Approximation of homomorphisms and derivations on non-Archimedean random Lie $C^{\ast }$-algebras via fixed point method

Abstract

In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of random homomorphisms in random $C^{\ast }$-algebras and random Lie $C^{\ast }$-algebras and of derivations on non-Archimedean random C^∗-algebras and non-Archimedean random Lie C^∗-algebras for an m-variable additive functional equation.

MSC:39A10, 39B52, 39B72, 46L05, 47H10, 46B03.

1 Introduction and preliminaries

By a non-Archimedean field we mean a field K equipped with a function (valuation) $|\cdot |$ from K into $[0,\mathrm{\infty })$ such that $|r|=0$ if and only if $r=0$, $|rs|=|r||s|$ and $|r+s|\le max\{|r|,|s|\}$ for all $r,s\in K$. Clearly, $|1|=|-1|=1$ and $|n|\le 1$ for all $n\in \mathbb{N}$. By the trivial valuation we mean the mapping $|\cdot |$ taking everything but 0 into 1 and $|0|=0$. Let X be a vector space over a field K with a non-Archimedean non-trivial valuation $|\cdot |$. A function $\parallel \cdot \parallel :X\to [0,\mathrm{\infty })$ is called a non-Archimedean norm if it satisfies the following conditions:

• (i) $\parallel x\parallel =0$ if and only if $x=0$;

• (ii) for any $r\in K$, $x\in X$, $\parallel rx\parallel =|r|\parallel x\parallel$;

• (iii) the strong triangle inequality (ultrametric) holds; namely

  $$\parallel x+y\parallel \le max\{\parallel x\parallel ,\parallel y\parallel \}(x,y\in X).$$

Then $(X,\parallel \cdot \parallel )$ is called a non-Archimedean normed space. From the fact that

$$\parallel x_n-x_m\parallel \le max\{\parallel x_{j+1}-x_j\parallel :m\le j\le n-1\}(n>m)$$

holds, a sequence $\{x_n\}$ is Cauchy if and only if $\{x_{n+1}-x_n\}$ converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean the one in which every Cauchy sequence is convergent.

For any nonzero rational number x, there exists a unique integer $n_x\in \mathbb{Z}$ such that $x=\frac{a}{b}{p}^{n_x}$, where a and b are integers not divisible by p. Then ${|x|}_p:=p^{-n_x}$ defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric $d(x,y)={|x-y|}_p$ is denoted by ${\mathbb{Q}}_p$, which is called the p-adic number field.

A non-Archimedean Banach algebra is a complete non-Archimedean algebra $\mathcal{A}$ which satisfies $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$ for all $a,b\in \mathcal{A}$. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [1, 2].

If $\mathcal{U}$ is a non-Archimedean Banach algebra, then an involution on $\mathcal{U}$ is a mapping $t\to t^{\ast }$ from $\mathcal{U}$ into $\mathcal{U}$ which satisfies

• (i) $t^{\ast \ast }=t$ for $t\in \mathcal{U}$;

• (ii) ${(\alpha s+\beta t)}^{\ast }=\overline{\alpha }{s}^{\ast }+\overline{\beta }{t}^{\ast }$;

• (iii) ${(st)}^{\ast }=t^{\ast }{s}^{\ast }$ for $s,t\in \mathcal{U}$.

If, in addition, $\parallel t^{\ast }t\parallel ={\parallel t\parallel }^2$ for $t\in \mathcal{U}$, then $\mathcal{U}$ is a non-Archimedean $C^{\ast }$-algebra.

The stability problem of functional equations was originated from a question of Ulam [3] concerning the stability of group homomorphisms. Let $(G_1,\ast )$ be a group and let $(G_2,\diamond ,d)$ be a metric group (a metric which is defined on a set with a group property) with the metric $d(\cdot ,\cdot )$. Given $ϵ>0$, does there exist a $\delta (ϵ)>0$ such that if a mapping $h:G_1\to G_2$ satisfies the inequality $d(h(x\ast y),h(x)\diamond h(y))<\delta$ for all $x,y\in G_1$, then there is a homomorphism $H:G_1\to G_2$ with $d(h(x),H(x))<ϵ$ for all $x\in G_1?$ If the answer is affirmative, we would say that the equation of a homomorphism $H(x\ast y)=H(x)\diamond H(y)$ is stable (see also [4–6]).

We recall a fundamental result in fixed point theory. Let Ω be a set. A function $d:\mathrm{\Omega }\times \mathrm{\Omega }\to [0,\mathrm{\infty }]$ is called a generalized metric on Ω if d satisfies

• (1) $d(x,y)=0$ if and only if $x=y$;

• (2) $d(x,y)=d(y,x)$ for all $x,y\in \mathrm{\Omega }$;

• (3) $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in \mathrm{\Omega }$.

Theorem 1.1 [7]

Let $(\mathrm{\Omega },d)$ be a complete generalized metric space, and let $J:\mathrm{\Omega }\to \mathrm{\Omega }$ be a contractive mapping with the Lipschitz constant $L<1$. Then for each given element $x\in \mathrm{\Omega }$, either $d(J^{n}x,J^{n+1}x)=\mathrm{\infty }$ for all nonnegative integers n or there exists a positive integer $n_0$ such that

• (1) $d(J^{n}x,J^{n+1}x)<\mathrm{\infty }$, $\mathrm{\forall }n\ge n_0$;

• (2) the sequence $\{J^{n}x\}$ converges to a fixed point $y^{\ast }$ of J;

• (3) $y^{\ast }$ is the unique fixed point of J in the set $\mathrm{\Gamma }=\{y\in \mathrm{\Omega }\mid d(J^{n_0}x,y)<\mathrm{\infty }\}$;

• (4) $d(y,y^{\ast })\le \frac{1}{1-L}d(y,Jy)$ for all $y\in \mathrm{\Gamma }$.

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in non-Archimedean random $C^{\ast }$-algebras and non-Archimedean random Lie $C^{\ast }$-algebras for the following additive functional equation (see [8]):

$$\sum _{i=1}^{m}f(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+f\left(\sum _{i=1}^m{x}_i\right)=2f\left(\sum _{i=1}^{m}m{x}_i\right)(m\in \mathbb{N},m\ge 2).$$

(1.1)

2 Random spaces

In the section, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces as in [9–21]. Throughout this paper, ${\mathrm{\Delta }}^{+}$ is the space of distribution functions, that is, the space of all mappings such that F is left-continuous and non-decreasing on , $F(0)=0$ and $F(+\mathrm{\infty })=1$. $D^{+}$ is a subset of ${\mathrm{\Delta }}^{+}$ consisting of all functions $F\in {\mathrm{\Delta }}^{+}$ for which $l^{-}F(+\mathrm{\infty })=1$, where $l^{-}f(x)$ denotes the left limit of the function f at the point x, that is, $l^{-}f(x)={lim}_{t\to x^{-}}f(t)$. The space ${\mathrm{\Delta }}^{+}$ is partially ordered by the usual point-wise ordering of functions, i.e., $F\le G$ if and only if $F(t)\le G(t)$ for all t in . The maximal element for ${\mathrm{\Delta }}^{+}$ in this order is the distribution function ${\epsilon }_0$ given by

$${\epsilon }_0(t)=\{\begin{array}{cc}0\hfill & \text{if }t\le 0,\hfill \\ 1\hfill & \text{if }t>0.\hfill \end{array}$$

Definition 2.1 [20]

A mapping $T:[0,1]\times [0,1]\to [0,1]$ is a continuous triangular norm (briefly, a continuous t-norm) if T satisfies the following conditions:

• (a) T is commutative and associative;

• (b) T is continuous;

• (c) $T(a,1)=a$ for all $a\in [0,1]$;

• (d) $T(a,b)\le T(c,d)$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.

Typical examples of continuous t-norms are $T_P(a,b)=ab$, $T_M(a,b)=min(a,b)$ and $T_L(a,b)=max(a+b-1,0)$ (the Lukasiewicz t-norm).

Definition 2.2 [21]

A non-Archimedean random normed space (briefly, NA-RN-space) is a triple $(X,\mu ,T)$, where X is a vector space, T is a continuous t-norm, and μ is a mapping from X into $D^{+}$ such that the following conditions hold:

(RN1) ${\mu }_x(t)={\epsilon }_0(t)$ for all $t>0$ if and only if $x=0$;

(RN2) ${\mu }_{\alpha x}(t)={\mu }_x(\frac{t}{|\alpha |})$ for all $x\in X$, $\alpha \ne 0$;

(RN3) ${\mu }_{x+y}(t)\ge T({\mu }_x(t),{\mu }_y(t))$ for all $x,y\in X$ and all $t\ge 0$.

Every normed space $(X,\parallel \cdot \parallel )$ defines a non-Archimedean random normed space $(X,\mu ,T_M)$, where

$${\mu }_x(t)=\frac{t}{t+\parallel x\parallel }$$

for all $t>0$, and $T_M$ is the minimum t-norm. This space is called the induced random normed space.

Definition 2.3 [22]

A non-Archimedean random normed algebra $(X,\mu ,T,T^{\prime })$ is a non-Archimedean random normed space $(X,\mu ,T)$ with an algebraic structure such that

(RN-4) ${\mu }_{xy}(t)\ge T^{\prime }({\mu }_x(t),{\mu }_y(t))$ for all $x,y\in X$ and all $t>0$, in which $T^{\prime }$ is a continuous t-norm.

Every non-Archimedean normed algebra $(X,\parallel \cdot \parallel )$ defines a non-Archimedean random normed algebra $(X,\mu ,T_M)$, where

$${\mu }_x(t)=\frac{t}{t+\parallel x\parallel }$$

for all $t>0$ iff

$$\parallel xy\parallel \le \parallel x\parallel \parallel y\parallel +t\parallel y\parallel +t\parallel x\parallel (x,y\in X;t>0).$$

This space is called an induced non-Archimedean random normed algebra.

Definition 2.4

• (1) Let $(X,\mu ,T_M)$ and $(Y,\mu ,T_M)$ be non-Archimedean random normed algebras. An ℝ-linear mapping $f:X\to Y$ is called a homomorphism if $f(xy)=f(x)f(y)$ for all $x,y\in X$.

• (2) An ℝ-linear mapping $f:X\to X$ is called a derivation if $f(xy)=f(x)y+xf(y)$ for all $x,y\in X$.

Definition 2.5 Let $(\mathcal{U},\mu ,T,T^{\prime })$ be a non-Archimedean random Banach algebra, then an involution on $\mathcal{U}$ is a mapping $u\to u^{\ast }$ from $\mathcal{U}$ into $\mathcal{U}$ which satisfies

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

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

• (iii) ${(uv)}^{\ast }=v^{\ast }{u}^{\ast }$ for $u,v\in \mathcal{U}$.

If, in addition, ${\mu }_{u^{\ast }u}(t)=T^{\prime }({\mu }_u(t),{\mu }_u(t))$ for $u\in \mathcal{U}$ and $t>0$, then $\mathcal{U}$ is a non-Archimedean random $C^{\ast }$-algebra.

Definition 2.6 Let $(X,\mu ,T)$ be an NA-RN-space.

• (1) A sequence $\{x_n\}$ in X is said to be convergent to x in X if, for every $ϵ>0$ and $\lambda >0$, there exists a positive integer N such that ${\mu }_{x_n-x}(ϵ)>1-\lambda$ whenever $n\ge N$.

• (2) A sequence $\{x_n\}$ in X is called a Cauchy sequence if, for every $ϵ>0$ and $\lambda >0$, there exists a positive integer N such that ${\mu }_{x_n-x_{n+1}}(ϵ)>1-\lambda$ whenever $n\ge m\ge N$.

• (3) An RN-space $(X,\mu ,T)$ is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.

3 Stability of homomorphisms and derivations in non-Archimedean random $C^{\ast }$-algebras

Throughout this section, assume that $\mathcal{A}$ is a non-Archimedean random $C^{\ast }$-algebra with the norm ${\mu }_{\cdot }^{\mathcal{A}}$ and that ℬ is a non-Archimedean random $C^{\ast }$-algebra with the norm ${\mu }_{\cdot }^{\mathcal{B}}$.

For a given mapping $f:\mathcal{A}\to \mathcal{B}$, we define

$$D_{\lambda }f(x_1,\dots ,x_m):=\sum _{i=1}^m\lambda f(m{x}_i+\sum _{j=1,j\ne i}^m{x}_j)+f\left(\lambda \sum _{i=1}^m{x}_i\right)-2f\left(\lambda \sum _{i=1}^{m}m{x}_i\right)$$

for all $\lambda \in {\mathbb{T}}^1:=\{\nu \in \mathbb{C}:|\nu |=1\}$ and all $x_1,\dots ,x_m\in \mathcal{A}$.

Note that a ℂ-linear mapping $H:\mathcal{A}\to \mathcal{B}$ is called a homomorphism in non-Archimedean random $C^{\ast }$-algebras if H satisfies $H(xy)=H(x)H(y)$ and $H(x^{\ast })=H{(x)}^{\ast }$ for all $x,y\in \mathcal{A}$.

We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random $C^{\ast }$-algebras for the functional equation $D_{\lambda }f(x_1,\dots ,x_m)=0$.

Theorem 3.1 Let $f:\mathcal{A}\to \mathcal{B}$ be a mapping for which there are functions $\phi :{\mathcal{A}}^m\to D^{+}$, $\psi :{\mathcal{A}}^2\to D^{+}$, and $\eta :\mathcal{A}\to D^{+}$ such that $|m|<1$ is far from zero and

(3.1)

(3.2)

(3.3)

for all $\lambda \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. If there exists an $L<1$ such that

(3.4)

(3.5)

(3.6)

for all $x,y,x_1,\dots ,x_m\in \mathcal{A}$ and $t>0$, then there exists a unique random homomorphism $H:\mathcal{A}\to \mathcal{B}$ such that

$${\mu }_{f(x)-H(x)}^{\mathcal{B}}(t)\ge {\phi }_{x,0,\dots ,0}\left((|m|-|m|L)t\right)$$

(3.7)

for all $x\in \mathcal{A}$ and $t>0$.

Proof It follows from (3.4), (3.5), (3.6), and $L<1$ that

(3.8)

(3.9)

(3.10)

for all $x,y,x_1,\dots ,x_m\in \mathcal{A}$ and $t>0$.

Let us define Ω to be the set of all mappings $g:\mathcal{A}⟶\mathcal{B}$ and introduce a generalized metric on Ω as follows:

$$d(g,h)=inf\{k\in (0,\mathrm{\infty }):{\mu }_{g(x)-h(x)}^{\mathcal{B}}(kt)>{\varphi }_{x,0,\dots ,0}(t),\mathrm{\forall }x\in \mathcal{A},t>0\}.$$

It is easy to show that $(\mathrm{\Omega },d)$ is a generalized complete metric space (see [23]).

Now, we consider the function $J:\mathrm{\Omega }⟶\mathrm{\Omega }$ defined by $Jg(x)=\frac{1}{m}g(mx)$ for all $x\in \mathcal{A}$ and $g\in \mathrm{\Omega }$. Note that for all $g,h\in \mathrm{\Omega }$, we have

$$\begin{array}{rcl}d(g,h)<k& ⟹& {\mu }_{g(x)-h(x)}^{\mathcal{B}}(kt)>{\varphi }_{x,0,\dots ,0}(t)\\ ⟹& {\mu }_{\frac{1}{m}g(mx)-\frac{1}{m}h(mx)}^{\mathcal{B}}(kt)>|m|{\varphi }_{mx,0,\dots ,0}(|m|t)\\ ⟹& {\mu }_{\frac{1}{m}g(mx)-\frac{1}{m}h(mx)}^{\mathcal{B}}(kLt)>{\varphi }_{mx,0,\dots ,0}(t)\\ ⟹& d(Jg,Jh)<kL.\end{array}$$

From this it is easy to see that $d(Jg,Jk)\le Ld(g,h)$ for all $g,h\in \mathrm{\Omega }$, that is, J is a self-function of Ω with the Lipschitz constant L.

Putting $\mu =1$, $x=x_1$ and $x_2=x_3=\cdots =x_m=0$ in (3.1), we have

$${\mu }_{f(mx)-mf(x)}^{\mathcal{B}}(t)\ge {\varphi }_{x,0,\dots ,0}(t)$$

for all $x\in \mathcal{A}$ and $t>0$. Then

$${\mu }_{f(x)-\frac{1}{m}f(mx)}^{\mathcal{B}}(t)\ge {\varphi }_{x,0,\dots ,0}(|m|t)$$

for all $x\in \mathcal{A}$ and $t>0$, that is, $d(Jf,f)\le \frac{1}{|m|}<\mathrm{\infty }$. Now, from the fixed point alternative, it follows that there exists a fixed point H of J in Ω such that

$$H(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{{|m|}^n}f\left(m^{n}x\right)$$

(3.11)

for all $x\in \mathcal{A}$ since ${lim}_{n\to \mathrm{\infty }}d(J^{n}f,H)=0$.

On the other hand, it follows from (3.1), (3.8), and (3.11) that

$$\begin{array}{rcl}{\mu }_{D_{\lambda }H(x_1,\dots ,x_m)}^{\mathcal{B}}(t)& =& \underset{n\to \mathrm{\infty }}{lim}{\mu }_{\frac{1}{m^n}Df(m^n{x}_1,\dots ,m^n{x}_m)}^{\mathcal{B}}(t)\\ \ge & \underset{n\to \mathrm{\infty }}{lim}{\varphi }_{m^n{x}_1,\dots ,m^n{x}_m}\left({|m|}^{n}t\right)=1.\end{array}$$

By a similar method to the above, we get $\lambda H(mx)=H(m\lambda x)$ for all $\lambda \in {\mathbb{T}}^1$ and all $x\in \mathcal{A}$. Thus, one can show that the mapping $H:\mathcal{A}\to \mathcal{B}$ is ℂ-linear.

It follows from (3.2), (3.9), and (3.11) that

$$\begin{array}{rcl}{\mu }_{H(xy)-H(x)H(y)}^{\mathcal{B}}(t)& =& \underset{n\to \mathrm{\infty }}{lim}{\mu }_{f(m^{2n}xy)-f(m^{n}x)f(m^{n}y)}^{\mathcal{B}}\left({|m|}^{2n}t\right)\\ \ge & \underset{n\to \mathrm{\infty }}{lim}{\psi }_{m^{n}x,m^{n}y}\left({|m|}^{2n}t\right)=1\end{array}$$

for all $x,y\in \mathcal{A}$. So, $H(xy)=H(x)H(y)$ for all $x,y\in \mathcal{A}$. Thus, $H:\mathcal{A}\to \mathcal{B}$ is a homomorphism satisfying (3.7) as desired.

Also by (3.3), (3.10), (3.11) and by a similar method, we have $H(x^{\ast })=H{(x)}^{\ast }$. □

Corollary 3.2 Let $r>1$ and θ be nonnegative real numbers, and let $f:\mathcal{A}\to \mathcal{B}$ be a mapping such that

for all $\lambda \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in \mathcal{A}$ and $t>0$. Then there exists a unique homomorphism $H:\mathcal{A}\to \mathcal{B}$ such that

$${\mu }_{f(x)-H(x)}^{\mathcal{B}}(t)\ge \frac{(|m|-{|m|}^r)t}{(|m|-{|m|}^r)t+\theta |m|-{|m|}^r{\parallel x\parallel }_{\mathcal{A}}^r}$$

for all $x\in \mathcal{A}$ and $t>0$.

Proof The proof follows from Theorem 3.1. By taking

for all $x_1,\dots ,x_m,x,y\in A$, $L={|m|}^{r-1}$ and $t>0$, we get the desired result. □

We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random $C^{\ast }$-algebras for the functional equation $D_{\lambda }f(x_1,\dots ,x_m)=0$.

Theorem 3.3 Let $f:\mathcal{A}\to \mathcal{A}$ be a mapping for which there are functions $\phi :{\mathcal{A}}^m\to D^{+}$, $\psi :{\mathcal{A}}^2\to D^{+}$, and $\eta :\mathcal{A}\to D^{+}$ such that $|m|<1$ is far from zero and

for all $\lambda \in {\mathbb{T}}^1$ and all $x_1,\dots ,x_m,x,y\in A$ and $t>0$. If there exists an $L<1$ such that (3.4), (3.5), and (3.6) hold, then there exists a unique derivation $\delta :\mathcal{A}\to \mathcal{A}$ such that

$${\mu }_{f(x)-\delta (x)}^{\mathcal{A}}(t)\ge {\phi }_{x,0,\dots ,0}\left((|m|-|m|L)t\right)$$

for all $x\in \mathcal{A}$ and $t>0$.

4 Stability of homomorphisms and derivations in non-Archimedean Lie $C^{\ast }$-algebras

A non-Archimedean random $C^{\ast }$-algebra $\mathcal{C}$, endowed with the Lie product

$$[x,y]:=\frac{xy-yx}{2}$$

on $\mathcal{C}$, is called a Lie non-Archimedean random $C^{\ast }$-algebra.

Definition 4.1 Let $\mathcal{A}$ and ℬ be random Lie $C^{\ast }$-algebras. A ℂ-linear mapping $H:\mathcal{A}\to \mathcal{B}$ is called a non-Archimedean Lie $C^{\ast }$-algebra homomorphism if $H([x,y])=[H(x),H(y)]$ for all $x,y\in \mathcal{A}$.

Throughout this section, assume that $\mathcal{A}$ is a non-Archimedean random Lie $C^{\ast }$-algebra with the norm ${\mu }^{\mathcal{A}}$ and that ℬ is a non-Archimedean random Lie $C^{\ast }$-algebra with the norm ${\mu }^{\mathcal{B}}$.

We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random Lie $C^{\ast }$-algebras for the functional equation $D_{\lambda }f(x_1,\dots ,x_m)=0$.

Theorem 4.2 Let $f:\mathcal{A}\to \mathcal{B}$ be a mapping for which there are functions $\phi :{\mathcal{A}}^m\to D^{+}$ and $\psi :{\mathcal{A}}^2\to D^{+}$ such that (3.1) and (3.3) hold and

$${\mu }_{f([x,y])-[f(x),f(y)]}^{\mathcal{B}}(t)\ge {\psi }_{x,y}(t)$$

(4.1)

for all $\lambda \in {\mathbb{T}}^1$, all $x,y\in \mathcal{A}$ and $t>0$. If there exists an $L<1$ and (3.4), (3.5), and (3.6) hold, then there exists a unique homomorphism $H:\mathcal{A}\to \mathcal{B}$ such that (3.7) holds.

Proof By the same reasoning as in the proof of Theorem 3.1, we can find the mapping $H:\mathcal{A}\to \mathcal{B}$ given by

$$H(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(m^{n}x)}{{|m|}^n}$$

(4.2)

for all $x\in \mathcal{A}$. It follows from (3.5) and (4.2) that

$$\begin{array}{rcl}{\mu }_{H([x,y])-[H(x),H(y)]}^{\mathcal{B}}(t)& =& \underset{n\to \mathrm{\infty }}{lim}{\mu }_{f(m^{2n}[x,y])-[f(m^{n}x),f(m^{n}y)]}^{\mathcal{B}}\left({|m|}^{2n}t\right)\\ \ge & \underset{n\to \mathrm{\infty }}{lim}{\psi }_{m^{n}x,m^{n}y}\left({|m|}^{2n}t\right)=1\end{array}$$

for all $x,y\in \mathcal{A}$ and $t>0$, then

$$H([x,y])=[H(x),H(y)]$$

for all $x,y\in \mathcal{A}$. Thus, $H:\mathcal{A}\to \mathcal{B}$ is a Lie $C^{\ast }$-algebra homomorphism satisfying (3.7), as desired. □

Corollary 4.3 Let $r>1$ and θ be nonnegative real numbers, and let $f:\mathcal{A}\to \mathcal{B}$ be a mapping such that

for all $\lambda \in {\mathbb{T}}^1$, all $x_1,\dots ,x_m,x,y\in \mathcal{A}$ and $t>0$. Then there exists a unique homomorphism $H:\mathcal{A}\to \mathcal{B}$ such that

$${\mu }_{f(x)-H(x)}^{\mathcal{B}}(t)\ge \frac{(|m|-{|m|}^r)t}{(|m|-{|m|}^r)t+\theta {\parallel x\parallel }_{\mathcal{A}}^r}$$

for all $x\in \mathcal{A}$ and $t>0$.

Proof The proof follows from Theorem 4.2 and a method similar to Corollary 3.2. □

Definition 4.4 Let $\mathcal{A}$ be a non-Archimedean random Lie $C^{\ast }$-algebra. A ℂ-linear mapping $\delta :\mathcal{A}\to \mathcal{A}$ is called a Lie derivation if $\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$ for all $x,y\in \mathcal{A}$.

We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random Lie $C^{\ast }$-algebras for the functional equation $D_{\lambda }f(x_1,\dots ,x_m)=0$.

Theorem 4.5 Let $f:\mathcal{A}\to \mathcal{A}$ be a mapping for which there are functions $\phi :A^m\to D^{+}$ and $\psi :A^2\to D^{+}$ such that (3.1) and (3.3) hold and

$${\mu }_{f([x,y])-[f(x),y]-[x,f(y)]}^{\mathcal{A}}(t)\ge {\psi }_{x,y}(t),$$

(4.3)

for all $x,y\in \mathcal{A}$. If there exists an $L<1$ and (3.4), (3.5), and (3.6) hold, then there exists a unique Lie derivation $\delta :\mathcal{A}\to \mathcal{A}$ such that (3.7) holds.

Proof By the same reasoning as the proof of Theorem 4.2, there exists a unique ℂ-linear mapping $\delta :\mathcal{A}\to \mathcal{A}$ satisfying (3.7); the mapping $\delta :\mathcal{A}\to \mathcal{A}$ is given by

$$\delta (x)=\underset{n\to \mathrm{\infty }}{lim}\frac{f(m^{n}x)}{{|m|}^n}$$

(4.4)

for all $x\in \mathcal{A}$.

It follows from (3.5) and (4.4) that

for all $x,y\in \mathcal{A}$ and $t>0$, then

$$\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$$

for all $x,y\in \mathcal{A}$. Thus, $\delta :\mathcal{A}\to \mathcal{A}$ is a Lie derivation satisfying (3.7). □