# 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.

## Introduction

Ulam  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 . He intelligently answered Ulam’s question when $$\varOmega_{1}$$ and $$\varOmega_{2}$$ are Banach spaces. Recently, Rassias  and others have obtained important results on stability and applied them to the investigations in the nonlinear sciences.

## 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

()

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

1. (a)

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

2. (b)

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

3. (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

()

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

1. (RN1)

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

2. (RN2)

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

3. (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:

1. (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

1. (i)

$$g^{**}=g$$ for $$g\in \mathcal{B}$$;

2. (ii)

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

3. (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 ).

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)

Firstly, Baker, Lawrence and Zorzitto  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. (1)

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

2. (2)

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

3. (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. (1)

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

2. (2)

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

3. (3)

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

4. (4)

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

5. (5)

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

## 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 .

### 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  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 $$\mathcal{B}$$. □

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

### Definition 4.1

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

1. (1)

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

2. (2)

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

3. (3)

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

4. (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  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. □

## 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 ):

1. (1)

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

2. (2)

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

3. (3)

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

4. (4)

$$[f, g, [h, k, j]]=[f, [k, h, g], j]=[[f, g, h], k, j]$$;

5. (5)

$$\Vert [f, g, h] \Vert \leq \Vert f \Vert \cdot \Vert g \Vert \cdot \Vert h \Vert$$;

6. (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  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. □

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## Author information

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### Contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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