Nearly Ring Homomorphisms and Nearly Ring Derivations on Non-Archimedean Banach Algebras

Abstract

We prove the generalized Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras. Moreover, we prove the superstability of homomorphisms on unital non-Archimedean Banach algebras and we investigate the superstability of derivations in non-Archimedean Banach algebras with bounded approximate identity.

1. Introduction and Preliminaries

In 1897, Hensel [1] has introduced a normed space which does not have the Archimedean property.

During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings, and superstrings [2]. Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are essentially different and require an entirely new kind of intuition [3–9].

Let 𝕂 be a field. A non-Archimedean absolute value on 𝕂 is a function |⋅|∶𝕂→ℝ such that for any 𝑎,𝑏∈𝕂 we have (i)|𝑎|≥0 and equality holds if and only if 𝑎=0,(ii)|𝑎𝑏|=|𝑎||𝑏|, (iii)|𝑎+𝑏|≤max{|𝑎|,|𝑏|}.

Condition (iii) is called the strict triangle inequality. By (ii), we have |1|=|−1|=1. Thus, by induction, it follows from (iii) that |𝑛|≤1 for each integer 𝑛. We always assume in addition that |⋅| is non trivial, that is, that there is an 𝑎₀∈𝕂 such that |𝑎₀|∉{0,1}.

Let 𝑋 be a linear space over a scalar field 𝕂 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→ℝ is a non-Archimedean norm (valuation) if it satisfies the following conditions: (NA1)‖𝑥‖=0 if and only if 𝑥=0; (NA2)‖𝑟𝑥‖=|𝑟|‖𝑥‖ for all 𝑟∈𝕂 and 𝑥∈𝑋; (NA3)the strong triangle inequality (ultrametric), namely, 𝑥+𝑦≤max‖𝑥‖,𝑦𝑥,𝑦∈𝑋. (1.1)

Then (𝑋,‖⋅‖) is called a non-Archimedean space.

It follows from (NA3) that 𝑥_𝑚−𝑥_𝑙≤max𝑥_𝚥+1−𝑥_𝚥∶𝑙≤𝚥≤𝑚−1(𝑚>𝑙), (1.2) therefore a sequence {𝑥_𝑚} is Cauchy in 𝑋 if and only if {𝑥_𝑚+1−𝑥_𝑚} converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. 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 can refer to [10].

The first stability problem concerning group homomorphisms was raised by Ulam [11] in 1960 and affirmatively solved by Hyers [12]. Perhaps Aoki was the first author who has generalized the theorem of Hyers (see [13]).

T. M. Rassias [14] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded.

Theorem 1.1 (T. M. Rassias). Let 𝑓∶𝐸→𝐸′ be a mapping from a normed vector space 𝐸 into a Banach space 𝐸^′ subject to the inequality 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜖‖𝑥‖^𝑝+𝑦^𝑝 (1.3) for all 𝑥,𝑦∈𝐸, where 𝜖 and 𝑝 are constants with 𝜖>0 and 𝑝<1. Then the limit 𝐿(𝑥)=lim_𝑛→∞𝑓2^𝑛𝑥2^𝑛 (1.4) exists for all 𝑥∈𝐸 and 𝐿∶𝐸→𝐸^′ is the unique additive mapping which satisfies 𝑓(𝑥)−𝐿(𝑥)≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.5) for all 𝑥∈𝐸. Also, if for each 𝑥∈𝐸 the mapping 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ, then 𝐿 is ℝ-linear.

Moreover, Bourgin [15] and Găvruţa [16] have considered the stability problem with unbounded Cauchy differences (see also [17–27]).

On the other hand, J. M. Rassias [28–33] considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa [34]. This stability phenomenon is called the Ulam-Găvruta-Rassias stability (see also [35]).

Theorem 1.2 (J. M. Rassias [28]). Let 𝑋 be a real normed linear space and 𝑌 a real complete normed linear space. Assume that 𝑓∶𝑋→𝑌 is an approximately additive mapping for which there exist constants 𝜃≥0 and 𝑝,𝑞∈ℝ such that 𝑟=𝑝+𝑞≠1 and 𝑓 satisfies the inequality 𝑓𝑥+𝑦−𝑓(𝑥)−𝑓𝑦≤𝜃‖𝑥‖^𝑝𝑦^𝑞 (1.6) for all 𝑥,𝑦∈𝑋. Then there exists a unique additive mapping 𝐿∶𝑋→𝑌 satisfying 𝑓(𝑥)(𝑥≤𝜃−𝐿)|2^𝑟|‖𝑥‖−2^𝑟 (1.7) for all 𝑥∈𝑋. If, in addition, 𝑓∶𝑋→𝑌 is a mapping such that the transformation 𝑡↦𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥∈𝑋, then 𝐿 is an ℝ-linear mapping.

Very recently, Ravi et al. [36] in the inequality (1.6) replaced the bound by a mixed one involving the product and sum of powers of norms, that is, 𝜃{‖𝑥‖^𝑝‖𝑦‖^𝑝+(‖𝑥‖^2𝑝+‖𝑦‖^2𝑝)}.

For more details about the results concerning such problems and mixed product-sum stability (J. M.-Rassias Stability) the reader is referred to [37–49].

Khodaei and T. M. Rassias [50] have established the general solution and investigated the Hyers-Ulam-Rassias stability of the following 𝑛-dimensional additive functional equation:^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑎_𝑖𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑎_𝑖𝑟𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖=2^𝑛−1𝑎₁𝑓𝑥₁, (1.8) where 𝑎₁,…,𝑎_𝑛∈ℤ−{0} with 𝑎₁≠±1.

In this paper, we investigate the Hyers-Ulam stability of homomorphisms and derivations associated with functional equation (1.8).

2. Main Results

Before taking up the main subject, for a given 𝑓∶𝒜→ℬ between vector spaces, we define the difference operator 𝐷𝑓𝑥₁,…,𝑥_𝑛∶=^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑎_𝑖𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑎_𝑖𝑟𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖−2^𝑛−1𝑎₁𝑓𝑥₁. (2.1)

Theorem 2.1. Let 𝒜,ℬ be two non-Archimedean Banach algebras and let 𝜓∶𝒜^𝑛→[0,∞), 𝜙∶𝒜²→[0,∞) be functions such that lim_𝑚→∞1𝑎₁^𝑚𝜓𝑎^𝑚₁𝑥₁,…,𝑎^𝑚₁𝑥_𝑛=lim_𝑘→∞1𝑘𝜙𝑘𝑥,𝑦=0 (2.2) for all 𝑥₁,…,𝑥_𝑛∈𝒜, and the limit 𝜓(𝑥)∶=lim_𝑚→∞1max𝑎₁^ℓ𝜓𝑎^ℓ₁𝑥,0,…,0∶0≤ℓ<𝑚 (2.3) exists and lim_𝑘→∞(1/𝑘)𝜓(𝑘𝑥)=0 for all 𝑥∈𝒜. Suppose that 𝑓∶𝒜→ℬ is a function satisfying 𝐷𝑓𝑥₁,…,𝑥_𝑛≤𝜓𝑥₁,…,𝑥_𝑛,𝑓𝑥𝑦−𝑓(𝑥)𝑓𝑦≤𝜙𝑥,𝑦 (2.4) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝒜. Then there exists a ring homomorphism 𝐻∶𝒜→ℬ such that 𝑓(𝑥)(𝑥≤1−𝐻)2^𝑛−1𝑎₁𝜓(𝑥) (2.5) for all 𝑥∈𝒜 and 𝐻(𝑥)𝐻𝑦−𝑓𝑦=𝑓(𝑥)−𝐻(𝑥)𝐻𝑦=0 (2.6) for all 𝑥,𝑦∈𝒜. Moreover, if lim_{0𝑥0𝑒𝑑03𝚥→∞}lim_𝑚→∞1max𝑎₁^ℓ𝜓𝑎^ℓ₁𝑥,0,…,0∶𝚥≤ℓ<𝑚+𝚥=0, (2.7) then 𝐻 is the unique ring homomorphism satisfying (2.5).

Proof. By [35, Theorem   4.4], there exists an additive function 𝐻∶𝒜→ℬ which satisfies (2.5). We have 𝐻(𝑥)∶=lim_𝑚→∞𝑎^𝑚₁𝑓𝑥𝑎^𝑚₁ (2.8) for all 𝑥∈𝒜. Now we show that 𝐻 is a multiplicative function. It follows from (2.5) that 𝑓()(≤1𝑘𝑥−𝐻𝑘𝑥)2^𝑛−1𝑎₁𝜓(𝑘𝑥) (2.9) for all 𝑥∈𝒜 and all 𝑘∈ℕ. On the other hand 𝐻 is additive then we have 1𝑘𝑓()(𝑥≤1𝑘𝑥−𝐻)2^𝑛−1𝑎₁𝑘𝜓(𝑘𝑥) (2.10) for all 𝑥∈𝒜 and all 𝑘∈ℕ. If 𝑘→∞, then by (2.3), the right hand side of above inequality tends to zero. It follows that 𝐻(𝑥)=lim_𝑘→∞1𝑘𝑓(𝑘𝑥) (2.11) for all 𝑥∈𝒜. Applying (2.3), (2.4), and (2.11) we have 𝐻𝑥𝑦−𝐻(𝑥)𝑓𝑦=lim_𝑘→∞1𝑘𝑓𝑘𝑥𝑦−𝑓(𝑘𝑥)𝑓𝑦=0 (2.12) for all 𝑥,𝑦∈𝒜. This means that 𝐻𝑥𝑦=𝐻(𝑥)𝑓𝑦 (2.13) for all 𝑥,𝑦∈𝒜. From (2.13) and additivity of 𝐻 we have 𝐻(𝑥)𝐻𝑦=𝐻(𝑥)lim_𝑘→∞1𝑘𝑓𝑘𝑦=lim_𝑘→∞1𝑘𝐻(𝑥)𝑓𝑘𝑦=lim_𝑘→∞1𝑘𝐻𝑥𝑘𝑦=𝐻𝑥𝑦 (2.14) for all 𝑥,𝑦∈𝒜. In other words, 𝐻 is multiplicative.It follows from (2.13) and (2.14) that 𝐻(𝑥)𝐻𝑦−𝑓𝑦=0 (2.15) for all 𝑥,𝑦∈𝒜. Similarly, we can show that 𝑓(𝑥)−𝐻(𝑥)𝐻𝑦=0 (2.16) for all 𝑥,𝑦∈𝒜. To prove the uniqueness property of 𝐻, let 𝑇∶𝒜→ℬ be another ring homomorphism which satisfies (2.5). Applying (2.11) and (2.5) we have ‖𝐻(𝑥)−𝑇(𝑥)‖=lim_𝑘→∞1𝑘𝑓𝑘𝑥𝑦−𝑇(𝑘𝑥)≤lim_𝑘→∞1𝑘12^𝑛−1𝑎₁𝜓(𝑘𝑥)=0 (2.17) for all 𝑥∈𝒜 which is the desired conclusion.

Now, we establish the superstability of homomorphisms as follows.

Corollary 2.2. Let 𝒜,ℬ be two unital non-Archimedean Banach algebras, and let 𝜓∶𝒜^𝑛→[0,∞),𝜙∶𝒜²→[0,∞),𝑓∶𝒜→ℬ be functions with conditions of Theorem 2.1. Suppose that lim_𝑚→∞𝑎^𝑚₁𝑓1_𝒜𝑎^𝑚₁=1_ℬ. (2.18) Then the mapping 𝑓∶𝒜→ℬ is a ring homomorphism.

Proof. It follows from (2.6) and (2.18) that 𝑓=𝐻 in Theorem 2.1. Hence, 𝑓 is a ring homomorphism.

Corollary 2.3. Let 𝜂∶[0,∞)→[0,∞) be a function satisfying (i)𝜂(|𝑎₁|𝑡)≤𝜂(|𝑎₁|)𝜂(𝑡) for all 𝑡≥0;(ii)𝜂(|𝑎₁|)<|𝑎₁|; (iii)lim_𝑘→∞(1/𝑘)𝜂(𝑘|𝑎₁|)=0. Suppose that 𝜀>0, and let 𝑓∶𝒜→ℬ satisfying 𝐷𝑓𝑥₁,…,𝑥_𝑛+𝑓𝑥𝑦−𝑓(𝑥)𝑓𝑦≤𝜀Min^𝑛_𝑖=1𝜂𝑥_𝑖,𝜂𝑥𝜂𝑦 (2.19) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝒜. Then there exists a unique ring homomorphism 𝐻∶𝒜→ℬ such that 𝑓(𝑥)(𝑥≤𝜀−𝐻)2^𝑛−1𝑎₁𝜂𝑥 (2.20) for all 𝑥∈𝒜.

Proof. Defining 𝜓∶𝒜^𝑛→[0,∞) and 𝜙∶𝒜²→[0,∞) by 𝜓𝑥₁,…,𝑥_𝑛∶=𝜀^𝑛_𝑖=1𝜂𝑥_𝑖,𝜙𝑥,𝑦∶=𝜂𝑥𝜂𝑦, (2.21) respectively, we have lim_𝑚→∞1𝑎₁^𝑚𝜓𝑎^𝑚₁𝑥₁,…,𝑎^𝑚₁𝑥_𝑛≤lim_𝑚→∞𝜂𝑎₁𝑎₁^𝑚𝜓𝑥₁,…,𝑥_𝑛=0 (2.22) for all 𝑥₁,…,𝑥_𝑛∈𝒜. Hence 𝜓(𝑥)∶=lim_𝑚→∞1max𝑎₁^ℓ𝜓𝑎^ℓ₁𝑥,0,…,0∶0≤ℓ<𝑚=𝜓(𝑥,0,…,0),lim_𝚥→∞lim_𝑚→∞1max𝑎₁^ℓ𝜓𝑎^ℓ₁𝑥,0,…,0∶𝚥≤ℓ<𝑚+𝚥=lim_𝚥→∞1𝑎₁^𝚥𝜓𝑎^𝚥₁𝑥,0,…,0=0 (2.23) for all 𝑥∈𝒜. On the other hand lim_𝑘→∞1𝑘𝜙𝑘𝑥,𝑦=lim_𝑘→∞1𝑘𝜂(𝑘‖𝑥‖)𝜂𝑦=0 (2.24) for all 𝑥,𝑦∈𝒜. The conclusion follows from Theorem 2.1.

Remark 2.4. The classical example of the function 𝜂 is the function 𝜂(𝑡)=𝑡^𝑝 for all 𝑡∈[0,∞), where 𝑝>1 with the further assumption that |𝑎₁|<1.

Now, we prove the stability of derivations non-Archimedean Banach algebras by using Theorem 2.1.

Theorem 2.5. Let 𝒜 be a non-Archimedean Banach algebra, and let 𝒳 be a non-Archimedean Banach 𝒜-module. Let 𝜓∶𝒜^𝑛→[0,∞),𝜙∶𝒜²→[0,∞) be a function such that lim_𝑚→∞1𝑎₁^𝑚𝜓𝑎^𝑚₁𝑥₁,…,𝑎^𝑚₁𝑥_𝑛=lim_𝑘→∞1𝑘𝜙𝑘𝑥,𝑦=0 (2.25) for all 𝑥₁,…,𝑥_𝑛∈𝒜, and the limit 𝜓(𝑥)∶=lim_𝑚→∞1max𝑎₁^ℓ𝜓𝑎^ℓ₁𝑥,0,…,0∶0≤ℓ<𝑚 (2.26) exists and lim_𝑘→∞(1/𝑘)𝜓(𝑘𝑥)=0 for all 𝑥∈𝒜. Suppose that 𝑓∶𝒜→𝒳 is a function satisfying 𝐷𝑓𝑥₁,…,𝑥_𝑛≤𝜓𝑥₁,…,𝑥_𝑛,𝑓𝑥𝑦−𝑓(𝑥)𝑦−𝑥𝑓𝑦≤𝜙𝑥,𝑦 (2.27) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝒜. Then there exists a ring derivation 𝐷∶𝒜→𝒳 such that 𝑓(𝑥)(𝑥≤1−𝐷)2^𝑛−1𝑎₁𝜓(𝑥) (2.28) for all 𝑥∈𝒜.

Proof. It is easy to see that 𝒳⊕₁𝒜 is a non-Archimedean Banach algebra equipped with the product 𝑥₁,𝑎₁𝑥₂,𝑎₂=𝑥₁⋅𝑎₂+𝑎₁⋅𝑥₂,𝑎₁𝑎₂𝑎₁,𝑎₂∈𝒜,𝑥₁,𝑥₂∈𝒳 (2.29) and with the following ℓ₁-norm: ‖(𝑥,𝑎)‖=‖𝑥‖+‖𝑎‖(𝑎∈𝒜,𝑥∈𝒳). (2.30) Let us define the mapping 𝜑_𝑓∶𝒜→𝒳⊕₁𝒜 by 𝑎↦(𝑓(𝑎),𝑎). It is easy to see that 𝜑_𝑓∶𝒜→𝒳⊕₁𝒜 satisfies the conditions of Theorem 2.1. By Theorem 2.1, there exists a unique ring homomorphism 𝐻∶𝒜→𝒳⊕₁𝒜 such that 𝐻(𝑎)−𝜑_𝑓(𝑎≤1)2^𝑛−1𝑎₁𝜓(𝑎)(𝑎∈𝒜). (2.31) We define projection maps 𝜋₁∶𝒳⊕₁𝒜→𝒳 and 𝜋₂∶𝒳⊕₁𝒜→𝒜 by (𝑥,𝑏)↦𝑥 and (𝑥,𝑏)↦𝑏, respectively.
It follows from (2.31) that 𝜋₂∘𝜑_𝑓(𝑘𝑎)−𝜋₂∘𝐻(𝑘𝑎)≤𝜑_𝑓()(≤1𝑘𝑎−𝐻𝑘𝑎)2^𝑛−1𝑎₁𝜓(𝑘𝑎)(𝑘∈ℕ,𝑎∈𝒜). (2.32) By the additivity of mappings under consideration 𝜋₂∘𝜑(𝑘𝑎)=𝑘𝜋₂(𝑎),𝜋∘𝜑₂∘𝜑_𝑓(𝑘𝑎)=𝜋₂𝑓(𝑘𝑎),𝑘𝑎=𝑘𝑎, (2.33) whence, by (2.32), 𝑎−𝜋₂(𝑎≤1∘𝐻)𝑘12^𝑛−1𝑎₁𝜓(𝑘𝑎) (2.34) for all 𝑘∈ℕ, 𝑎∈𝒜. By letting 𝑘 tend to ∞ in (2.34), we obtain by (2.25) that 𝜋₂∘𝐻(𝑎)=𝑎(𝑎∈𝒜). (2.35) Put 𝐷∶=𝜋₁∘𝐻. Then we have 𝜋₁∘𝐻(𝑎𝑏),𝑎𝑏=𝜋₁(𝐻(𝑎𝑏)),𝜋₂(𝐻(()(𝑎)𝐻(𝑏)=𝜋𝑎𝑏))=𝐻𝑎𝑏=𝐻₁(𝐻(𝑎)),𝜋₂(𝐻(𝑎))𝜋₁(𝐻(𝑏)),𝜋₂(𝐻(𝑏=𝜋))₁(𝐻(𝑎)),𝑎𝜋₁(𝐻(𝑏=)),𝑏𝑎𝜋₁(𝐻(𝑏))+𝜋₁(𝐻(𝑎))𝑏,𝑎𝑏 (2.36) for all 𝑎,𝑏∈𝒜. It follows that 𝐷 is a derivation. On the other hand, by (2.31) we have 𝐷(𝑎)−𝑓(𝑎)=𝜋₁(𝐻(𝑎))−𝜋₁𝜑_𝑓(𝑎)≤𝐻(𝑎)−𝜑_𝑓(𝑎≤1)2^𝑛−1𝑎₁𝜓(𝑎) (2.37) for all 𝑎∈𝒜.
To prove the uniqueness property of 𝐷, assume that 𝐷^∗ is another derivation from 𝒜 into 𝒳 satisfying 𝐷^∗(𝑎)(𝑎≤1−𝑓)2^𝑛−1𝑎₁𝜓(𝑎)(𝑎∈𝒜). (2.38) Then by (2.25), we have 𝐷(𝑎)−𝐷^∗(𝑎)=lim_𝑘→∞1𝑘𝐷(𝑘𝑎)−𝐷^∗(𝑘𝑎)≤lim_𝑘→∞1𝑘𝐷^∗(𝑎)(𝑎+1−𝑓)𝑘𝐷(𝑎)−𝑓(𝑎)≤lim_𝑘→∞2𝑘12^𝑛−1𝑎₁𝜓(𝑘𝑎)=0 (2.39) for all 𝑎∈𝒜. This means that 𝐷(𝑎)=𝐷^∗(𝑎) for all 𝑎∈𝒜.

Corollary 2.6. Let 𝜂∶[0,∞)→[0,∞) be a function satisfying (i)𝜂(|𝑎₁|𝑡)≤𝜂(|𝑎₁|)𝜂(𝑡) for all 𝑡≥0;(ii)𝜂(|𝑎₁|)<|𝑎₁|; (iii)lim_𝑘→∞(1/𝑘)𝜂(𝑘|𝑎₁|)=0.Suppose that 𝜀>0, and let 𝑓∶𝒜→𝒳 satisfying 𝐷𝑓𝑥₁,…,𝑥_𝑛+𝑓𝑥𝑦−𝑓(𝑥)𝑦−𝑥𝑓𝑦≤𝜀Min^𝑛_𝑖=1𝜂𝑥_𝑖,𝜂𝑥𝜂𝑦 (2.40) for all 𝑥₁,…,𝑥_𝑛,𝑥,𝑦∈𝒜. Then there exists a unique ring derivation 𝐷∶𝒜→𝒳 such that 𝑓(𝑥)(𝑥≤𝜀−𝐷)2^𝑛−1𝑎₁𝜂𝑥 (2.41) for all 𝑥∈𝒜.

Now, we would like to prove the superstability of derivations on non-Archimedean Banach algebras.

Theorem 2.7. Let 𝒜 be a non-Archimedean Banach algebra with bounded approximate identity. Let 𝜓∶𝒜^𝑛→[0,∞), 𝜙∶𝒜²→[0,∞), 𝑓∶𝒜→𝒜 be functions satisfying the conditions of Theorem 2.5. Then 𝑓∶𝒜→𝒜 is a ring derivation.

Proof. In the proof of Theorem 2.5, we can see that 𝐻(𝑏)𝐻(𝑎)−𝜑_𝑓(𝑎)=𝐻(𝑎)−𝜑_𝑓(𝑎)𝐻(𝑏)=0 (2.42) for all 𝑎,𝑏∈𝒜𝑓(𝑎)−𝐷(𝑎)𝑏=𝜋₁𝑓(𝑎)−𝐷(𝑎)𝑏,0=𝜋₁𝑓(𝑎)−𝐷(𝑎),0(𝐷(𝑏),𝑏)=𝜋₁𝜋₁𝐻(𝑎)−𝜑_𝑓(𝑎),0𝜋₁(𝐻(𝑏)),𝑏=𝜋₁𝜋₁𝐻(𝑎)−𝜑_𝑓(𝑎),0𝐻(𝑏)=𝜋₁𝜋₁(𝐻(𝑎)),𝑎−𝜋₁𝜑_𝑓(𝑎),𝑎𝐻(𝑏)=𝜋₁(0,0)by(2.42)=0 (2.43) for all 𝑎,𝑏∈𝒜. Since 𝒜 has a bounded approximate identity, then by above equation, we have 𝑓(𝑎)=𝐷(𝑎) for all 𝑎∈𝒜. 𝑓 is a ring derivation on 𝒜.