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Approximation of derivations and the superstability in random Banach -algebras

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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 [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.

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:

  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

([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

  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 [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. (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 [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 \(\mathcal{B}\). □

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 [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. □

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. (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 [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. □

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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.

Correspondence to Reza Saadati.

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MSC

  • 46S50
  • 47H10
  • 26E60

Keywords

  • Derivation
  • Quadratic derivation
  • Superstability
  • Fixed point method
  • Random Banach -algebra
  • Random \(C^{*}\)-ternary algebra