In this section, we prove the stability of quadratic *-derivations on a Banach *-algebra .
Definition 4.1 Letbe a *-normed algebra. A mappingis a quadratic *-derivation onif δ satisfies the following properties:
(1) δ is a quadratic mapping,
(2) δ is quadratic homogeneous, that is, δ(λa) = λ2δ(a) for alland all λ ∈ ℂ,
(3) δ(a b) = δ(a)b2 + a2δ(b) for all,
(4) δ(a*) = δ(a)* for all.
Theorem 4.2 Suppose thatis a mapping with f(0) = 0 for which there exists a functionsuch that
(4.1)
(4.2)
for alland all. Also, if for each fixedthe mapping t → f(ta) from ℝ tois continuous, then there exists a unique quadratic *-derivation δ onsatisfying
for all.
Proof. Putting a = b, c = d = 0, , and λ = 1 in (4.1), we have
for all . One can use induction to show that
(4.3)
for all n > m ≥ 0 and all . It follows from (4.3) that the sequence is Cauchy. Since is complete, this sequence is convergent. Define
Since f(0) = 0, we have δ(0) = 0. Replacing a and b by 2 na and 2 nb, c = d = 0, respectively, in (4.1), we get
Taking the limit as n → ∞, we obtain
(4.4)
for all and all . Putting λ = 1 in (4.4), we obtain that δ is a quadratic mapping. Setting b: = a in (4.4), we get
for all and all . Hence,
for all and all . Under the assumption that f(ta) is continuous in t ∈ ℝ for each fixed , by the same reasoning as in the proof of [10], we obtain that δ(λa) = λ2δ(a) for all and all λ ∈ ℝ. Hence,
for all and all λ ∈ ℂ (λ ≠ 0). This means that δ is quadratic homogeneous.
Replacing c and d by 2 nc and 2 nd, respectively, and putting a = b = 0 in (4.1), we get
for all .
Hence, we have
Thus, δ is a quadratic *-derivation on .
The rest of the proof is similar to the proof of Theorem 2.1. □
Corollary 4.3 Let ε, p be positive real numbers with p < 2. Suppose thatis a mapping such that
(4.5)
for alland all. Also, if for each fixedthe mapping t → f(ta) is continuous, then there exists a unique derivation δ onsatisfying
for all.
Proof. Putting φ(a, b, c, d) = ε(||a|| p + ||b||p+ ||c||p+ ||d||p) in Theorem 4.2, we get the desired result.
Similarly, we can obtain the following. We will omit the proof.
Theorem 4.4 Suppose thatis a mapping with f(0) = 0 for which there exists a functionsatisfying (4.1), (4.2) and
for all. Also, if for each fixedthe mapping t → f(ta) from ℝ tois continuous, then there exists a unique quadratic *-derivation δ onsatisfying
for all, where
Corollary 4.5 Let ε, p be positive real numbers with p > 4. Suppose thatis a mapping satisfying (4.5). Also, if for each fixedthe mapping t → f(ta) is continuous, then there exists a unique derivation δ onsatisfying
for all.
Proof. Putting φ(a, b, c, d) = ε(||a||p+ ||b||p+ ||c||p+ ||d||p) in Theorem 4.4, we get the desired result. □