Approximation of derivations and the superstability in random Banach ∗-algebras

Abstract

We prove that approximations of derivations on random Banach ∗-algebras are exactly derivations by using a fixed point method. Furthermore, we show that approximations of quadratic ∗-derivations on random Banach ∗-algebras are exactly quadratic ∗-derivations. We, moreover, prove that approximations of derivations on random $C^{*}$-ternary algebras are exactly derivations by using a fixed point method.

1 Introduction

Ulam [1] presented an effective lecture at the University of Wisconsin in which he stated a number of essential unsolved problems, in the fall of 1940. The next question concerning the stability of homomorphisms was among those:

Assume that $\varOmega_{1}$ is a group and suppose that $\varOmega_{2}$ is a metric group with a metric $\Delta (\cdot,\cdot)$. Let $\xi > 0$, is there $\eta > 0$ such that if a function $\varphi : \varOmega_{1}\to \varOmega _{2}$ satisfies the inequality $\Delta (\varphi (uv), \varphi (u) \varphi (v)) <\eta$ for all $u,v \in \varOmega_{1}$ then there is a homomorphism $\varPhi : \varOmega_{1}\to \varOmega_{2}$ with $\Delta (\varphi (u),\varPhi (u)) <\xi$ for all $u \in \varOmega_{1}$?

When the answer is established, the functional equation for homomorphisms is stable.

The first mathematician who presented the result concerning the stability of functional equations was Hyers [2]. He intelligently answered Ulam’s question when $\varOmega_{1}$ and $\varOmega_{2}$ are Banach spaces. Recently, Rassias [3] and others have obtained important results on stability and applied them to the investigations in the nonlinear sciences.

2 Preliminaries

Assume that $\Delta^{+}$ is the family of distribution functions, i.e., the family of all left-continuous functions $G:[-\infty ,\infty ] \to [0,1]$ such that G is increasing on $[-\infty ,\infty ]$, $G(0)=0$ and $G(+\infty )=1$. $D^{+}\subseteq \Delta^{+}$ contains each function $G \in \Delta^{+}$ for which $\ell^{-}G(+\infty )=1$ and $\ell^{-}g(x)$ is the left limit of the map g at x, i.e., $\ell^{-}g(x)=\lim_{t\to x^{-}}g(t)$. In $\Delta^{+}$, we have $H \leq F$ if and only if $H(s) \leq F(s)$ for all s in $\mathbb{R}$ (partially ordered). Note that the function $\varepsilon_{u}$ defined by

$$ \varepsilon_{u}(s)= \textstyle\begin{cases} 0, & \text{if } s\leq u, \\ 1, & \text{if } s>u , \end{cases} $$

is an element of $\Delta^{+}$ and $\varepsilon_{0}$ is the maximal element in this space. For more details see [4,5,6].

Definition 2.1

([6])

Let $I=[0,1]$. A continuous triangular norm (briefly, ct-norm) is a function T from I to I with continuity property such that:

• (a) $T(\theta ,\vartheta )=T(\vartheta ,\theta )$ and $T(\theta ,T( \vartheta ,\iota ))=T(T(\theta ,\vartheta ),\iota )$ for all $\theta ,\vartheta ,\iota \in I$;

• (b) $T(\theta ,1)=\theta$ for $0\leq \theta \leq 1$;

• (c) $T(\theta ,\vartheta )\leq T(\iota ,\kappa )$ whenever $\theta \leq \iota$ and $\vartheta \leq \kappa$ for each $\theta ,\vartheta ,\iota ,\kappa \in I$.

$T_{P}(\theta ,\vartheta )=\theta \vartheta$, $T_{M}(\theta ,\vartheta )=\min (\theta ,\vartheta )$ and $T_{L}(\theta ,\vartheta )=\max ( \theta +\vartheta -1,0)$ (the Lukasiewicz t-norm) are some examples of t-norms. Also, we define $\prod^{n}_{j=1}\theta_{j}=T^{n-1}( \theta_{1},\ldots,\theta_{n})$.

Definition 2.2

([6])

Suppose that T is a ct-norm, V is a vector space and let μ be a map from V to $D^{+}$. In this case, the ordered triple $(V,\mu ,T)$ with the properties

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

• (RN2) $\mu_{\alpha v}(\theta )=\mu_{v}(\frac{\theta }{ \vert \alpha \vert })$ for all $v\in V$, $\alpha \neq 0$;

• (RN3) $\mu_{u+v}(\theta +\vartheta )\geq T(\mu_{u}(\theta ),\mu _{v}(\vartheta ))$ for all $u,v\in V$ and all $\theta ,\vartheta \geq 0$,

is said to be a random normed space (in short, RN-space).

Let $(V, \Vert \cdot \Vert )$ be a linear normed space. Then

$$ \mu_{v}(\vartheta )=\frac{\vartheta }{\vartheta + \Vert v \Vert } $$

for all $\vartheta >0$, defines a random norm, and the ordered triple $(V,\mu ,T_{M})$ is an RN-space.

Definition 2.3

Assume that the following algebraic structure on an RN-space $(V,\mu ,T)$ holds:

• (RN-4) $\mu_{uv}(\theta \vartheta )\geq T'(\mu_{u}(\theta ), \mu _{v}(\vartheta ))$ for each $u,v\in V$ and all $\theta ,\vartheta >0$, where $T'$ is a ct-norm.

Then $(V,\mu ,T,T')$ is called a random normed algebra.

Suppose that $(V, \Vert \cdot \Vert )$ is a normed algebra. Then $(V,\mu ,T _{M},T_{P})$ is a random normed algebra, where

$$ \mu_{v}(\vartheta )=\frac{\vartheta }{\vartheta + \Vert v \Vert } $$

for all $\vartheta >0$ if and only if

$$ \Vert uv \Vert \le \Vert v \Vert \Vert u \Vert +\theta \Vert u \Vert +\vartheta \Vert v \Vert \quad (v,u \in V; \theta ,\vartheta >0). $$

For more details, see [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].

Definition 2.4

A random Banach ∗-algebra $\mathcal{B}$ is a random complex Banach algebra $({\mathcal{B}},\mu ,T,T')$, together with an involution on $\mathcal{B}$ which is a mapping $g\mapsto g^{*}$ from $\mathcal{B}$ into $\mathcal{B}$ that satisfies

• (i) $g^{**}=g$ for $g\in \mathcal{B}$;

• (ii) $(a g+b h)^{*}=\overline{a} g^{*}+\overline{b} h^{*}$;

• (iii) $(gh)^{*}=h^{*}g^{*}$ for $g,h\in \mathcal{B}$.

If, in addition, $\mu_{g^{*}g}(\theta \vartheta )=T'(\mu_{g}(\theta ), \mu_{g}(\vartheta ))$ for $g\in \mathcal{B}$ and $\theta ,\vartheta >0$, then $\mathcal{B}$ is called a random $C^{*}$-algebra.

Assume that $\mathcal{B}$ is a random Banach ∗-algebra. A derivation on $\mathcal{B}$ is a mapping δ from $\mathcal{B}$ to $\mathcal{B}$ such that:

$$\begin{aligned}& \delta (\lambda g+h)=\lambda \delta (g)+\delta (h), \end{aligned}$$

(2.1)

$$\begin{aligned}& \delta (gh)=\delta (g)h+g\delta (h) \end{aligned}$$

(2.2)

for all $g,h\in \mathcal{B}$ and all $\lambda \in \mathbb{C}$. A derivation δ is called a ∗-derivation on $\mathcal{B}$ if $\delta (g^{*})=\delta (g)^{*}$ for all $g \in \mathcal{B}$ (see [23]).

Recall that

$$\begin{aligned}& \omega (u+v)=\omega (u)+\omega (v), \end{aligned}$$

(2.3)

$$\begin{aligned}& \omega (u+v)+\omega (u-v)=2\omega (u)+2\omega (v) , \end{aligned}$$

(2.4)

respectively, are Cauchy additive and Cauchy quadratic functional equations.

Firstly, Baker, Lawrence and Zorzitto [24] defined the concept of superstability. Let $(\mathcal{B},\mu ,T,T')$ be an RN algebra. The random norm is multiplicative if $\mu_{uv}(\theta \vartheta )=T'(\mu _{u}(\theta ), \mu_{v}(\vartheta ))$ for all $u,v\in \mathcal{B}$ and all $\theta ,\vartheta >0$.

Suppose that $\varGamma \neq \emptyset$. A function $\Delta : \varGamma \times \varGamma \rightarrow [0, \infty ]$ is a generalized metric (GM) on Γ if

• (1) $\Delta (\rho ,\varrho )=0$ if and only if $\rho =\varrho$;

• (2) $\Delta (\rho ,\varrho )=\Delta (\varrho ,\rho )$ for all $\rho , \varrho \in \varGamma$;

• (3) $\Delta (\rho ,\varrho )\leq \Delta (\rho ,\sigma )+\Delta (\sigma , \varrho )$ for all $\rho ,\varrho ,\sigma \in \varGamma$.

Theorem 2.1

([25, 26])

Suppose that $(\varGamma , \Delta )$ is a complete GM space and assume that the selfmapping ϒ on Γ with Lipschitz constant $0< L<1$ is strictly contractive. Then, for $\varrho \in \varGamma$, either

$$ \Delta \bigl(\varUpsilon^{n}\varrho ,\varUpsilon^{n+1}\varrho \bigr)=\infty $$

for each $0\leq n\in \mathcal{Z}$, or there exists $n_{0}\in \mathbb{N}$ such that

• (1) $\Delta (\varUpsilon^{n}\varrho , \varUpsilon^{n+1}\varrho )<\infty$, $\forall n \geq n_{0}$;

• (2) the sequence $\{\varUpsilon^{n} \varrho \}$ tends to $\sigma^{*}$ in Γ;

• (3) $\varUpsilon (\sigma^{*})=\sigma^{*}$;

• (4) $\varUpsilon (\sigma^{*})=\sigma^{*}$ and is unique in $\mathbb{E}=\{ \sigma \in \varGamma| \Delta (\varUpsilon^{n_{0}} \varrho , \sigma )< \infty \}$

• (5) $(1-L)\Delta (\sigma , \sigma^{*}) \leq \Delta (\sigma ,\varUpsilon \sigma )$ for all $\sigma \in \varGamma$.

3 Approximation of derivations on random Banach ∗-algebras

Assume that a random ∗-Banach algebra $\mathcal{B}$ has unit e. Our results improve and expand the result presented by Jang [27].

Theorem 3.1

Let $\psi_{1}: \mathcal{B} \times \mathcal{B} \rightarrow D^{+}$ and $\psi_{2}: \mathcal{B} \rightarrow D^{+}$ be distribution functions. Assume that $f: \mathcal{B} \rightarrow \mathcal{B}$ is a mapping such that

$$\begin{aligned}& \mu_{ f( \xi p+q)-\xi f(p)-f(q)}(t) \geq \psi_{1} (p,q,t), \end{aligned}$$

(3.1)

$$\begin{aligned}& \mu_{ f(pq)-pf(q)-f(p)q}(t) \geq \psi_{1} (p, q,t), \end{aligned}$$

(3.2)

$$\begin{aligned}& \mu_{ f(p^{*} )-f(p)^{*}}(t) \geq \psi_{2} (p,t), \end{aligned}$$

(3.3)

for all $\xi \in \mathbb{T}$, $p,q\in \mathcal{B}$ and $t>0$. If there exist $n\in \mathbb{N}$ and $0< L<1$ such that $\psi_{1} (sp,sq,Lst)> \psi_{1}(p,q,t)$, $\psi_{1} (sp,q,Lst)>\psi_{1}(p,q,t)$, $\psi_{1} (p,sq,Lst)> \psi_{1}(p,q,t)$ and $\psi_{2} (sp,Lst)>\psi_{2}(p,t)$ for all $p,q \in \mathcal{B}$ and $t>0$. Then f on $\mathcal{B}$ is a ∗-derivation.

Proof

Putting $p=q$ and $\xi =1$ in (3.1), we get

$$ \mu_{ f(2p)-2f(p)}(t) \geq \psi_{1} (p,p,t) $$

(3.4)

for all $p \in \mathcal{B}$ and $t>0$. By induction, we can prove that

$$ \mu_{ f(np)-nf(p)}(t) \geq \prod^{n-1}_{j=1} \psi_{1} (jp,p,t_{j}) $$

(3.5)

for all $p,q \in \mathcal{B}$, $t>0$ and $n\geq 2$ where $\sum^{n-1} _{j=1} t_{j}=t$.

Define

$$ \varPsi (p,t)=\prod^{s-1}_{j=1} \psi_{1} (jp,p,t_{j}) $$

for $p \in \mathcal{B}$, $t>0$ and $s\geq 2$ where $\sum^{s-1}_{j=1} t _{j}=t$. So

$$ \mu_{ f(sp)-sf(p)}(t) \geq \varPsi (p,t). $$

(3.6)

Put $\varGamma =\{g; g:\mathcal{B} \rightarrow \mathcal{B}\}$. Define a function $\Delta : \varGamma \times \varGamma \to [0, \infty ]$ such that

$$ \Delta (\vartheta , \upsilon )=\inf \bigl\{ \nu >0: \mu_{\vartheta (p)-\upsilon (p)}(\nu t) \geq \varPsi (p,t), \forall p \in \mathcal{B}, t>0\bigr\} , $$

where $\vartheta , \upsilon \in \varGamma$. Miheţ and Radu [28] proved that $(\varGamma , \Delta )$ is a complete GM space. Define a mapping $H: \varGamma \rightarrow \varGamma$ by $H(\vartheta )(p)=s ^{-1} \upsilon (sp)$. Put

$$ \Delta (\vartheta ,\upsilon )=\nu , $$

where $\vartheta ,\upsilon \in \varGamma$. Then

$$ \mu_{ H(\vartheta )(p)-H(\upsilon )(p)}(t)= \mu_{ \vartheta (sp)-\upsilon (sp)}(st) \geq \varPsi \biggl( sp, \frac{s}{ \alpha }t \biggr) \geq \varPsi \biggl( p,\frac{t}{L\alpha } \biggr) . $$

So, for $\vartheta ,\upsilon \in S$, we have

$$ \Delta \bigl(H(\vartheta ), H(\upsilon )\bigr)\leq L\Delta ( \vartheta ,\upsilon ). $$

(3.7)

Then the mapping H on Γ with Lipschitz constant L is strictly contractive. From (3.6), we have

$$ \mu_{ (Hf)(p)-f(p)}(t)=\mu_{f(sp)-f(p)}(st)=\mu_{ f(sp)-sf(p)}(st) \geq \varPsi (p,st), $$

which implies that $\Delta (H(f), f)\leq 1/ \vert s \vert$. Theorem 2.1 implies that, in the set

$$ U=\bigl\{ \vartheta \in \varGamma : \Delta \bigl(\vartheta , H(f)\bigr)< \infty \bigr\} , $$

$h: \mathcal{B} \rightarrow \mathcal{B}$ is a unique fixed point of H. Also for every $p \in \mathcal{A}$

$$ h(p)=\lim_{m \rightarrow \infty } H^{m} \bigl(f(p) \bigr)= \lim_{m \rightarrow \infty } s^{-m} f\bigl(s^{m}p \bigr). $$

(3.8)

Using (3.6), we get

$$\begin{aligned} \mu_{ h(\xi p+q)-\xi h(p)-h(q)}(t) =& \lim_{n \rightarrow \infty } \mu_{ f(s^{n} (\xi p+q))-\xi f(s^{n}p)-f(s^{n} q)} \bigl(s^{n}t\bigr) \\ \geq & \lim_{n \rightarrow \infty } \psi_{1} \bigl(s^{n} p, s^{n} q,s^{n}t\bigr) \\ \geq & \lim_{n \rightarrow \infty } \psi_{1} \biggl( p,q, \frac{t}{L^{n}} \biggr) =1 \end{aligned}$$

for all $p,q \in \mathcal{B}$, $\xi \in T$ and $t>0$. Let $\xi =\xi _{1}+i \xi_{2} \in \mathbb{C}$, $\xi_{1},\xi_{2} \in \mathbb{R}$ and let $\mu_{1}=\xi_{1}-[\xi_{1}]$ and $\mu_{2}=\xi_{2}-[\xi_{2}]$ where $[\xi ]$ denotes the integer part of ξ. So $0\leq \mu_{i}<1$ ($1 \leq i \leq 2$). Now, we represent $\mu_{i}$ as $\mu_{i}=\frac{\xi _{i,1}+\xi_{i,2}}{2}$ such that $\xi_{ i,j} \in \mathbb{T}$ ($1\leq i$, $j\leq 2$). Since $h(\xi p+q)=\lambda h(p)+h(q)$ for $\xi \in T$, we conclude that

$$\begin{aligned} h(\xi p) =& h(\xi_{1} p)+ih(\xi_{2} p) \\ =& \bigl([\xi_{1}]h(p)+\delta (\mu_{1} p)\bigr)+i\bigl([ \xi_{2}]h(p)+h(\mu_{2} p)\bigr) \\ =& \biggl([\xi_{1}]h(p)+\frac{1}{2} h(\xi_{1,1} p+ \xi_{1,2}p)\biggr)+i\biggl([\xi_{2}]h(p)+ \frac{1}{2} h( \xi_{2,1} p+\xi_{2,2}p)\biggr) \\ =& \biggl([\xi_{1}]h(p)+\frac{1}{2} \xi_{1,1}h(p)+ \frac{1}{2} \xi_{1,2}h( p)\biggr)+i\biggl([ \xi_{2}]h(p)+ \frac{1}{2} \xi_{2,1} h(p)+\frac{1}{2} \xi_{2,2}h(p) \biggr) \\ =& \xi_{1} h(p)+i\xi_{2} h(p) \\ =& h(p) \end{aligned}$$

for all $p \in \mathcal{B}$ and $\xi \in \mathbb{C}$. So, on $\mathcal{B}$, h is a $\mathbb{C}$-linear mapping. For the involution of h, we have

$$\begin{aligned} \mu_{ h(p^{*} )-h(p)^{*} }(t) =& \lim_{n\rightarrow \infty } \mu_{ f(s^{n} p^{*} )-f(s^{n}p)^{*} } \bigl(s^{n}t\bigr) \\ \geq & \lim_{n\rightarrow \infty } \psi_{2} \bigl(s^{n} p,s^{n}t\bigr) \\ \geq & \lim_{n\rightarrow \infty } \psi_{2} \biggl( p, \frac{t}{L^{n}} \biggr) \\ =& 1. \end{aligned}$$

Now, we prove the derivation property of h. In (3.2), we replace p by $s^{n} p$, q by $s^{n} q$, divide by $s^{2n}$ and get

$$\begin{aligned} \mu_{\frac{f(s^{n} ps^{n} q)}{s^{2n}}-p\frac{f(s^{n} q)}{s^{n}}-\frac{f(s ^{n} p)}{s^{n}}p}(t) \geq \psi_{1} \bigl(s^{n} p, s^{n} q,s^{2n}t\bigr)\geq \psi _{1} \biggl( p,q,\frac{t}{L^{2n}} \biggr) . \end{aligned}$$

(3.9)

In (3.9), letting $n\rightarrow \infty$, we get

$$ h(pq)=ph(q)+h(p)q $$

(3.10)

for all $p,q \in \mathcal{B}$. So h is a ∗-derivation on $\mathcal{B}$. Now, in (3.2), replacing p by $s^{n} p$ and dividing by $s^{n}$, we get

$$ \mu_{\frac{f(s^{n} pq)}{s^{n}}-pf(q)-\frac{f(s^{n} p)}{s^{n}} q}(t) \geq \psi_{1} \bigl(s^{n} p, q,s^{n}t\bigr)\geq \psi_{1} \biggl( p,q,\frac{t}{L ^{n}} \biggr) $$

for all $p,q \in \mathcal{B}$, $n\in \mathbb{N}$ and $t>0$. Letting $n\rightarrow \infty$, we get

$$ h(pq)=pf(q)+h(p)q $$

(3.11)

for all $p,q \in \mathcal{B}$. Fix $m \in \mathbb{N}$. From

$$\begin{aligned} p f\bigl(s^{m} q\bigr) =& h\bigl(s^{m} pq \bigr)-h(p)s^{m} q \\ =& s^{m} p f(q) \end{aligned}$$

(3.12)

for all $p,q \in \mathcal{B}$, we have $pf(q)=p\frac{f(s^{m} q)}{s ^{m}}$ for all $p,q \in \mathcal{B}$ and $m \in \mathbb{N}$. Letting $m\rightarrow \infty$, we get $p f(q)=p h(q)$. Putting $p=e$, we get $h(q)=f(q)$ for all $q \in \mathcal{B}$. Hence f is a ∗-derivation on $\mathcal{B}$. □

4 Approximation of quadratic ∗-derivations on random Banach ∗-algebras

Definition 4.1

Assume that a mapping $\delta : \mathcal{B} \rightarrow \mathcal{B}$ satisfies

• (1) $\delta (\eta +\kappa )+\delta (\eta -\kappa )-2\delta (\eta )-2 \delta (\kappa )=0$;

• (2) δ is quadratic homogeneous, that is, $\delta (\lambda \eta )= \lambda^{2} \delta (\eta )$;

• (3) $\delta (\eta \kappa )=\delta (\eta )\kappa^{2}+\eta^{2} \delta ( \kappa )$;

• (4) $\delta (\eta^{*} )=\delta (\eta )^{*}$;

for all $\eta , \kappa \in \mathcal{B}$ and $\lambda \in \mathbb{C}$. Then it is called a ∗-quadratic derivation on $\mathcal{B}$.

Theorem 4.2

Assume that $\psi_{1}:\mathcal{B} \times \mathcal{B} \rightarrow D ^{+}$ and $\psi_{2}:\mathcal{B} \rightarrow D^{+}$ are distribution functions. Let $f:\mathcal{B} \rightarrow \mathcal{B}$ be a function such that

$$\begin{aligned}& \mu_{f(p+q)+f(p-q)-2f(p)-2f(q) }(t) \geq \psi_{1}(p,q,t), \end{aligned}$$

(4.1)

$$\begin{aligned}& \mu_{f(pq)-p^{2} f(q)-f(p)q^{2} }(t) \geq \psi_{1}(p,q,t), \end{aligned}$$

(4.2)

$$\begin{aligned}& \mu_{f(\xi p)-\lambda^{2} f(p) }(t) \geq \psi_{2}(p,t), \end{aligned}$$

(4.3)

$$\begin{aligned}& \mu_{f(p^{*} )-f(p)^{*} }(t) \geq \psi_{2}(p,t), \end{aligned}$$

(4.4)

for all $\xi \in \mathbb{C}$, $p,q\in \mathcal{B}$ and $t>0$. If there exist $s\in \mathbb{N}$ and $0< L<1$ such that $\psi_{1} (2^{s} p, 2^{s} q, 2^{2s}Lt)>\psi_{1} (p,q,t)$, $\psi_{1} (2^{s} p,q,2^{2s}Lt )>\psi _{1}(p,q,t)$, $\psi_{1}(p, 2^{s} q,2^{2s}Lt)>\psi_{1}(p,q,t)$ and $\psi_{2} (2^{s} p, 2^{2s}Lt)>\psi_{2}(p,t)$ for all $p,q \in \mathcal{B}$ and $t>0$. Then, on $\mathcal{B}$, f is a ∗-quadratic derivation.

Proof

Putting $p=q$ and $\xi =1$ in (4.1), we get

$$ \mu_{ f(2p)-4f(p) }(t) \geq \psi_{1} (p,p,t) $$

for all $p \in \mathcal{B}$ and $t>0$. Induction on n yields

$$ \mu_{ f(2^{n} p)-2^{2n}f(p)}(t)\geq \prod^{n-1}_{i=0} \psi_{1} \biggl( 2^{i} p, 2^{i} p, \frac{t_{i}}{2^{2(n-i)}} \biggr) $$

(4.5)

for all $p,q \in \mathcal{B}$, $n\geq 2$ and $t>0$ where $\sum^{n-1} _{i=0}t_{i}=t$. Define

$$ \varPsi (p,t)=\prod^{s-1}_{i=0} 2^{2(s-i)} \psi_{1} \biggl( 2^{i} p, 2^{i} p,\frac{t_{i}}{2^{2(n-i)}} \biggr) . $$

(4.6)

Then we have

$$ \mu_{ f(2^{s} p)-2^{2s}f(p)}(t) \geq \varPsi (p,t). $$

The set of all mappings $\zeta : \mathcal{B} \rightarrow \mathcal{B}$ is denoted by Γ. Define a function $\Delta : \varGamma \times \varGamma \rightarrow [0, \infty ]$ by

$$ \Delta (\zeta ,\eta )=\inf \biggl\{ \nu >0: \mu_{ \zeta (p)-\eta (p)}(t) \geq \varPsi \biggl( p,\frac{t}{\nu } \biggr) , \forall p\in \mathcal{B} \biggr\} . $$

Miheţ and Radu [28] proved that $(\varGamma , \Delta )$ is a complete GM space. Now, define a mapping $H: \varGamma \rightarrow \varGamma$ by $H(\zeta )(p)=2^{-2s} \zeta (2^{s} p)$. Putting

$$ \Delta (\zeta ,\eta )=\nu \quad (\zeta ,\eta \in \varGamma ), $$

we obtain

$$ \mu_{ H(\zeta )(p)-H(\eta )(p)}(t)=\mu_{ \zeta (2^{s}p)-\eta (2^{s} p)} \biggl( \frac{t}{2^{2s}} \biggr) \geq \varPsi \biggl( 2^{s} p, \frac{t}{ \nu 2^{2s}} \biggr) \geq \varPsi \biggl( p, \frac{t}{L\alpha } \biggr) . $$

Then, for $\zeta ,\eta \in S$, we have

$$ \Delta \bigl(H(\zeta ), H(\eta )\bigr)\leq L\Delta (\zeta ,\eta ), $$

(4.7)

which means that H on Γ, with Lipschitz constant L is a strictly contractive mapping. Also, for $p \in \mathcal{B}$, we have

$$ \mu_{ (Hf)(p)-f(p)}(t)=\mu_{ 2^{-2s} f(2^{s} p)-f(p)}(t)= \mu_{ f(2^{s}) 2^{2s}f(p)} \bigl(2^{2s}t\bigr) \geq \varPsi \bigl(p,2^{2s}t\bigr), $$

which implies that $\Delta (H(f), f)\leq 1/2^{2s}$. Using Theorem 2.1, we conclude that, in the set

$$ U=\bigl\{ \zeta \in \varGamma : \Delta \bigl(\zeta , H(f)\bigr)< \infty \bigr\} $$

(4.8)

and for each $p \in \mathcal{B}$, $h: \mathcal{B}\rightarrow \mathcal{B}$ is a unique fixed point of H and

$$ h(p)=\lim_{m\rightarrow \infty } H^{m} \bigl(f(p) \bigr)=\lim 2^{-2sm} f\bigl(2^{sm}p\bigr). $$

(4.9)

By (4.9), we have

$$\begin{aligned}& \mu_{ h(p+q)+h(p-q)-2h(p)-2h(q)}(t) \\& \quad =\lim_{n\rightarrow \infty } \mu_{ f(2^{sn}(p+q)+f(2^{sn}(p-q))-2f(2^{sn}p)-2f(2^{sn}q)}\bigl(2^{2sn}t \bigr) \\& \quad \geq \lim_{n\rightarrow \infty } \psi_{1} \bigl(2^{ns}p, 2^{ns}q,2^{2ns}t\bigr) \geq \lim _{n\rightarrow \infty } \psi_{1} \biggl( p,q,\frac{t}{L^{n}} \biggr) =1 \end{aligned}$$

for all $p,q \in \mathcal{B}$ and $t>0$. Then h is a quadratic mapping on $\mathcal{B}$. Also, we have

$$\begin{aligned} \mu_{ h(\xi p)-\lambda^{2} h(p)}(t) =& \lim_{n\rightarrow \infty } \mu_{ f(2^{ns}(\xi p)-\lambda^{2} f(2^{ns}p)} \bigl(2^{2ns}t\bigr) \\ \geq &\lim_{n\rightarrow \infty } \psi_{2}\bigl(2^{ns}p,2^{ 2ns}t \bigr) \\ \geq &\lim_{n\rightarrow \infty } \psi_{2} \biggl( p, \frac{t}{L^{n}} \biggr) \\ =&1, \end{aligned}$$

which implies that h is quadratic homogeneous.

Now, replacing p by $2^{ns}p$ in (4.2) and dividing by $2^{-2sn}$, we get

$$ \mu_{\frac{f(2^{ns}pq)}{2^{2ns}}-p^{2} f(q)- \frac{f(2^{ns}p)}{2^{2ns}}q^{2} }(t) \geq \psi_{1} \bigl(2^{ns}p,q, 2^{2ns}t\bigr) \geq \psi_{1} \biggl( p,q,\frac{t}{L^{n}} \biggr) $$

(4.10)

for all $p,q \in \mathcal{B}$, $n\in \mathbb{N}$ and $t>0$. Letting $n\rightarrow \infty$, we get

$$ h(pq)=p^{2} f(q)+h(p)q^{2}, $$

(4.11)

for all $p,q \in \mathcal{B}$. Let $m \in \mathbb{N}$. We have

$$\begin{aligned} p^{2} f\bigl(2^{ms}q\bigr) =& h\bigl(2^{ms}pq \bigr)-h\bigl(2^{ms}p\bigr)q^{2} \\ =& 2^{2ms}p^{2} f(q)+h\bigl(2^{ms}p \bigr)q^{2}-h\bigl(2^{ms}p\bigr)q^{2} \\ =&2^{2ms} p^{2} f(q) \end{aligned}$$

(4.12)

for all $p,q\in \mathcal{B}$, and so $p^{2} f(q)=p^{2} \frac{f(2^{ms}q)}{2^{2ms}}$ for all $p,q\in \mathcal{B}$ and $m \in \mathbb{N}$. Letting $m \rightarrow \infty$ yields $p^{2}f(q)=p ^{2} h(q)$. Putting $p=e$, we get $h(q)=f(q)$ for all $q \in \mathcal{B}$. Hence, on $\mathcal{B}$, f is a ∗-quadratic derivation. □

5 Derivations on random $C^{*}$-ternary algebras

A complex random Banach space $(\mathcal{B},\mu ,T,T')$, which has a ternary product $(f, g, h) \longmapsto [f, g, h]$ of $\mathcal{B}^{3}$ into $\mathcal{B}$, is a random $C^{*}$-ternary algebra if (see [29]):

• (1) $[\xi f+v, g, h]=\xi [f, g, h]+[v, g, h]$ for all $\xi \in \mathbb{C}$;

• (2) $[ f, \xi g+v, h]=\xi [f, g, h]+[f, v, h]$ for all $\xi \in \mathbb{C}$;

• (3) $[ f, g, \xi h+v]=\xi [f, g, h]+[f, g, v]$ for all $\xi \in \mathbb{C}$;

• (4) $[f, g, [h, k, j]]=[f, [k, h, g], j]=[[f, g, h], k, j]$;

• (5) $\Vert [f, g, h] \Vert \leq \Vert f \Vert \cdot \Vert g \Vert \cdot \Vert h \Vert$;

• (6) $\Vert [f,f,f] \Vert = \Vert f \Vert ^{3}$;

for $f,g,h,v,k,j \in \mathcal{B}$.

If $(\mathcal{B},\mu ,T,T')$ has the unit e satisfying $f=[f, e, e]=[e, e, f]$ for all $f \in \mathcal{B}$, then the random $C^{*}$-ternary algebra has unit e. If for $f \in \mathcal{B}$, we have $[e,f,e]=f^{*}$, then ∗ is an involution on the $C^{*}$-ternary algebra. A $C^{*}$-ternary derivation is a mapping $\delta : \mathcal{B}\longrightarrow \mathcal{B}$ such that

$$\begin{aligned}& \delta \bigl([f, g, h]\bigr) = \bigl[\delta (f), g, h\bigr]+\bigl[f, \delta (g), h\bigr]+\bigl[f, g, \delta (h)\bigr], \\& \delta (\xi f+g) = \xi \delta (f)+\delta (g) \end{aligned}$$

for all $f,g,h\in \mathcal{B}$ and $\xi \in \mathbb{C}$. Recall that $\delta ([e, f, e])=[e, \delta (f), e]$ implies that δ is an involution.

Theorem 5.1

Assume that $\mathcal{B}$ is a random $C^{*}$-ternary algebra which has the unit e. Suppose that $\psi_{1}: \mathcal{B}^{2} \longrightarrow [0,\infty )$ and $\psi_{2}: \mathcal{B}^{3} \longrightarrow [0, \infty )$ are functions. Let $f: \mathcal{B} \longrightarrow \mathcal{B}$ be a mapping such that

$$\begin{aligned}& \mu_{ f(\xi p+q)-\lambda f(p)-f(q)}(t) \geq \psi_{1}(p,q,t), \end{aligned}$$

(5.1)

$$\begin{aligned}& \mu_{ f([p,q,r])-[f(p), q, r]-[p, f(q), r] [p, q, f(r)]}(t) \geq \psi _{2}(p, q, r,t), \end{aligned}$$

(5.2)

$$\begin{aligned}& \mu_{ f([e, q, e])-[e, f(q), e]}(t) \geq \psi_{2}(e, q, e,t) \end{aligned}$$

(5.3)

for all $\lambda \in \mathbb{C}$, $p,q,r\in \mathcal{B}$ and $t>0$. Assume there exist $s\in \mathbb{N}$ and $0< L<1$ such that $\psi_{1} (s ^{i} p, s^{j} q,s^{(i+j)}L^{(i+j)}t)>\psi_{1} (p,q,t)$, $\psi_{2} (s ^{i} p, s^{j} q, s^{k} r,s^{(i+j+k)}L^{(i+j+k)}t)>\psi_{2} (p,q,r,t)$ for all $p,q,r \in \mathcal{B}$ and $i, j, k=0, 1$. Then on $\mathcal{B}$, f is a ∗-derivation.

Proof

Put

$$ \varPsi (p,t)=\prod^{s-1}_{j=1} \psi_{1} (jp,p,t_{j}) $$

for $p\in \mathcal{B}$ and $t>0$ where $\sum^{s-1}_{j=1} t_{j}=t$. Then we have

$$ \mu_{ f(sp)-sf(p)}(t) \geq \varPsi (p,t). $$

(5.4)

We use similar method presented in the proof of Theorem 3.1. Let Γ be the set of all mappings $r: \mathcal{B}\longrightarrow \mathcal{B}$. Define a function $\Delta : \varGamma \times \varGamma \longrightarrow [0, \infty ]$ by

$$ \Delta (\zeta ,\eta )=\inf \bigl\{ \nu >0: \mu_{ \zeta (z)-\eta (z)}(\nu s) \geq \varPsi (z,s) \bigr\} $$

for $\zeta ,\eta \in \varGamma$, $z \in \mathcal{B}$ and $t>0$. Miheţ and Radu [28] proved that $(\varGamma , \Delta )$ is a complete GM space. Define a mapping $H: \varGamma \longrightarrow \varGamma$ by $H(\zeta )(z)=s^{-1} \zeta (sz)$. Now

$$ \Delta (\zeta ,\eta )=\nu (\zeta ,\eta \in \varGamma ) $$

implies that

$$ \mu_{ H(\zeta )(z)-H(\eta )(z)}(t)=\mu_{ \zeta (sz)-\eta (sz)}(\nu s t) \geq \varPsi (sz,st)\geq \varPsi \biggl( z,\frac{t}{L\nu } \biggr) $$

and for $\zeta ,\eta \in \varGamma$

$$ \Delta \bigl(H(\zeta ), H(\eta )\bigr)\leq L\Delta (\zeta ,\eta ). $$

(5.5)

Therefore H on Γ with Lipschitz constant L is a strictly contractive function. From (5.4), we have

$$ \mu { (Hf) (z)-f(z)}(t)=\mu_{ s^{-1} f(sz)-f(z)}(t)=\mu_{f(sz)-sf(z)}(st) \geq \varPsi (z,st). $$

So $\Delta (H(f), f)\leq 1/ \vert s \vert$. Using Theorem 2.1, we conclude that, in the set

$$ U=\bigl\{ \zeta \in \varGamma : \Delta \bigl(\zeta , H(f)\bigr)< \infty \bigr\} , $$

$h: \mathcal{B} \longrightarrow \mathcal{B}$ is a unique fixed point of H.

Now, for every $z \in \mathcal{B}$, we have

$$ h(z)=\lim_{m\rightarrow \infty } H^{m} \bigl(f(z) \bigr)=\lim_{m\rightarrow \infty } s^{-m} f\bigl(s^{m} z \bigr) $$

(5.6)

which implies that h is a $\mathbb{C}$-linear mapping on $\mathcal{B}$. Also, we can show that h has the $C^{*}$-ternary derivation property,

$$\begin{aligned}& \mu_{ h([p,q,r]) [h(p), q, r] [p, h(q), r] [p, q, h(r)]}(t) \\ & \quad =\lim_{n\rightarrow \infty } \mu_{ f(s^{3n}[p,q,r])-s^{2n} [f(s^{n}p), q, r]-s^{2n}[p, f(s^{n} q), r]-s^{2n}[p,q, f(s^{n}r)]}\bigl(s^{3n}t \bigr) \\ & \quad \geq \lim_{n\rightarrow \infty } \psi_{1}\bigl(s^{n} p, s^{n}q, s ^{n} r,s^{3n}t\bigr) \geq \lim _{n\rightarrow \infty } \psi_{1} \biggl( p,q,r,\frac{t}{L ^{3n}} \biggr) =1. \end{aligned}$$

So

$$ h\bigl([p,q,r]\bigr)=\bigl[h(p), q,r\bigr]+\bigl[p, h(q), r \bigr]+\bigl[p,q, h(r)\bigr] $$

(5.7)

for all $p,q,r \in \mathcal{B}$. Also,

$$\begin{aligned} \mu_{ h([e, p, e])-[e, h(p), e]}(t) =& \lim_{n\rightarrow \infty } \mu_{ f(s^{3n}[e, p, e])-s^{2n}[e, f(s^{n} p), e]} \bigl(s^{3n}t\bigr) \\ \geq & \lim_{n\rightarrow \infty } \psi_{1}\bigl(s^{n} e, s^{n}p, s^{n} e,s ^{3n}t\bigr) \\ \geq & \lim_{n\rightarrow \infty } L^{3n} \psi_{1} \biggl( e, p, e,\frac{t}{L ^{3n}} \biggr) \\ =& 1, \end{aligned}$$

which implies that, on $\mathcal{B}$, h is a ∗-derivation.

Now, in (5.2), we replace q by $s^{n} q$, r by $s^{n} r$ and divide by $s^{2n}$. Letting $n\to \infty$, we get

$$\begin{aligned}& \lim_{n\rightarrow \infty } \mu_{ s^{-2n} ( f([p, s^{n} q, s^{n} r])-[f(p), s^{n} q, s^{n} r]-s ^{n}[p, f(s^{n} q), r]-s^{n} [p,q, f(s^{n} r)] ) }(t) \\& \quad =\lim_{n\rightarrow \infty } \mu_{f( s^{2n}[p,q,r])-s^{2n}[f(p), q,r]-s^{n} [p, f(s^{n} q), r]-s ^{n} [p,q, f(s^{n} r)]}\bigl(s^{ 2n}t \bigr) \\& \quad \geq \lim_{n\rightarrow \infty } \psi_{1}\bigl(p, s^{n} q, s^{n} r,s ^{ 2n}\bigr) \geq \lim _{n\rightarrow \infty } \psi_{1} \biggl( p,q,r\frac{t}{L ^{2n}} \biggr) =1, \end{aligned}$$

which implies that

$$ h\bigl([p,q,r]\bigr)=\bigl[f(p), q,r\bigr]+\bigl[p, h(q), r \bigr]+\bigl[p,q, h(r)\bigr] $$

(5.8)

for all $p,q,r \in \mathcal{B}$. Putting $f(p)-h(p)$ instead of q and r in (5.7) and (5.8), we obtain $\mu_{ h(p)-f(p)}(t)=1$. Hence, on $\mathcal{B}$, f is a ∗-derivation. □