Approximation of Homomorphisms and Derivations on non-Archimedean Lie 𝐶^∗-Algebras via Fixed Point Method

Abstract

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in 𝐶^∗-algebras and Lie 𝐶^∗-algebras and of derivations on non-Archimedean 𝐶^∗-algebras and Non-Archimedean Lie 𝐶^∗-algebras for an 𝑚-variable additive functional equation.

1. Introduction and Preliminaries

By a non-Archimedean field we mean a field 𝐾 equipped with a function (valuation) |⋅| from 𝐾 into [0,∞) such that |𝑟|=0 if and only if 𝑟=0, |𝑟𝑠|=|𝑟||𝑠|, and |𝑟+𝑠|≤max{|𝑟|,|𝑠|} for all 𝑟,𝑠∈𝐾. Clearly |1|=|−1|=1 and |𝑛|≤1 for all 𝑛∈ℕ. By the trivial valuation we mean the mapping |⋅| taking everything but 0 into 1 and |0|=0. Let 𝑋 be a vector space over a field 𝐾 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→[0,∞) is called a non-Archimedean norm if it satisfies the following conditions:(i)‖𝑥‖=0 if and only if 𝑥=0;(ii)for any 𝑟∈𝐾, and 𝑥∈𝑋, ‖𝑟𝑥‖=|𝑟|‖𝑥‖;(iii)the strong triangle inequality (ultrametric) holds; namely,𝑥+𝑦≤max‖𝑥‖,𝑦. (1.1) for all 𝑥,𝑦∈𝑋.

Then (𝑋,‖⋅‖) is called a non-Archimedean normed space. From the fact that𝑥_𝑛−𝑥_𝑚≤max𝑥_𝑗+1−𝑥_𝑗∶𝑚≤𝑗≤𝑛−1(𝑛>𝑚) (1.2)

holds, a sequence {𝑥_𝑛} is a Cauchy sequence 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 one in which every Cauchy sequence is convergent.

For any nonzero rational number 𝑥, there exists a unique integer 𝑛_𝑥∈ℤ such that 𝑥=(𝑎/𝑏)𝑝^𝑛𝑥, where 𝑎 and 𝑏 are integers not divisible by 𝑝. Then |𝑥|_𝑝∶=𝑝^−𝑛𝑥 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 ℚ_𝑝, which is called the 𝑝-adic number field.

A non-Archimedean Banach algebra is a complete non-Archimedean algebra 𝒜 which satisfies ‖𝑎𝑏‖≤‖𝑎‖⋅‖𝑏‖ for all 𝑎,𝑏∈𝒜. For more detailed definitions of non-Archimedean Banach algebras, we refer the reader to [1, 2].

If 𝒰 is a non-Archimedean Banach algebra, then an involution on 𝒰 is a mapping 𝑡→𝑡^∗ from 𝒰 into 𝒰 which satisfies(i)𝑡^∗∗=𝑡 for 𝑡∈𝒰;(ii)(𝛼𝑠+𝛽𝑡)^∗=𝛼𝑠^∗+𝛽𝑡^∗;(iii)(𝑠𝑡)^∗=𝑡^∗𝑠^∗ for 𝑠,𝑡∈𝒰.

If, in addition, ‖𝑡^∗𝑡‖=‖𝑡‖² for 𝑡∈𝒰, then 𝒰 is a non-Archimedean 𝐶^∗-algebra.

The stability problem of functional equations was originated from a question of Ulam [3] concerning the stability of group homomorphisms: let (𝐺₁,∗) be a group and let (𝐺₂,⋄,𝑑) be a metric group (a metric which is defined on a set with group property) with the metric 𝑑(⋅,⋅). Given 𝜖>0, does there exist a 𝛿(𝜖)>0 such that, if a mapping ℎ∶𝐺₁→𝐺₂ satisfies the inequality 𝑑(ℎ(𝑥∗𝑦),ℎ(𝑥)⋄ℎ(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then there is a homomorphism 𝐻∶𝐺₁→𝐺₂ with 𝑑(ℎ(𝑥),𝐻(𝑥))<𝜖 for all 𝑥∈𝐺₁? If the answer is affirmative, we would say that the equation of homomorphism 𝐻(𝑥∗𝑦)=𝐻(𝑥)⋄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∶Ω×Ω→[0,∞] 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∈Ω;(3)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) for all 𝑥,𝑦,𝑧∈Ω.

Theorem 1.1 (see [7]). Let (Ω,𝑑) be a complete generalized metric space and let 𝐽∶Ω→Ω be a contractive mapping with Lipschitz constant 𝐿<1. Then for each given element 𝑥∈Ω, either 𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)=∞ for all nonnegative integers n or there exists a positive integer n₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞ for all 𝑛≥𝑛₀;(2)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(3)𝑦^∗ is the unique fixed point of 𝐽 in the set Γ={𝑦∈Ω∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(4)𝑑(𝑦,𝑦^∗)≤(1/(1−𝐿))𝑑(𝑦,𝐽𝑦) for all 𝑦∈Γ.

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in non-Archimedean 𝐶^∗-algebras and non-Archimedean Lie 𝐶^∗-algebras for the following additive functional equation (see [8]):^𝑚_𝑖=1𝑓𝑚𝑥_𝑖+^𝑚_{𝑗=1,𝑗≠𝑖}𝑥_𝑗+𝑓^𝑚_𝑖=1𝑥_𝑖=2𝑓^𝑚_𝑖=1𝑚𝑥_𝑖(𝑚∈ℕ,𝑚≥2). (1.3)

2. Stability of Homomorphisms and Derivations in 𝐶^∗-Algebras

Throughout this section, assume that 𝒜 is a non-Archimedean 𝐶^∗-algebra with norm ‖⋅‖_𝒜 and that ℬ is a non-Archimedean 𝐶^∗-algebra with norm ‖⋅‖_ℬ.

For a given mapping 𝑓∶𝒜→ℬ, we define𝐷_𝜇𝑓𝑥₁,…,𝑥_𝑚∶=^𝑚_𝑖=1𝜇𝑓𝑚𝑥_𝑖+^𝑚_{𝑗=1,𝑗≠𝑖}𝑥_𝑗+𝑓𝜇^𝑚_𝑖=1𝑥_𝑖−2𝑓𝜇^𝑚_𝑖=1𝑚𝑥_𝑖 (2.1)

for all 𝜇∈𝕋¹∶={𝜈∈ℂ∶|𝜈|=1} and all 𝑥₁,…,𝑥_𝑚∈𝒜.

Note that a ℂ-linear mapping 𝐻∶𝒜→ℬ is called a homomorphism in non-Archimedean 𝐶^∗-algebras if 𝐻 satisfies 𝐻(𝑥𝑦)=𝐻(𝑥)𝐻(𝑦) and 𝐻(𝑥^∗)=𝐻(𝑥)^∗ for all 𝑥,𝑦∈𝒜.

We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean 𝐶^∗-algebras for the functional equation 𝐷_𝜇𝑓(𝑥₁,…,𝑥_𝑚)=0.

Theorem 2.1. Let 𝑓∶𝒜→ℬ be a mapping for which there are functions 𝜑∶𝒜^𝑚→[0,∞), 𝜓∶𝒜²→[0,∞) and 𝜂∶𝒜→[0,∞) such that |𝑚|<1 is far from zero and 𝐷_𝜇𝑓𝑥₁,…,𝑥_𝑚ℬ≤𝜑𝑥₁,…,𝑥_𝑚,𝑓𝑥𝑦−𝑓(𝑥)𝑓𝑦_ℬ≤𝜓𝑥,𝑦,𝑓𝑥^∗−𝑓(𝑥)^∗_ℬ≤𝜂(𝑥) (2.2)(2.3)(2.4) for all 𝜇∈𝕋¹ and 𝑥₁,…,𝑥_𝑚,𝑥,𝑦∈𝐴. If there exists an 𝐿<1 such that 𝜑𝑚𝑥₁,…,𝑚𝑥_𝑚≤|𝑚|𝐿𝜑𝑥₁,…,𝑥_𝑚,𝜓𝑚𝑥,𝑚𝑦≤|𝑚|²𝐿𝜓𝑥,𝑦,𝜂(𝑚𝑥)≤|𝑚|𝐿𝜂(𝑥) (2.5)(2.6)(2.7) for all 𝑥,𝑦,𝑥₁,…,𝑥_𝑚∈𝒜, then there exists a unique homomorphism 𝐻∶𝒜→ℬ such that 𝑓(𝑥)−𝐻(𝑥)_ℬ≤1|𝑚|−|𝑚|𝐿𝜑(𝑥,0,…,0) (2.8) for all 𝑥∈𝒜.

Proof. It follows from (2.5), (2.6), (2.7) and 𝐿<1 that lim_𝑛→∞1|𝑚|^𝑛𝜑𝑚^𝑛𝑥₁,…,𝑚^𝑛𝑥_𝑚=0,lim_𝑛→∞1|𝑚|^2𝑛𝜓𝑚^𝑛𝑥,𝑚^𝑛𝑦=0,lim_𝑛→∞1|𝑚|^𝑛𝜂𝑚^𝑛𝑥=0 (2.9)(2.10)(2.11) for all 𝑥,𝑦,𝑥₁,…,𝑥_𝑚∈𝒜.
Let us define Ω to be the set of all mappings 𝑔∶𝒜→ℬ and introduce a generalized metric on Ω as follows𝑑𝑔,ℎ=inf𝑘∈(0,∞)∶𝑔(𝑥)−ℎ(𝑥)_ℬ<𝑘𝜙(𝑥,0,…,0),∀𝑥∈𝒜. (2.12) It is easy to show that (Ω,𝑑) is a generalized complete metric space (see [9]).
Now we consider the function 𝐽∶Ω→Ω defined by 𝐽𝑔(𝑥)=(1/𝑚)𝑔(𝑚𝑥) for all 𝑥∈𝒜 and 𝑔∈Ω. Note that for all 𝑔,ℎ∈Ω we have 𝑑𝑔,ℎ<𝑘⟹𝑔(𝑥)−ℎ(𝑥)_ℬ()⟹1<𝑘𝜙𝑥,0,…,0𝑚𝑔()−1𝑚𝑥𝑚ℎ(𝑚𝑥)_ℬ<𝑘|𝑚|𝜙()⟹1𝑚𝑥,0,…,0𝑚𝑔()−1𝑚𝑥𝑚ℎ(𝑚𝑥)_ℬ<𝑘𝐿𝜙(𝑚𝑥,0,…,0)⟹𝑑𝐽𝑔,𝐽ℎ<𝑘𝐿. (2.13) From this it is easy to see that 𝑑(𝐽𝑔,𝐽𝑘)≤𝐿𝑑(𝑔,ℎ) for all 𝑔,ℎ∈Ω, that is, 𝐽 is a self-function of Ω with the Lipschitz constant 𝐿.
Putting 𝜇=1, 𝑥=𝑥₁ and 𝑥₂=𝑥₃=⋯=𝑥_𝑚=0 in (2.2), we have𝑓(𝑚𝑥)−𝑚𝑓(𝑥)_ℬ≤𝜙(𝑥,0,…,0) (2.14) for all 𝑥∈𝒜. Then 1𝑓(𝑥)−𝑚𝑓(𝑚𝑥)_ℬ≤1|𝑚|𝜙(𝑥,0,…,0) (2.15) for all 𝑥∈𝒜, that is, 𝑑(𝐽𝑓,𝑓)≤1/|𝑚|<∞. Now, from the fixed point alternative, it follows that there exists a fixed point 𝐻 of 𝐽 in Ω such that 𝐻(𝑥)=lim_𝑛→∞1|𝑚|^𝑛𝑓𝑚^𝑛𝑥 (2.16) for all 𝑥∈𝒜 since lim_𝑛→∞𝑑(𝐽^𝑛𝑓,𝐻)=0.
On the other hand, it follows from (2.2), (2.9), and (2.16) that𝐷_𝜇𝐻𝑥₁,…,𝑥_𝑚ℬ=lim_𝑛→∞1𝑚^𝑛𝐷𝑓𝑚^𝑛𝑥₁,…,𝑚^𝑛𝑥_𝑚ℬ≤lim_𝑛→∞1|𝑚|^𝑛𝜙𝑚^𝑛𝑥₁,…,𝑚^𝑛𝑥_𝑚=0. (2.17)
By a similar method to the above, we get 𝜇𝐻(𝑚𝑥)=𝐻(𝑚𝜇𝑥) for all 𝜇∈𝕋¹ and 𝑥∈𝒜. Thus one can show that the mapping 𝐻∶𝒜→ℬ is ℂ-linear.
It follows from (2.3), (2.10) and (2.16) that 𝐻𝑥𝑦−𝐻(𝑥)𝐻𝑦_ℬ=lim_𝑛→∞1|𝑚|^2𝑛𝑓𝑚^2𝑛𝑥𝑦−𝑓𝑚^𝑛𝑥𝑓𝑚^𝑛𝑦_ℬ≤lim_𝑛→∞1|𝑚|^2𝑛𝜓𝑚^𝑛𝑥,𝑚^𝑛𝑦=0 (2.18)for all 𝑥,𝑦∈𝒜. So 𝐻(𝑥𝑦)=𝐻(𝑥)𝐻(𝑦) for all 𝑥,𝑦∈𝒜. Thus 𝐻∶𝒜→ℬ is a homomorphism, satisfying (2.8), as desired.
Also, by (2.4), (2.11), (2.16) and by a similar method, we have 𝐻(𝑥^∗)=𝐻(𝑥)^∗.

Corollary 2.2. Let 𝑟>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝒜→ℬ be a mapping such that 𝐷_𝜇𝑓𝑥₁,…,𝑥_𝑚ℬ≤𝜃⋅𝑥₁^𝑟_𝒜+𝑥₂^𝑟_𝒜+⋯+𝑥_𝑚^𝑟_𝒜,𝑓𝑥𝑦−𝑓(𝑥)𝑓𝑦_ℬ≤𝜃⋅‖𝑥‖^𝑟_𝒜⋅𝑦^𝑟_𝒜,𝑓𝑥^∗−𝑓(𝑥)^∗_ℬ≤𝜃⋅‖𝑥‖^𝑟_𝒜, (2.19) for all 𝜇∈𝕋¹ and 𝑥₁,…,𝑥_𝑚, 𝑥,𝑦∈𝒜. Then there exists a unique homomorphism 𝐻∶𝒜→ℬ such that 𝑓(𝑥)−𝐻(𝑥)_ℬ≤𝜃|𝑚|−|𝑚|^𝑟‖𝑥‖^r_𝒜 (2.20) for all 𝑥∈𝒜.

Proof. The proof follows from Theorem 2.1 by taking 𝜑𝑥₁,…,𝑥_𝑚=𝜃⋅𝑥₁^𝑟_𝒜+𝑥₂^𝑟_𝒜+⋯+𝑥_𝑚^𝑟_𝒜,𝜓𝑥,𝑦∶=𝜃⋅‖𝑥‖^𝑟_𝒜⋅𝑦^𝑟_𝒜,𝜂(𝑥)=𝜃⋅‖𝑥‖^𝑟_𝒜 (2.21)for all 𝑥₁,…,𝑥_𝑚,𝑥,𝑦∈𝐴, 𝐿=|𝑚|^𝑟−1 and so we get the desired result.

Note that a ℂ-linear mapping 𝛿∶𝒜→𝒜 is called a derivation on 𝒜 if 𝛿 satisfies 𝛿(𝑥𝑦)=𝛿(𝑥)𝑦+𝑥𝛿(𝑦) for all 𝑥,𝑦∈𝒜.

We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean 𝐶^∗-algebras for the functional equation 𝐷_𝜇𝑓(𝑥₁,…,𝑥_𝑚)=0.

Theorem 2.3. Let 𝑓∶𝒜→𝒜 be a mapping for which there are functions 𝜑∶𝒜^𝑚→[0,∞), 𝜓∶𝒜²→[0,∞) and 𝜂∶𝒜→[0,∞) such that |𝑚|<1 is far from zero and 𝐷_𝜇𝑓𝑥₁,…,𝑥_𝑚𝒜≤𝜑𝑥₁,…,𝑥_𝑚,𝑓𝑥𝑦−𝑓(𝑥)𝑦−𝑥𝑓𝑦_𝒜≤𝜓𝑥,𝑦,𝑓(𝑥^∗)−𝑓(𝑥)^∗_𝒜≤𝜂(𝑥) (2.22) for all 𝜇∈𝕋¹ and 𝑥₁,…,𝑥_𝑚,𝑥,𝑦∈𝐴. If there exists an 𝐿<1 such that (2.5), (2.6) and (2.7) hold, then there exists a unique derivation 𝛿∶𝒜→𝒜 such that ‖𝑓(𝑥)−𝛿(𝑥)‖_𝒜≤1𝑚|−|𝑚|𝐿)𝜑()(|𝑥,0,…,0 (2.23) for all 𝑥∈𝒜.

3. Stability of Homomorphisms and Derivations in Non-Archimedean Lie 𝐶^∗-Algebras

A non-Archimedean 𝐶^∗-algebra 𝒞, endowed with the Lie product𝑥,𝑦∶=𝑥𝑦−𝑦𝑥2 (3.1)

on 𝒞, is called a Lie non-Archimedean 𝐶^∗-algebra.

Definition 3.1. Let 𝒜 and ℬ be Lie 𝐶^∗-algebras. A ℂ-linear mapping 𝐻∶𝒜→ℬ is called a non-Archimedean Lie 𝐶^∗-algebra homomorphism if 𝐻([𝑥,𝑦])=[𝐻(𝑥),𝐻(𝑦)] for all 𝑥,𝑦∈𝒜.
Throughout this section, assume that 𝒜 is a non-Archimedean Lie 𝐶^∗-algebra with norm ‖⋅‖_𝒜 and ℬ is a non-Archimedean Lie 𝐶^∗-algebra with norm ‖⋅‖_ℬ.
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean Lie 𝐶^∗-algebras for the functional equation 𝐷_𝜇𝑓(𝑥₁,…,𝑥_𝑚)=0.

Theorem 3.2. Let 𝑓∶𝒜→ℬ be a mapping for which there are functions 𝜑∶𝒜^𝑚→[0,∞) and 𝜓∶𝒜²→[0,∞) such that (2.2) and (2.4) hold and 𝑓𝑥,𝑦−𝑓(𝑥),𝑓𝑦_ℬ≤𝜓𝑥,𝑦 (3.2) for all 𝜇∈𝕋¹ and 𝑥,𝑦∈𝒜. If there exists an 𝐿<1 and (2.5), (2.6), and (2.7) hold, then there exists a unique homomorphism 𝐻∶𝒜→ℬ such that (2.8) holds.

Proof. By the same reasoning as in the proof of Theorem 2.1, we can find the mapping 𝐻∶𝒜→ℬ given by 𝐻(𝑥)=lim_𝑛→∞𝑓𝑚^𝑛𝑥|𝑚|^𝑛 (3.3) for all 𝑥∈𝒜. It follows from (2.6) and (3.3) that 𝐻𝑥,𝑦−𝐻(𝑥),𝐻𝑦_ℬ=lim_𝑛→∞1|𝑚|^2𝑛𝑓𝑚^2𝑛𝑥,𝑦−[𝑓𝑚^𝑛𝑥,𝑓𝑚^𝑛𝑦_ℬ≤lim_𝑛→∞1|𝑚|^2𝑛𝜓𝑚^𝑛𝑥,𝑚^𝑛𝑦=0 (3.4) for all 𝑥,𝑦∈𝒜 and so 𝐻𝑥,𝑦=𝐻(𝑥),𝐻𝑦, (3.5) for all 𝑥,𝑦∈𝒜. Thus 𝐻∶𝒜→ℬ is a Lie 𝐶^∗-algebra homomorphism satisfying (2.8), as desired.

Corollary 3.3. Let 𝑟>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝒜→ℬ be a mapping such that 𝐷_𝜇𝑓𝑥₁,…,𝑥_𝑚ℬ≤𝜃𝑥₁^r_𝐴+𝑥₂^r_𝐴+⋯+𝑥_𝑚^r_𝐴,𝑓𝑥,𝑦−𝑓(𝑥),𝑓𝑦_ℬ≤𝜃⋅‖𝑥‖^r_𝐴⋅𝑦^r_𝐴,𝑓𝑥^∗−𝑓(𝑥)^∗_ℬ≤𝜃⋅‖𝑥‖^r_𝒜 (3.6) all 𝜇∈𝕋¹ and 𝑥₁,…,𝑥_𝑚,𝑥,𝑦∈𝒜. Then there exists a unique homomorphism 𝐻∶𝒜→ℬ such that 𝑓(𝑥)−𝐻(𝑥)_ℬ≤𝜃|𝑚|−|𝑚|^𝑟‖𝑥‖^r_𝒜 (3.7) for all 𝑥∈𝒜.

Proof. The proof follows from Theorem 3.2 and a method similar to Corollary 3.3.

Definition 3.4. Let 𝒜 be a non-Archimedean Lie 𝐶^∗-algebra. A ℂ-linear mapping 𝛿∶𝒜→𝒜 is called a Lie derivation if 𝛿([𝑥,𝑦])=[𝛿(𝑥),𝑦]+[𝑥,𝛿(𝑦)] for all 𝑥,𝑦∈𝒜.
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean Lie 𝐶^∗-algebras for the functional equation 𝐷_𝜇𝑓(𝑥₁,…,𝑥_𝑚)=0.

Theorem 3.5. Let 𝑓∶𝒜→𝒜 be a mapping for which there are functions 𝜑∶𝐴^𝑚→[0,∞) and 𝜓∶𝐴²→[0,∞) such that (2.2) and (2.4) hold and 𝑓𝑥,𝑦−𝑓(𝑥),𝑦−𝑥,𝑓𝑦_𝒜≤𝜓𝑥,𝑦 (3.8) for all 𝑥,𝑦∈𝒜. If there exists an 𝐿<1 and (2.5), (2.6) and (2.7) hold, then there exists a unique Lie derivation 𝛿∶𝒜→𝒜 such that such that (2.8) holds.

Proof. By the same reasoning as the proof of Theorem 2.3, there exists a unique ℂ-linear mapping 𝛿∶𝒜→𝒜 satisfying (2.8) and the mapping 𝛿∶𝒜→𝒜 is given by 𝛿(𝑥)=lim_𝑛→∞𝑓𝑚^𝑛𝑥|𝑚|^𝑛 (3.9) for all 𝑥∈𝒜.
It follows from (2.6) and (3.9) that 𝛿𝑥,𝑦−𝛿(𝑥),𝑦−𝑥,𝛿𝑦_𝒜=lim_𝑛→∞1|𝑚|^2𝑛𝑓𝑚^2𝑛𝑥,𝑦−𝑓𝑚^𝑛𝑥,𝑚^𝑛𝑦−𝑚^𝑛𝑥,𝑓𝑚^𝑛𝑦_𝒜≤lim_𝑛→∞1|𝑚|^2𝑛𝜓𝑚^𝑛𝑥,𝑚^𝑛𝑦=0, (3.10) for all 𝑥,𝑦∈𝒜 and so 𝛿𝑥,𝑦=𝛿(𝑥),𝑦+𝑥,𝛿𝑦 (3.11) for all 𝑥,𝑦∈𝒜. Thus 𝛿∶𝒜→𝒜 is a Lie derivation satisfying (2.8).

Acknowledgment

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0021821).