Random $C^{*}$-ternary algebras and application

Abstract

In this paper, we introduce the concept of random $C^{*}$-ternary algebras and consider some properties of them. As an application we approximate a random $C^{*}$-ternary algebra homomorphism in these spaces.

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) 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) 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:

• (a) T is commutative and associative;

• (b) T is continuous;

• (c) $T(a,1)=a$ for all $a\in[0,1]$;

• (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:

• (RN1) $\mu_{x}(t)=\varepsilon_{0}(t)$ for all $t>0$ if and only if $x=0$;

• (RN2) $\mu_{\alpha x}(t)=\mu_{x}(\frac{t}{|\alpha|})$ for all $x\in X$, $\alpha\neq0$;

• (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

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

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

• (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) 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) 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) 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. □