorphism.

Proof

By using the proof of Theorem 3.2, there exists a unique \(C^{*}\)-ternary algebra homomorphism \(H : A \rightarrow B\) satisfying (5). Now,

$$ H(x)=\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f\bigl( \lambda^{n} x\bigr)\qquad \biggl(H(x)=\lim_{n\rightarrow\infty} \lambda^{n} f \biggl(\frac{x}{\lambda^{n}} \biggr) \biggr) $$
(8)

for all \(x\in A\) and all real numbers \(\lambda>1\) (\(0<\lambda<1\)). Therefore, by the assumption, we get that \(H(x_{0})=e'\). Let \(\lambda>1\) and \(\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f(\lambda^{n} x_{0}) = e'\). It follows from (4) and (8) that

$$\begin{aligned}& \mu_{[H(x),H(y),H(z)]-[H(x), H(y), f(z)]}(t) \\& \quad = \mu_{H[x,y,z]-[H(x), H(y), f(z)]}(t) \\& \quad = \lim_{n\rightarrow\infty}\mu_{ \frac{1}{\lambda^{2n}}(f([\lambda^{n} x,\lambda^{n} y, z]) - [f(\lambda^{n} x),f(\lambda^{n} y),f(z) ])}(t) \\& \quad = \lim_{n\rightarrow\infty}\mu_{ f([\lambda^{n} x,\lambda^{n} y, z]) - [f(\lambda^{n} x),f(\lambda^{n} y),f(z) ]}\bigl( \lambda^{2n}t\bigr) \\& \quad \ge \lim_{n\rightarrow\infty} \psi_{\lambda^{x}, \lambda^{y}, \lambda^{z}}\bigl( \lambda^{2n}t\bigr) \\& \quad = \psi_{x,y,z} \biggl(\frac{\lambda^{2n}}{\lambda^{2ns}}t \biggr) \\& \quad = 1 \end{aligned}$$

for all \(x\in A\) and \(t>0\) and so

$$\bigl[H(x),H(y),H(z)\bigr]=\bigl[H(x), H(y), f(z)\bigr] $$

for all \(x,y,z\in A\). Letting \(x=y=x_{0}\) in the last equality, we get \(f(z)=H(z)\) for all \(z\in A\).

Similarly, one can show that \(H(x)=f(x)\) for all \(x\in A\) when \(0<\lambda<1\) and \(\lim_{n\rightarrow\infty} \lambda^{n} f(\frac{x_{0}}{\lambda^{n}})=e'\). Therefore, the mapping \(f : A \rightarrow B\) is a \(C^{*}\)-ternary algebra homomorphism. This completes the proof. □

Theorem 3.4

Let \(r>1\), \(s>3\), θ be non-negative real numbers and let \(f:A \rightarrow B\) be a mapping satisfying (3) and (4). If there exists a real number \(0<\lambda<1\) (\(\lambda>1\)) and an element \(x_{0}\in A\) such that

$$\lim_{n\rightarrow\infty} \frac{1}{\lambda^{n}} f\bigl(\lambda^{n} x_{0}\bigr) = e' \qquad \biggl(\lim_{n\rightarrow\infty} \lambda^{n} f\biggl(\frac{x_{0}}{\lambda^{n}}\biggr) = e' \biggr), $$

then the mapping \(f : A \rightarrow B\) is a \(C^{*}\)-ternary algebra homomorphism.

Proof

The proof is similar to the proof of Theorem 3.3 and we omit it. □