Nearly Derivations on Banach Algebras

Abstract

Let 𝑛 be a fixed integer greater than 3 and let 𝜆 be a real number with 𝜆≠(𝑛²−𝑛+4)/2. We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality ‖∑_{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝑓((𝑥_𝑖+𝑥_𝑗)/2+∑^𝑛−2_𝑙=1𝑥_𝑘𝑙)+𝑓(∑^𝑛_𝑖=2𝑥_𝑖)+𝑓(𝑥₁)‖≤‖𝜆𝑓(∑^𝑛_𝑖=1𝑥_𝑖)‖.

1. Introduction and Preliminaries

Let 𝒜 ba a Banach algebra and let 𝑋 be a Banach 𝒜-bimodule. Then 𝑋^∗, the dual space of 𝑋, is also a Banach 𝒜-bimodule with module multiplications defined by 𝑥,𝑎⋅𝑥^∗=𝑥⋅𝑎,𝑥^∗,𝑥,𝑥^∗⋅𝑎=𝑎⋅𝑥,𝑥^∗,𝑎∈𝒜,𝑥∈𝑋,𝑥^∗∈𝑋^∗. (1.1) A bounded linear operator 𝐷∶𝒜→𝑋 is called a derivation if𝐷(𝑎𝑏)=𝑎⋅𝐷(𝑏)+𝐷(𝑎)⋅𝑏(𝑎,𝑏∈𝒜). (1.2)

Let 𝑥∈𝑋. We define 𝛿_𝑥(𝑎)=𝑎⋅𝑥−𝑥⋅𝑎 for all 𝑎∈𝒜. 𝛿_𝑥 is a derivation from 𝒜 into 𝑋, which is called inner derivation. A Banach algebra 𝒜 is amenable if every derivation from 𝒜 into every dual 𝒜-bimodule 𝑋^∗ is inner. This definition was introduced by Johnson in [1]. A Banach algebra 𝒜 is weakly amenable if every derivation from 𝒜 into 𝒜^∗ is inner. Bade et al. [2] have introduced the concept of weak amenability for commutative Banach algebras.

The stability problem of functional equations originated from a question of Ulam [3, 4] concerning the stability of group homomorphisms.

A famous talk presented by Ulam in 1940 triggered the study of stability problems for various functional equations.

We are given a group 𝐺₁ and a metric group 𝐺₂ with metric 𝜌(⋅,⋅). Given 𝜖>0, does there exist a 𝛿>0 such that if 𝑓∶𝐺₁→𝐺₂ satisfies 𝜌(𝑓(𝑥𝑦),𝑓(𝑥)𝑓(𝑦))<𝛿 for all 𝑥,𝑦∈𝐺₁, then a homomorphism ℎ∶𝐺₁→𝐺₂ exists with 𝜌(𝑓(𝑥),ℎ(𝑥))<𝜖 for all 𝑥∈𝐺₁?

In the following year, Hyers was able to give a partial solution to Ulam’s question that was the first significant breakthrough and step toward more solutions in this area (see [5]). Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’ theorem.

Let 𝑛 be a fixed integer greater than 3 and let 𝜆 be a real number with |𝜆|≠(𝑛²−𝑛+4)/2. We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality:_{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝑓𝑥_𝑖+𝑥_𝑗2+^𝑛−2_𝑙=1𝑥_𝑘𝑙+𝑓^𝑛_𝑖=2𝑥_𝑖+𝑓𝑥₁≤𝜆𝑓^𝑛_𝑖=1𝑥_𝑖. (1.3)

2. Main Results

Let 𝐴 be a Banach algebra and let 𝑋 be a Banach 𝐴-module. From now on, the sum of 𝑓(𝑥) and 𝑓(−𝑥) will be denoted by 𝑓(𝑥). Also, 𝑓(𝑎𝑏)−𝑓(𝑎)𝑏−𝑎𝑓(𝑏) will be denoted by Δ𝑓(𝑎,𝑏). In the following, we will use the Pascal formula: 𝐶(𝑟,𝑘)=𝐶(𝑟−1,𝑘)+𝐶(𝑟−1,𝑘−1) (2.1) here, 𝐶(𝑟,𝑘) denotes 𝑟!/𝑘!(𝑟−𝑘)! Moreover, we assume that 𝑛₀∈ℕ is a positive integer and suppose that 𝕋¹_1/𝑛𝑜∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛_𝑜}.

Lemma 2.1. Let 𝑓∶𝐴→𝑋 be a mapping such that _{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝑓𝑥_𝑖+𝑥_𝑗2+^𝑛−2_𝑙=1𝑥_𝑘𝑙+𝑓^𝑛_𝑖=2𝑥_𝑖+𝑓𝑥₁≤𝜆𝑓^𝑛_𝑖=1𝑥_𝑖 (2.2) for all 𝑥₁,…,𝑥_𝑛∈𝐴. Then 𝑓 is Cauchy additive.

Proof. Substituting 𝑥₁,…,𝑥_𝑛=0 in the functional inequality (2.2), we get (𝐶(𝑛,2)+2)𝑓(0)≤𝜆𝑓(0). (2.3) Since 𝑛≥3 and |𝜆|≠(𝑛²−𝑛+4)/2, 𝑓(0)=0. Letting 𝑥₁=𝑥, 𝑥₂=−𝑥 and 𝑥₃=⋯=𝑥_𝑛=0 in (2.2) and using Pascal formula, we get )𝑓𝑥(𝑛−22+(𝐶(𝑛−2,2)+1)𝑓(0)+𝑓(𝑥)≤𝜆𝑓(0), (2.4) for all 𝑥∈𝐴. Hence ()𝑓𝑥𝑛−22+𝑓(𝑥)=0 (2.5) for all 𝑥∈𝐴. Letting 𝑥₁=2𝑥, 𝑥₂=−𝑥, 𝑥₃=−𝑥 and 𝑥₄=⋯=𝑥_𝑛=0 in (2.2), we get 2𝑓−𝑥2+()𝑓()(𝑥)()𝑓𝑥𝑛−3−𝑥+𝑓+2𝑛−32+𝐶(𝑛−3,2)𝑓(0)+𝑓(2𝑥)≤𝜆𝑓(0) (2.6) for all 𝑥∈𝐴. Hence 2𝑓−𝑥2+()𝑓()(𝑥)()𝑓𝑥𝑛−3−𝑥+𝑓+2𝑛−32+𝑓()𝑥2𝑥=0,2𝑓2+(𝑛−3)𝑓(𝑥)+𝑓(−𝑥)+2(𝑛−3)𝑓−𝑥2+𝑓(−2𝑥)=0 (2.7) for all 𝑥∈𝐴. Since 𝑓(−𝑥)=𝑓(𝑥), we obtain from (2.7) and (2.4) that 2()𝑓𝑥𝑛−22+(𝑛−2)𝑓(𝑥)+2𝑓(2𝑥)=0 (2.8) for all 𝑥∈𝐴. It follows from (2.5) and (2.8) that 2𝑓𝑥2−𝑓(𝑥)=0 (2.9) for all 𝑥∈𝐴. By using (2.5) and (2.9), we get 𝑛𝑓(𝑥/2)=0 and so 𝑓(−𝑥)=−𝑓(𝑥) for all 𝑥∈𝐴. Hence, we obtain from (2.7) that 𝑓(𝑥/2)=(1/2)𝑓(𝑥) for all 𝑥∈𝐴. Letting 𝑥₁=𝑥+𝑦, 𝑥₂=−𝑥, 𝑥₃=−𝑦 and 𝑥₄=⋯=𝑥_𝑛=0 in (2.2), we get 𝑓−𝑦2+𝑓−𝑥2+(𝑛−3)𝑓−𝑥−𝑦2+𝑓𝑥+𝑦2+()𝑓𝑥𝑛−32+()𝑓𝑦𝑛−32+𝐶(𝑛−3,2)𝑓(0)+𝑓𝑥+𝑦≤𝜆𝑓(0) (2.10) for all 𝑥,𝑦∈𝐴. Next, notice that, using oddness of 𝑓 and 𝑓(𝑥/2)=(1/2)𝑓(𝑥), we have 𝑓𝑥+𝑦=𝑓(𝑥)+𝑓𝑦 (2.11) for all 𝑥,𝑦∈𝐴, as desired.

We can prove the following lemma by the same reasoning as in the proof of Theorem  2.2 of [6].

Lemma 2.2. Let 𝑓∶𝐴→𝑋 be an additive mapping such that 𝑓(𝜇𝑥)=𝜇𝑓(𝑥) for all 𝜇∈𝑇¹_1/𝑛𝑜 and all 𝑥∈𝐴. Then the mapping 𝑓 is ℂ-linear.

Theorem 2.3. Let 𝑓∶𝐴→𝑋 be a mapping satisfying 𝑓(0)=0 and the inequality _{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝑓𝜇𝑥_𝑖+𝜇𝑥_𝑗2+^𝑛−2_𝑙=1𝜇𝑥_𝑘𝑙+𝑓^𝑛_𝑖=2𝜇𝑥_𝑖+𝜇𝑓𝑥₁+Δ𝑓(𝑎,𝑏)≤𝜆𝑓^𝑛_𝑖=1𝜇𝑥_𝑖+𝛿 (2.12) for some 𝛿>0, for all 𝜇∈𝑇¹_1/𝑛𝑜 and all 𝑎,𝑏,𝑥₁,…,𝑥_𝑛∈𝐴. Then there exists a unique derivation 𝒟∶𝐴→𝑋 such that 𝑓(𝑥)−𝒟(𝑥)≤13𝑛−24𝑛(𝑛−4)𝛿 (2.13) for all 𝑥∈𝐴.

Proof. Letting 𝑎=𝑏=0, 𝑥₁=2𝑥, 𝑥₂=−2𝑥, 𝑥₃=⋯=𝑥_𝑛=0 and 𝜇=1 in (2.12), we get (𝑛−2)𝑓(𝑥)+𝑓(2𝑥)≤𝛿 (2.14) for all 𝑥∈𝑋. Letting 𝑎=𝑏=0, 𝑥₁=2𝑥, 𝑥₂=−𝑥, 𝑥₃=−𝑥, 𝑥₄=⋯=𝑥_𝑛=0 and 𝜇=1 in (2.12), we get 2𝑓−𝑥2+()𝑓()(𝑥)()𝑓𝑥𝑛−3−𝑥+𝑓+2𝑛−32+𝑓(2𝑥)≤𝛿 (2.15) for all 𝑥∈𝑋. Letting 𝑎=𝑏=0, 𝑥₁=−2𝑥, 𝑥₂=𝑥, 𝑥₃=𝑥, 𝑥₄=⋯=𝑥_𝑛=0 and 𝜇=1 in (2.12), we get 𝑥2𝑓2+(𝑛−3)𝑓(𝑥)+𝑓(−𝑥)+2(𝑛−3)𝑓−𝑥2+𝑓(−2𝑥)≤𝛿 (2.16) for all 𝑥∈𝑋. It follows from (2.15) and (2.16) that )𝑓𝑥(𝑛−22+(𝑛−2)2𝑓(𝑥)+𝑓(2𝑥)≤𝛿 (2.17) for all 𝑥∈𝑋. It follows from (2.14) and (2.17) that 𝑓(𝑥≤6)𝑛𝛿 (2.18) for all 𝑥∈𝑋. It follows from (2.15) and (2.18) that 2𝑓𝑥2+𝑓(𝑥)+()𝑓()()𝑓𝑥𝑛−4−𝑥+2𝑛−42≤𝑛+6𝑛𝛿 (2.19) for all 𝑥∈𝑋. From the last two inequalities, we have 𝑓(2𝑥)+2𝑓(−𝑥)≤𝑛+24𝑛(𝑛−4)𝛿 (2.20) for all 𝑥∈𝑋. It follows from (2.18) and (2.20) that 𝑓(𝑥)−12𝑓(2𝑥)≤13𝑛−242𝑛(𝑛−4)𝛿 (2.21) for all 𝑥∈𝑋. Hence 12^𝑟𝑓2^𝑟𝑥−12^𝑚𝑓2^𝑚𝑥≤13𝑛−242𝑛(𝑛−4)^𝑚−1_𝑘=𝑟𝛿2^𝑘 (2.22) for all 𝑥∈𝑋 and integers 𝑚>𝑟≥0. Thus it follows that a sequence {(1/2^𝑚)𝑓(2^𝑚𝑥)} is Cauchy in 𝑌 and so it converges. Therefore we can define a mapping 𝒟∶𝑋→𝑌 by 𝒟(𝑥)∶=lim_𝑚→∞(1/2^𝑚)𝑓(2^𝑚𝑥) for all 𝑥∈𝑋. In addition it is clear from (2.12) that the following inequality: _{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝒟𝜇𝑥_𝑖+𝜇𝑥_𝑗2+^𝑛−2_𝑙=1𝜇𝑥_𝑘𝑙+𝒟^𝑛_𝑖=2𝜇𝑥_𝑖+𝜇𝒟𝑥₁=lim_𝑚→∞12^𝑚_{1≤𝑖<𝑗≤𝑛1≤𝑘𝑙≠𝑖,𝑗≤𝑛}𝑓2^𝑚−1𝜇𝑥_𝑖+𝑥_𝑗+^𝑛−2_𝑙=12^𝑚𝜇𝑥_𝑘𝑙+𝑓^𝑛_𝑖=22^𝑚𝜇𝑥_𝑖+𝜇𝑓2^𝑚𝑥₁≤lim_𝑚→∞12^𝑚𝜆𝑓^𝑛_𝑖=12^𝑚𝜇𝑥_𝑖+lim_𝑚→∞𝛿2^𝑚=𝜆𝒟^𝑛_𝑖=1𝜇𝑥_𝑖 (2.23) holds for all 𝜇∈𝑇¹_1/𝑛𝑜 and all 𝑥₁,…,𝑥_𝑛∈𝑋. If we put 𝜇=1 in the last inequality, then 𝒟 is additive by Lemma 2.1. Letting 𝑥₁=𝑥, 𝑥₂=−𝑥 and 𝑥₃=⋯=𝑥_𝑛=0 in last inequality and using Lemma 2.1, we get (𝑛−2)𝒟𝜇𝑥2+𝒟−𝜇𝑥+𝜇𝒟(𝑥)=𝜇𝒟(𝑥)−𝒟𝜇𝑥. (2.24) So 𝒟(𝜇𝑥)=𝜇𝒟(𝑥) for all 𝑥∈𝑋 and all 𝜇∈𝑇¹_1/𝑛𝑜. Now by using Lemmas 2.1 and 2.2, we infer that the mapping 𝒟∶𝑋→𝑌 is ℂ-linear. Taking the limit as 𝑚→∞ in (2.22) with 𝑟=0, we get (2.13).
To prove the afore-mentioned uniqueness, we assume now that there is another ℂ-linear mapping 𝔏∶𝐴→𝑋 which satisfies the inequality (2.13). Then we get 12^𝑚𝑓2^𝑚𝑥(𝑥=1−𝔏)2^𝑚𝑓2^𝑚𝑥−𝔏2^𝑚𝑥≤13𝑛−242^𝑚𝑛(𝑛−4)𝛿 (2.25) for all 𝑥∈𝐴 and integers 𝑚≥1. Thus from 𝑚→∞, one establishes 𝒟(𝑥)−𝔏(𝑥)=0 (2.26) for all 𝑥∈𝐴, completing the proof of uniqueness.
Now, we have to show that 𝒟 is a derivation. To this end, let 𝑥₁=𝑥₂=⋯=𝑥_𝑛=0 in (2.12), we get 𝑓(𝑎𝑏)−𝑓(𝑎)𝑏−𝑎𝑓(𝑏)≤𝛿 (2.27) for all 𝑎,𝑏∈𝐴. It follows from linearity of 𝒟 and (2.27) that ‖𝒟()(𝑎)(𝑏=1𝑎𝑏−𝒟𝑏−𝑎𝒟)‖2^𝑚𝒟2^𝑚(𝑎)1𝑎𝑏−𝒟2^𝑚2^𝑚𝑏−12^𝑚2^𝑚𝑎𝒟(𝑏)=lim_𝑚→∞14^𝑚𝑓4^𝑚𝑎𝑏−𝑓2^𝑚𝑎14^𝑚2^𝑚𝑏−14^𝑚2^𝑚𝑎𝑓2^𝑚𝑏=lim_𝑚→∞14^𝑚𝑓2^𝑚𝑎2^𝑚𝑏−𝑓2^𝑚𝑎2^𝑚𝑏−2^𝑚𝑎𝑓2^𝑚𝑏≤lim_𝑚→∞14^𝑚𝛿=0 (2.28) for all 𝑎,𝑏∈𝐴. This means that 𝒟 is a derivation from 𝐴 into 𝑋. Therefore the mapping 𝒟∶𝐴→𝑋 is a unique derivation satisfying (2.13), as desired.

Theorem 2.4. Let 𝐴 be an amenable Banach algebra and let 𝑓∶𝐴→𝑋^∗ be a mapping such that 𝑓(0)=0 and (2.12). If sup𝑓(𝑥)∶‖𝑥‖≤1<∞, (2.29) then there exists 𝑥₀∈𝑋^∗ such that 𝑓(𝑎)−𝑎𝑥₀−𝑥₀𝑎≤13𝑛−24𝑛(𝑛−4)𝛿 (2.30) for all 𝑎∈𝐴.

Proof. Let sup{‖𝑓(𝑥)‖∶‖𝑥‖≤1}=𝑀_𝑓. Then by (2.29), we have 𝑀_𝑓<∞. By Theorem 2.3, there exists a derivation 𝐷∶𝐴→𝑋^∗ satisfying (2.13). Then we have sup{‖𝐷(𝑥)‖∶‖𝑥‖≤1}≤𝑀_𝑓+13𝑛−24𝑛(𝑛−4)𝛿. (2.31) This means that 𝐷 is bounded, and hence 𝐷 is continuous. On the other hand, 𝐴 is amenable. Then every continuous derivation from 𝐴 into 𝑋^∗ is an inner derivation. It follows that 𝐷 is and an inner derivation. In the other words, there exists 𝑥₀∈𝑋^∗ such that 𝐷(𝑎)=𝑎𝑥₀−𝑥₀𝑎 for all 𝑎∈𝐴. This completes the proof.

We know that every nuclear 𝐶^∗-algebra is amenable (see [7]). Then we have the following result.

Corollary 2.5. Let 𝐴 be a nuclear 𝐶^∗-algebra and let 𝑓∶𝐴→𝑋^∗ be a mapping such that 𝑓(0)=0, and (2.12) and (2.29). Then there exists 𝑥₀∈𝑋^∗ such that 𝑓(𝑎)−𝑎𝑥₀−𝑥₀𝑎≤13𝑛−24𝑛(𝑛−4)𝛿 (2.32) for all 𝑎∈𝐴.

Theorem 2.6. Let 𝐴 be a 𝐶^∗-algebra and let 𝑓∶𝐴→𝐴^∗ be a mapping such that 𝑓(0)=0, and (2.12) and (2.29). Then there exists 𝑎^′∈𝐴^∗ such that 𝑓(𝑎)(𝑏)−𝑎^′(𝑏𝑎−𝑎𝑏)≤13𝑛−24𝑛()𝛿‖𝑏‖𝑛−4 (2.33) for all 𝑎,𝑏∈𝐴.

Proof. We know that every 𝐶^∗-algebra is weakly amenable (see, e.g., [7]). Then every continuous derivation from 𝐴 into 𝐴^∗ is an inner derivation. By the same reasoning as in the proof of Theorem 2.4, there exists a 𝑎′∈𝐴^∗ such that 𝐷(𝑎)=𝑎𝑎′−𝑎′𝑎 for all 𝑎∈𝐴, and 𝑓(𝑎)−𝑎𝑎′−𝑎′𝑎≤13𝑛−24𝑛(𝑛−4)𝛿 (2.34) for all 𝑎∈𝐴. By definition of mudule actions of 𝐴 on 𝐴^∗, we have 𝑓(𝑎)(𝑏)−𝑎^′(𝑏𝑎−𝑎𝑏)≤13𝑛−24𝑛()𝛿‖𝑏‖𝑛−4 (2.35) for all 𝑎,𝑏∈𝐴.

Corollary 2.7. Let 𝐴 be a commutative 𝐶^∗-algebra and let 𝑓∶𝐴→𝐴^∗ be a mapping such that 𝑓(0)=0, and (2.12) and (2.29). Then lim_𝑚→∞12^𝑚𝑓2^𝑚𝑎𝑓(𝑎≤=0,)13𝑛−24𝑛()𝛿𝑛−4 (2.36) for all 𝑎∈𝐴.