Approximately Ternary Homomorphisms and Derivations on 𝐶^∗-Ternary Algebras

Abstract

We investigate the stability and superstability of ternary homomorphisms between 𝐶^∗-ternary algebras and derivations on 𝐶^∗-ternary algebras, associated with the following functional equation 𝑓((𝑥₂−𝑥₁)/3)+𝑓((𝑥₁−3𝑥₃)/3)+𝑓((3𝑥₁+3𝑥₃−𝑥₂)/3)=𝑓(𝑥₁).

1. Introduction

A 𝐶^∗-ternary algebra is a complex Banach space 𝐴, equipped with a ternary product (𝑥,𝑦,𝑧)↣[𝑥,𝑦,𝑧] of 𝐴³ into 𝐴, which is ℂ-linear in the outer variables, conjugate ℂ-linear in the middle variable, and associative in the sense that [𝑥,𝑦,[𝑧,𝑤,𝑣]]=[𝑥,[𝑤,𝑧,𝑦],𝑣]=[[𝑥,𝑦,𝑧],𝑤,𝑣], and satisfies ‖[𝑥,𝑦,𝑧]‖≤‖𝑥‖⋅‖𝑦‖⋅‖𝑧‖ and ‖[𝑥,𝑥,𝑥]‖=‖𝑥‖³. If a 𝐶^∗-ternary algebra (𝐴,[⋅,⋅,⋅]) has an identity, that is, an element 𝑒∈𝐴 such that 𝑥=[𝑥,𝑒,𝑒]=[𝑒,𝑒,𝑥] for all 𝑥∈𝐴, then it is routine to verify that 𝐴, endowed with 𝑥𝑜𝑦∶=[𝑥,𝑒,𝑦] and 𝑥^∗∶=[𝑒,𝑥,𝑒], is a unital 𝐶^∗-algebra. Conversely, if (𝐴,𝑜) is a unital 𝐶^∗-algebra, then [𝑥,𝑦,𝑧]∶=𝑥𝑜𝑦^∗𝑜𝑧 makes 𝐴 into a 𝐶^∗-ternary algebra. A ℂ-linear mapping 𝐻∶𝐴→𝐵 is called a 𝐶^∗-ternary algebra homomorphism if𝐻𝑥,𝑦,𝑧=𝐻(𝑥),𝐻𝑦,𝐻(𝑧), (1.1) for all 𝑥, 𝑦, 𝑧∈𝐴. A ℂ-linear mapping 𝛿∶𝐴→𝐴 is called a 𝐶^∗-ternary algebra derivation if𝛿𝑥,𝑦,𝑧=𝛿(𝑥),𝑦,𝑧+𝑥,𝛿𝑦,𝑧+𝑥,𝑦,𝛿(𝑧), (1.2) for all 𝑥, 𝑦, 𝑧∈𝐴.

Ternary structures and their generalization the so-called 𝑛-ary structures raise certain hopes in view of their applications in physics (see [1–8]).

We say a functional equation 𝜁 is stable if any function 𝑔 satisfying the equation 𝜁 approximately is near to true solution of 𝜁. Moreover, 𝜁 is superstable if every approximately solution of 𝜁 is an exact solution of it.

The study of stability problems originated from a famous talk given by Ulam [9] in 1940: “Under what condition does there exist a homomorphism near an approximate homomorphism?” In the next year 1941, Hyers [10] answered affirmatively the question of Ulam for additive mappings between Banach spaces.

A generalized version of the theorem of Hyers for approximately additive maps was given by Rassias [11] in 1978 as follows.

Theorem 1.1. Let 𝑓∶𝐸₁→𝐸₂ be a mapping from a normed vector space 𝐸₁ into a Banach space 𝐸₂ subject to the inequality: 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝, (1.3) for all 𝑥, 𝑦∈𝐸₁, where 𝜖 and p are constants with 𝜖>0 and 𝑝<1. Then, there exists a unique additive mapping 𝑇∶𝐸₁→𝐸₂ such that 𝑓(𝑥)−𝑇(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝, (1.4) for all 𝑥∈E₁.

The stability phenomenon that was introduced and proved by Rassias is called Hyers-Ulam-Rassias stability. And then the stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [12–27]).

Throughout this paper, we assume that 𝐴 is a 𝐶^∗-ternary algebra with norm ‖⋅‖_𝐴 and that 𝐵 is a 𝐶^∗-ternary algebra with norm ‖⋅‖_𝐵. Moreover, we assume that 𝑛₀∈ℕ is a positive integer and suppose that 𝕋¹_1/𝑛𝑜∶={𝑒^𝑖𝜃;0≤𝜃≤2𝜋/𝑛_𝑜}.

2. Superstability

In this section, first we investigate homomorphisms between 𝐶^∗-ternary algebras. We need the following Lemma in the main results of the paper.

Lemma 2.1. Let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝑥₃3+𝑓3𝑥₁+3𝑥₃−𝑥₂3_𝐵≤𝑓𝑥_1𝐵, (2.1) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Then 𝑓 is additive.

Proof. Letting 𝑥₁=𝑥₂=𝑥₃=0 in (2.1), we get 3𝑓(0)_𝐵≤𝑓(0)_𝐵. (2.2) So 𝑓(0)=0. Letting 𝑥₁=𝑥₂=0 in (2.1), we get 𝑓−𝑥₃+𝑓𝑥_3𝐵≤𝑓(0)_𝐵=0, (2.3) for all 𝑥₃∈𝐴. Hence 𝑓(−𝑥₃)=−𝑓(𝑥₃) for all 𝑥₃∈𝐴. Letting 𝑥₁=0 and 𝑥₂=6𝑥₃ in (2.1), we get 𝑓2𝑥₃−2𝑓𝑥_3𝐵≤𝑓(0)_𝐵=0, (2.4) for all 𝑥₃∈𝐴. Hence 𝑓2𝑥₃=2𝑓𝑥₃, (2.5) for all 𝑥₃∈𝐴. Letting 𝑥₁=0 and 𝑥₂=9𝑥₃ in (2.1), we get 𝑓3𝑥₃−𝑓𝑥₃−2𝑓𝑥_3𝐵≤𝑓(0)_𝐵=0, (2.6) for all 𝑥₃∈𝐴. Hence 𝑓3𝑥₃=3𝑓𝑥₃, (2.7) for all 𝑥₃∈𝐴. Letting 𝑥₁=0 in (2.1), we get 𝑓𝑥₂3+𝑓−𝑥₃+𝑓𝑥₃−𝑥₂3_𝐵≤𝑓(0)_𝐵=0, (2.8) for all 𝑥₂, 𝑥₃∈𝐴. So 𝑓𝑥₂3+𝑓−𝑥₃+𝑓𝑥₃−𝑥₂3=0, (2.9) for all 𝑥₂, 𝑥₃∈𝐴. Let 𝑡₁=𝑥₃−(𝑥₂/3) and 𝑡₂=𝑥₂/3 in (2.9). Then 𝑓𝑡₂−𝑓𝑡₁+𝑡₂+𝑓𝑡₁=0, (2.10) for all 𝑡₁, 𝑡₂∈𝐴, this means that 𝑓 is additive.

Now, we prove the first result in superstability as follows.

Theorem 2.2. Let 𝑝≠1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3_𝐵≤𝑓𝑥_1𝐵,𝑓𝑥₁,𝑥₂,𝑥₃−[𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃]_𝐵≤𝜃𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝, (2.11)(2.12) for all 𝜇∈𝕋¹_1/𝑛𝑜 and all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Then, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra homomorphism.

Proof. Assume 𝑝>1.
Let 𝜇=1 in (2.11). By Lemma 2.1, the mapping 𝑓∶𝐴→𝐵 is additive. Letting 𝑥₁=𝑥₂=0 in (2.11), we get 𝑓−𝜇𝑥₃+𝜇𝑓𝑥_3𝐵≤𝑓(0)_𝐵=0, (2.13) for all 𝑥₃∈𝐴 and 𝜇∈𝕋¹. So −𝑓𝜇𝑥₃+𝜇𝑓𝑥₃=𝑓−𝜇𝑥₃+𝜇𝑓𝑥₃=0, (2.14) for all 𝑥₃∈𝐴 and all 𝜇∈𝕋¹. Hence 𝑓(𝜇𝑥₃)=𝜇𝑓(𝑥₃) for all 𝑥₃∈𝐴 and all 𝜇∈𝕋¹_1/𝑛𝑜. By same reasoning as proof of Theorem 2.2 of [28], the mapping 𝑓∶𝐴→𝐵 is ℂ-linear. It follows from (2.12) that 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥_3𝐵=lim_𝑛→∞8^𝑛𝑓𝑥₁,𝑥₂,𝑥₃2^𝑛⋅2^𝑛⋅2^𝑛−𝑓𝑥₁2^𝑛𝑥,𝑓²2^𝑛𝑥,𝑓³2^𝑛_𝐵≤lim_𝑛→∞8^𝑛𝜃8^𝑛𝑝𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝=0, (2.15) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Thus, 𝑓𝑥₁,𝑥₂,𝑥₃=𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₃, (2.16) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Hence, the mapping 𝑓∶𝐴→𝐵 is a 𝐶^∗-ternary algebra homomorphism. Similarly, one obtains the result for the case 𝑝<1.

Now, we establish the superstability of derivations on 𝐶^∗-ternary algebras as follows.

Theorem 2.3. Let 𝑝≠1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (2.11) such that 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑥₂,x₃−𝑥₁,𝑓𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝑓𝑥_3𝐴≤𝜃𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝, (2.17) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Then the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation.

Proof. Assume 𝑝>1.
By the Theorem 2.2, the mapping 𝑓∶𝐴→𝐴 is ℂ-linear. It follows from (2.17) that 𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝑓𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝑓𝑥_3𝐴=lim_𝑛→∞8^𝑛𝑓𝑥₁,𝑥₂,𝑥₃8^𝑛−𝑓𝑥₁2^𝑛,𝑥₂2^𝑛,𝑥₃2^𝑛−𝑥₁2^𝑛𝑥,𝑓²2^𝑛,𝑥₃2^𝑛−𝑥₁2^𝑛,𝑥₂2^𝑛𝑥,𝑓³2^𝑛_𝐴≤lim_𝑛→∞8^𝑛𝜃8^𝑛𝑝𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝=0, (2.18) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. So 𝑓𝑥₁,𝑥₂,𝑥₃=𝑓𝑥₁,𝑥₂,𝑥₃+𝑥₁,𝑓𝑥₂,𝑥₃+𝑥₁,𝑥₂,𝑓𝑥₃ (2.19) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Thus, the mapping 𝑓∶𝐴→𝐴 is a 𝐶^∗-ternary derivation. Similarly, one obtains the result for the case 𝑝<1.

3. Stability

First we prove the generalized Hyers-Ulam-Rassias stability of homomorphisms in 𝐶^∗-ternary algebras.

Theorem 3.1. Let 𝑝>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping such that 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝑓𝑥_1𝐵≤𝜃𝑥₁^𝑝_𝐴+𝑥₂^𝑝_𝐴+𝑥₃^𝑝_𝐴,𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥_3𝐵≤𝜃x_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝, (3.1)(3.2) for all 𝜇∈𝕋¹_1/𝑛𝑜, and all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Then there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥_1𝐵≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴1−3^1−𝑝, (3.3) for all 𝑥₁∈𝐴.

Proof. Let us assume 𝜇=1, 𝑥₂=2𝑥₁ and 𝑥₃=0 in (3.1). Then we get 𝑥3𝑓¹3−𝑓𝑥_1𝐵≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴, (3.4) for all 𝑥₁∈𝐴. So by induction, we have 3^𝑛𝑓𝑥₁3^𝑛−𝑓𝑥_1𝐵≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴^𝑛−1_𝑖=03^{𝑖(1−𝑝)}, (3.5) for all 𝑥₁∈𝐴. Hence 3^𝑛+𝑚𝑓𝑥₁3^𝑛+𝑚−3^𝑚𝑓𝑥₁3^𝑚_𝐵≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴^𝑛−1_𝑖=03^{(𝑖+𝑚)(1−𝑝)}≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴^{𝑛+𝑚−1}_𝑖=𝑚3^{𝑖(1−𝑝)}, (3.6) for all nonnegative integers 𝑚 and 𝑛 with 𝑛≥𝑚, and all 𝑥₁∈𝐴. It follows that the sequence {3^𝑛𝑓(𝑥₁/3^𝑛)} is a Cauchy sequence for all 𝑥₁∈𝐴. Since 𝐵 is complete, the sequence {3^𝑛𝑓(𝑥₁/3^𝑛)} converges. Thus, one can define the mapping 𝐻∶𝐴→𝐵 by 𝐻𝑥₁∶=lim_𝑛→∞3^𝑛𝑓𝑥₁3^𝑛, (3.7) for all 𝑥₁∈𝐴. Moreover, letting 𝑚=0 and passing the limit 𝑛→∞ in (3.6), we get (3.3). It follows from (3.1) that 𝐻𝑥₂−𝑥₁3𝑥+𝐻¹−3𝜇𝑥₃3+𝜇𝐻3𝑥₁+3𝑥₃−𝑥₂3−𝐻𝑥_1𝐵=lim_𝑛→∞3^𝑛𝑓𝑥₂−𝑥₁3^𝑛+1𝑥+𝑓¹−3𝜇𝑥₃3^𝑛+1+𝑓3𝑥₁+3𝑥₃−𝑥₂3^𝑛+1𝑥−𝑓¹3^𝑛_𝐵≤lim_𝑛→∞3^𝑛𝜃3^𝑛𝑝𝑥₁^𝑝_𝐴+𝑥₂^𝑝_𝐴+𝑥₃^𝑝_𝐴=0, (3.8) for all 𝜇∈𝕋¹_1/𝑛𝑜, and all 𝑥₁,𝑥₂,𝑥₃∈𝐴. So 𝐻𝑥₂−𝑥₁3𝑥+𝐻¹−3𝜇𝑥₃3+𝜇𝐻3𝑥₁+3𝑥₃−𝑥₂3=𝐻𝑥₁, (3.9) for all 𝜇∈𝕋¹_1/𝑛𝑜, and all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. By the same reasoning as proof of Theorem 2.2 of [28], the mapping 𝐻∶𝐴→𝐵 is ℂ-linear.
Now, let 𝐻^′∶𝐴→𝐵 be another additive mapping satisfying (3.3). Then, we have 𝐻𝑥₁−𝐻^′𝑥_1𝐵=3^𝑛𝐻𝑥₁3^𝑛−𝐻^′𝑥₁3^𝑛_𝐵≤3^𝑛𝐻𝑥₁3^𝑛𝑥−𝑓¹3^𝑛_𝐵+𝐻^′𝑥₁3^𝑛𝑥−𝑓¹3^𝑛_𝐵≤2⋅3^𝑛𝜃1+2^𝑝3^𝑛𝑝1−3^1−𝑝‖𝑥‖^𝑝_𝐴, (3.10) which tends to zero as 𝑛→∞ for all 𝑥₁∈𝐴. So we can conclude that 𝐻(𝑥₁)=𝐻^′(𝑥₁) for all 𝑥₁∈𝐴. This proves the uniqueness of 𝐻.
It follows from (3.2) that 𝐻𝑥₁,𝑥₂,𝑥₃−𝐻𝑥₁,𝐻𝑥₂,𝐻𝑥_3𝐵=lim_𝑛→∞27^𝑛𝑓𝑥₁,𝑥₂,𝑥₃3^𝑛⋅3^𝑛⋅3^𝑛−𝑓𝑥₁3^𝑛𝑥,𝑓²3^𝑛𝑥,𝑓³3^𝑛_𝐵≤lim_𝑛→∞27^𝑛𝜃27^𝑛𝑝𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝=0, (3.11) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴.
Thus, the mapping 𝐻∶𝐴→𝐵 is a unique 𝐶^∗-ternary homomorphism satisfying (3.3).

Theorem 3.2. Let 𝑝<1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐵 be a mapping satisfying (3.1) and (3.2). Then, there exists a unique 𝐶^∗-ternary homomorphism 𝐻∶𝐴→𝐵 such that 𝐻𝑥₁−𝑓𝑥_1𝐵≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴3^1−𝑝,−1 (3.12) for all 𝑥₁∈𝐴.

Proof. The proof is similar to the proof of Theorem 3.1.

Now, we prove the generalized Hyers-Ulam-Rassias stability of derivations on 𝐶^∗-ternary algebras.

Theorem 3.3. Let 𝑝>1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping such that 𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝜇𝑥₃3+𝜇𝑓3𝑥₁+3𝑥₃−𝑥₂3−𝑓𝑥_1𝐴≤𝜃𝑥₁^𝑝_𝐴+𝑥₂^𝑝_𝐴+𝑥₃^𝑝_𝐴,𝑓𝑥₁,𝑥₂,𝑥₃−𝑓𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝑓𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝑓𝑥_3𝐴≤𝜃𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝, (3.13)(3.14) for all 𝜇∈𝕋¹_1/𝑛𝑜, and all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥_1𝐴≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴1−3^1−𝑝, (3.15) for all 𝑥₁∈𝐴.

Proof. By the same reasoning as in the proof of the Theorem 3.1, there exists a unique ℂ-linear mapping 𝐷∶𝐴→𝐴 satisfying (3.15). The mapping 𝐷∶𝐴→𝐴 is defined by 𝐷𝑥₁∶=lim_𝑛→∞3^𝑛𝑓𝑥₁3^𝑛, (3.16) for all 𝑥₁∈𝐴. It follows from (3.14) that 𝐷𝑥₁,𝑥₂,𝑥₃−𝐷𝑥₁,𝑥₂,𝑥₃−𝑥₁,𝐷𝑥₂,𝑥₃−𝑥₁,𝑥₂,𝐷𝑥_3𝐴=lim_𝑛→∞27^𝑛𝑥₁,𝑥₂,𝑥₃3^𝑛⋅3^𝑛⋅3^𝑛−𝑓𝑥₁3^𝑛,𝑥₂3^𝑛,𝑥₃3^𝑛−𝑥₁3^𝑛𝑥,𝑓²3^𝑛,𝑥₃3^𝑛−𝑥₁3^𝑛,𝑥₂3^𝑛𝑥,𝑓³3^𝑛_𝐴≤lim_𝑛→∞27^𝑛𝜃27^𝑛𝑝𝑥_1𝐴^3𝑝+𝑥_2𝐴^3𝑝+𝑥_3𝐴^3𝑝=0, (3.17) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴. So 𝐷𝑥₁,𝑥₂,𝑥₃=𝐷𝑥₁,𝑥₂,𝑥₃+𝑥₁,𝐷𝑥₂,𝑥₃+𝑥₁,𝑥₂,𝐷𝑥₃ (3.18) for all 𝑥₁, 𝑥₂, 𝑥₃∈𝐴.
Thus, the mapping 𝐷∶𝐴→𝐴 is a unique 𝐶^∗-ternary derivation satisfying (3.15).

Theorem 3.4. Let 𝑝<1 and 𝜃 be nonnegative real numbers, and let 𝑓∶𝐴→𝐴 be a mapping satisfying (3.13)and (3.14). Then, there exists a unique 𝐶^∗-ternary derivation 𝐷∶𝐴→𝐴 such that 𝐷𝑥₁−𝑓𝑥_1𝐴≤𝜃1+2^𝑝𝑥₁^𝑝_𝐴3^1−𝑝,−1 (3.19) for all 𝑥₁∈𝐴.

Proof. The proof is similar to the proof of Theorems 3.1 and 3.3.

4. Conclusions

In this paper, we have analyzed some detail 𝐶^∗-ternary algebras and derivations on 𝐶^∗-ternary algebras, associated with the following functional equation:𝑓𝑥₂−𝑥₁3𝑥+𝑓¹−3𝑥₃3+𝑓3𝑥₁+3𝑥₃−𝑥₂3=𝑓𝑥₁. (4.1) A detailed study of how we can have the generalized Hyers-Ulam-Rassias stability of homomorphisms and derivations on 𝐶^∗-ternary algebras is given.

Acknowledgment

This paper was partially supported by Research Program of Payame Noor University of Tabriz.