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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 [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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Correspondence to Reza Saadati.

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Saadati, R., Park, C. Approximation of derivations and the superstability in random Banach -algebras. Adv Differ Equ 2018, 418 (2018). https://doi.org/10.1186/s13662-018-1882-6

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MSC

  • 46S50
  • 47H10
  • 26E60

Keywords

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