Approximate *-derivations and approximate quadratic *-derivations on C *-algebras

Sun Young Jang; Choonkil Park

doi:10.1186/1029-242X-2011-55

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Approximate -derivations and approximate quadratic -derivations on C*-algebras

Sun Young Jang¹ and
Choonkil Park²Email author

Journal of Inequalities and Applications20112011:55

https://doi.org/10.1186/1029-242X-2011-55

Received: 17 March 2011

Accepted: 14 September 2011

Published: 14 September 2011

Abstract

In this paper, we prove the stability of *-derivations and of quadratic *-derivations on Banach *-algebras. We moreover prove the superstability of *-derivations and of quadratic *-derivations on C*-algebras.

2000 Mathematics Subject Classification: 39B52; 47B47; 46L05; 39B72.

Keywords

*-derivation quadratic *-derivation C*-algebra; stability superstability

1 Introduction and preliminaries

Suppose that $A$ is a complex Banach *-algebra. A ℂ-linear mapping $δ : D (δ) \to A$ is said to be a derivation on $A$ if δ(ab) = δ(a)+ b + aδ(b) for all $a, b \in A$ , where D(δ) is a domain of δ and D(δ) is dense in $A$ . If δ satisfies the additional condition δ(a*) = δ(a)* for all $a \in A$ , then δ is called a *-derivation on $A$ . It is well known that if $A$ is a C*-algebra and D(δ) is $A$ , then the derivation δ is bounded.

A C*-dynamical system is a triple (

A

, G, α) consisting of a C*-algebra

A

, a locally compact group G, and a pointwise norm continuous homomorphism α of G into the group Aut(

A

) of *-automorphisms of

A

. Every bounded *-derivation δ arises as an infinitesimal generator of a dynamical system for ℝ. In fact, if δ is a bounded *-derivation of

A

on a Hilbert space

H

, then there exists an element h in the enveloping von Neumann algebra

A^{^{″}}

such that

δ (x) = a d_{i h} (x)

for all $x \in A$ .

If, for each t ∈ ℝ, α_t is defined by α_t (x) = eⁱ th xe^-ith for all $x \in A$ , then α_t is a *-automorphism of $A$ induced by unitaries U_t = eⁱ th for each t ∈ ℝ. The action $α : ℝ \to A u t (A)$ , t → α_t , is a strongly continuous one-parameter group of *-automorphisms of $A$ . For several reasons, the theory of bounded derivations of C*-algebras is important in the quantumn mechanics (see [1–3]).

A functional equation is called stable if any function satisfying the functional equation "approximately" is near to a true solution of the functional equation. We say that a functional equation is superstable if every approximate solution is an exact solution of it (see [4]).

In 1940, Ulam [5] proposed the following question concerning stability of group homomorphisms: under what condition does there exist an additive mapping near an approximately additive mapping? Hyers [6] answered the problem of Ulam for the case where G₁ and G₂ are Banach spaces. A generalized version of the theorem of Hyers for an approximately linear mapping was given by Rassias [7]. Since then, the stability problems of various functional equations have been extensively investigated by a number of authors (see [8–19]). In particular, those of the important functional equations are the following functional equations

f (x + y) = f (x) + f (y),

(1.1)

2 f (\frac{x + y}{2}) = f (x) + f (y),

(1.2)

which are called the Cauchy functional equation and the Jensen functional equation, respectively. The function f(x) = bx is a solution of these functional equations. Every solution of the functional equations (1.1) and (1.2) is said to be an additive mapping.

In this paper, we introduce functional equations of *-derivations and of quadratic *-derivations. we prove the stability of *-derivations associated with the Cauchy functional equation and the Jensen functional equation and of quadratic *-derivations on Banach *-algebra. We moreover prove the superstability of *-derivations and of quadratic *-derivations on C*-algebras.

2 Stability of -derivations on Banach -algebras

In this section, let $A$ be a Banach *-algebra. We prove the stability of *-derivations on $A$ .

Theorem 2.1 Suppose that

f : A \to A

is a mapping with f(0) = 0 for which there exists a function

φ : A^{4} \to [0, \infty)

such that

\tilde{φ} (a, b, c, d) : = \sum_{n = 0}^{\infty} \frac{1}{2^{n + 1}} φ (2^{n} a, 2^{n} b, 2^{n} c, 2^{n} d) < \infty,

(2.1)

∥ f (λ a + b + c d) - λ f (a) - f (b) - f (c) d - c f (d) ∥ \leq φ (a, b, c, d),

(2.2)

∥ f (a^{*}) - f {(a)}^{*} ∥ \leq φ (a, a, a, a)

(2.3)

for all

λ \in T : = {λ \in ℂ : | λ | = 1}

and all

a, b, c, d \in A

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \tilde{φ} (a, a, 0, 0),

(2.4)

for all $a \in A$ .

Proof. Setting a = b, c = d = 0 and λ = 1 in (2.2), we have

∥ f (2 a) - 2 f (a) ∥ \leq φ (a, a, 0, 0)

for all

a \in A

. One can use induction to show that

‖ \frac{f {(2}^{n} a)}{2^{n}} - \frac{f {(2}^{m} a)}{2^{m}} ‱ = \sum_{k = m}^{n - 1} \frac{1}{2^{k + 1}} μ {(2}^{k} a {,2}^{k} a,0,0)

(2.5)

for all n > m ≥ 0 and all

a \in A

. It follows from (2.5) and (2.1) that the sequence

{\frac{f {(2}^{n} a)}{2^{n}}}

is Cauchy. Due to the completeness of

A

, this sequence is convergent. Define

δ (a) : = \lim_{n \to \infty} \frac{f {(2}^{n} a)}{2^{n}}

(2.6)

for all

a \in A

. Then, we have

δ (\frac{1}{2^{k}} a) = \lim_{n \to \infty} \frac{1}{2^{k}} \frac{f {(2}^{n - k} a)}{2^{n - k}} = \frac{1}{2^{k}} δ (a)

(2.7)

for each k ∈ ℕ. Putting c = d = 0 and replacing a and b by 2 ⁿa and 2 ⁿb, respectively, in (2.2), we get

‖ \frac{1}{2^{n}} f {(2}^{n} (λ a + b)) - λ \frac{1}{2^{n}} f {(2}^{n} a) - \frac{1}{2^{n}} f {(2}^{n} b) ‱ \leq \frac{1}{2^{n}} μ {(2}^{n} a {,2}^{n} b,0,0).

Taking the limit as n → ∞, we obtain

δ (λ a + b) = λ δ (a) + δ (b)

(2.8)

for all

a, b \in A

and all

λ \in T

. Putting a = b = 0 and replacing c and d by 2 ⁿc and 2 ⁿd, respectively, in (2.2), we get

\begin{array}{l} ‖ \frac{1}{2^{2 n}} f {(2}^{2 n} c d) - \frac{1}{2^{2 n}} f {(2}^{n} c {) (2}^{n} d) - \frac{1}{2^{2 n}} {(2}^{n} c) f {(2}^{n} d) ‱ \\ \leq \frac{1}{2^{2 n}} μ ({0,0,2}^{n} c {,2}^{n} d) \leq \frac{1}{2^{n}} μ ({0,0,2}^{n} c {,2}^{n} d). \end{array}

Taking the limit as n → ∞, we obtain

δ (c d) = δ (c) d + c δ (d)

(2.9)

for all $c, d \in A$ .

Next, let λ = λ₁ +iλ₂ ∈ ℂ where λ₁, λ₂, ∈ ℝ. Let γ₁ = λ₁ - [λ₁] and γ₂ = λ₂ - [λ₂], where [λ] denotes the integer part of λ. Then, 0 ≤ γ₁ < 1(1 ≤ i ≤ 2). One can represent γ_i as

γ_{i} = \frac{λ_{i, 1} + λ_{i, 2}}{2}

such that

λ_{i, j} \in T (1 \leq i, j \leq 2)

. From (2.7) and (2.8), it follows that

\begin{align} δ (λ a) & = δ (λ_{1} a) + i δ (λ_{2} a) & (1) \\ = ([λ_{1}] δ (a) + δ (γ_{1} a)) + i ([λ_{2}] δ (a) + δ (γ_{2} a)) & (2) \\ = ([λ_{1}] δ (a) + \frac{1}{2} δ (λ_{1, 1} a + λ_{1, 2} a)) + i ([λ_{2}] δ (a) + \frac{1}{2} δ (λ_{2, 1} a + λ_{2, 2} a)) & (3) \\ = ([λ_{1}] δ (a) + \frac{1}{2} λ_{1, 1} δ (a) + \frac{1}{2} λ_{1, 2} δ (a)) + i ([λ_{2}] δ (a) + \frac{1}{2} λ_{2, 1} δ (a) + \frac{1}{2} λ_{2, 2} δ (a)) & (4) \\ = λ_{1} δ (a) + i λ_{2} δ (a) = λ δ (a) & (5) \\ (6) \end{align}

for all

a \in A

. Hence, δ is ℂ-linear, and so it is a derivation on

A

. Moreover, it follows from (2.5) with m = 0 and (2.6) that

∥ δ (a) - f (a) ∥ \leq \tilde{φ} (a, a, 0, 0)

for all

a \in A

. It is well known that the additive mapping δ satisfying (2.4) is unique (see [3] or [19]). Replacing a and a* by 2 ⁿa and 2 ⁿa*, respectively, in (2.3), we get

‖ \frac{1}{2^{n}} f {(2}^{n} a^{*}) - \frac{1}{2^{n}} f {{(2}^{n} a)}^{*} ‱ \leq \frac{1}{2^{n}} μ {(2}^{n} a {,2}^{n} a {,2}^{n} a {,2}^{n} a).

Passing to the limit as n → ∞, we get the δ(a*) = δ(a)* for all $a \in A$ . So δ is a *-derivation on $A$ , as desired. □

Corollary 2.2 Let ε, p be positive real numbers with p < 1. Suppose that

f : A \to A

is a mapping satisfying

∥ f (λ a + b + c d) - λ f (a) - f (b) - c f (d) - f (c) d ∥ \leq ε (∥ a ∥^{p} + ∥ b ∥^{p} + ∥ c ∥^{p} + ∥ d ∥^{p}),

(2.10)

∥ f (a^{*}) - f {(a)}^{*} ∥ \leq 4 ε ∥ a ∥^{p}

(2.11)

for all

λ \in T

and all

a, b, c, d \in A

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{2 ε}{2 - 2^{p}} ∥ a ∥^{p}

for all $a \in A$ .

Proof. Putting φ(a, b, c, d) = ε(||a|| ^p + ||b|| ^p + ||c|| ^p + ||d| ^p ) in Theorem 2.1, we get the desired result. □

Similarly, we can obtain the following. We will omit the proof.

Theorem 2.3 Suppose that

f : A \to A

is a mapping with f (0) = 0 for which there exists a function

φ : A^{4} \to [0, \infty)

satisfying (2.2), (2.3) and

\sum_{n = 1}^{\infty} 2^{2 n - 1} φ (\frac{a}{2^{n}}, \frac{b}{2^{n}}, \frac{c}{2^{n}}, \frac{d}{2^{n}}) < \infty

for all

a, b, c, d \in A

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \tilde{φ} (a, a, 0, 0),

for all

a \in A

, where

\tilde{φ} (a, b, c, d) : = \sum_{n = 1}^{\infty} 2^{n - 1} φ (\frac{a}{2^{n}}, \frac{b}{2^{n}}, \frac{c}{2^{n}}, \frac{d}{2^{n}}) .

Corollary 2.4 Let ε, p be positive real numbers with p > 2. Suppose that

f : A \to A

is a mapping satisfying (2.10) and (2.11). Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{2 ε}{2^{p} - 2} ∥ a ∥^{p}

for all $a \in A$ .

Proof. Putting φ(a, b, c, d) = ε(||a|| ^p + ||b|| ^p + ||c|| ^p + ||d| ^p ) in Theorem 2.3, we get the desired result. □

3 Stability of *-derivations associated with the Jensen functional equation

The stability of the Jensen functional equation has been studied first by Kominek and then by several other mathematicians (see [11, 20]).

In this section, we study the stability of *-derivation associated with the Jensen functional equation in a Banach *-algebra $A$ .

Theorem 3.1 Let

A

be a Banach *-algebra. Suppose that

f : A \to A

is a mapping with f (0) = 0 for which there exists a function

φ : A \times A \to [0, \infty)

such that

\tilde{φ} (a, b) : = \sum_{n = 0}^{\infty} \frac{1}{3^{n}} φ (3^{n} a, 3^{n} b) < \infty,

(3.1)

∥2 f (\frac{λ a + λ b}{2}) - λ f (a) - λ f (b)∥ \leq φ (a, b),

(3.2)

∥ f (a^{*}) - f {(a)}^{*} ∥ \leq φ (a, a),

(3.3)

∥ f (a b) - a f (b) - f (a) b ∥ \leq φ (a, b)

(3.4)

for all

a, b \in A

and all

λ \in T

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{1}{3} (\tilde{φ} (a, - a) + \tilde{φ} (- a, 3 a))

(3.5)

for all $a \in A$ .

Proof. Letting λ = 1 and b = -a in (3.2), we get

∥ - f (a) - f (- a) ∥ \leq φ (a, - a)

for all

a \in A

. Letting λ = 1 and replacing a and b by -a and 3a, respectively, in (3.2), we get

∥ 2 f (a) - f (- a) - f (3 a) ∥ \leq φ (- a, 3 a)

for all

a \in A

. Thus,

\begin{align} ∥f (a) - \frac{1}{3} f (3 a)∥ & \leq \frac{1}{3} (∥ f (a) + f (- a) ∥ + ∥ 2 f (a) - f (- a) - f (3 a) ∥) & (1) \\ \leq \frac{1}{3} (φ (a, - a) + φ (- a, 3 a)) & (2) \\ (3) \end{align}

for all

a \in A

. So

\begin{matrix} ‖ \frac{1}{3^{n}} f {(3}^{n} a) - \frac{1}{3^{m}} f {(3}^{m} a) ‱ \leq \sum_{j = m}^{n - 1} ‖ \frac{1}{3^{j}} f {(3}^{j} a) - \frac{1}{3^{j + 1}} f {(3}^{j + 1} a) ‱ \\ \leq \frac{1}{3} \sum_{j = m}^{n - 1} \frac{1}{3^{j}} (μ {(3}^{j} a, - 3^{j} a) + μ (- 3^{j} a {,3}^{j + 1} a)) \end{matrix}

(3.6)

for all nonnegative integers n, m with n > m and all

a \in A

. It follows from (3.6) that the sequence

{\frac{1}{3^{n}} f (3^{n} a)}

is a Cauchy sequence for all

a \in A

. Since

A

is complete, the sequence

{\frac{1}{3^{n}} f (3^{n} a)}

is convergent. So one can define the mapping

δ : A \to A

by

δ (a) = lim_{n \to \infty} \frac{1}{3^{n}} f (3^{n} a)

for all

a \in A

. By (3.2),

\begin{matrix} ‖ 2 δ (\frac{a + b}{2}) - δ (a) - δ (b) ‱ = \lim_{n \to \infty} \frac{1}{3^{n}} ‖ 2 f (3^{n} \frac{a + b}{2}) - f {(3}^{n} a) - f {(3}^{n} b) ‱ \\ \leq \lim_{n \to \infty} \frac{1}{3^{n}} μ (3^{n} a {,3}^{n} b) = 0 \end{matrix}

for all

a, b \in A

. Thus

2 δ (\frac{a + b}{2}) = δ (a) + δ (b)

(3.7)

for all $a, b \in A$ . Since f(0) = 0, we have δ(0) = 0. Putting b = 0 in (3.7), we get $2 δ (\frac{a}{2}) = δ (a)$ for all $a \in A$ and therefore $δ (a) + δ (b) = 2 δ (\frac{a + b}{2}) = δ (a + b)$ for all $a, b \in A$ . Moreover, letting m = 0 and passing the limit n → ∞ in (3.6), we get (3.5).

Replacing both a and b in (3.2) by 3 ⁿa and then dividing both sides of the obtained inequality by 3 ⁿ , we get

‖ \frac{1}{3^{n}} f (λ 3^{n} a) - \frac{λ}{3^{n}} f {(3}^{n} a) ‱ \leq \frac{1}{3^{n}} μ {(3}^{n} a {,3}^{n} a).

Passing the limit as n → ∞, we get δ(λa) = λδ(a) for all $λ \in T$ . Thus we can get δ(λa) = λδ(a) for all λ ∈ ℂ by the similar discussion in the proof of Theorem 2.1.

Replacing a in (3.3) by 3 ⁿa and then dividing the both sides of the obtained inequality by 3ⁿ, we get

‖ \frac{1}{3^{n}} f {(3}^{n} a^{*}) - \frac{1}{3^{n}} f {{(3}^{n} a)}^{*} ‱ \leq \frac{1}{3^{n}} μ {(3}^{n} a {,3}^{n} a).

Passing the limit as n tends to infinity, we get δ(a*) = δ(a)*.

Similarly, replacing a and b in (3.4) by 3 ⁿa and 3 ⁿb, respectively, we get

‖ \frac{f {(3}^{2 n} a b)}{3^{2 n}} - \frac{3^{n} a f {(3}^{n} b)}{3^{2 n}} - \frac{f {(3}^{n} a {) (3}^{n} b)}{3^{2 n}} ‱ \leq \frac{1}{3^{2 n}} μ {(3}^{n} a {,3}^{n} b) \leq \frac{1}{3^{n}} μ {(3}^{n} a {,3}^{n} b),

which tends to zero, as n tends to ∞. So we get δ(ab) = δ(a)d + aδ(b) for all $a, b \in A$ . Hence, δ is a *-derivation on $A$ .

Corollary 3.2 Let ε, p be positive real numbers with p < 1. Suppose that

f : A \to A

is a mapping satisfying

∥2 f (\frac{λ a + λ b}{2}) - λ f (a) - λ f (b)∥ \leq ε (∥ a ∥^{p} + ∥ b ∥^{p}),

(3.8)

∥ f (a^{*}) - f {(a)}^{*} ∥ \leq 2 ε ∥ a ∥^{p},

(3.9)

∥ f (a b) - a f (b) - f (a) b ∥ \leq ε (∥ a ∥^{p} + ∥ b ∥^{p})

(3.10)

for all

λ \in T

and all

a, b \in A

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{3 + 3^{p}}{3 - 3^{p}} ε ∥ a ∥^{p}

for all $a \in A$ .

Proof. Putting φ(a, b) = ε(||a||^p+ ||b||^p) in Theorem 3.1, we get the desired result. □

Similarly, we can obtain the following. We will omit the proof.

Theorem 3.3 Let

A

be a Banach *-algebra. Suppose that

f : A \to A

is a mapping with f(0) = 0 for which there exists a function

∥ f (a) - δ (a) ∥ \leq \frac{2 ε}{2^{p} - 2} ∥ a ∥^{p}

satisfying (3.2), (3.3), (3.4) and

\sum_{n = 1}^{\infty} 3^{2 n} φ (\frac{a}{3^{n}}, \frac{b}{3^{n}}) < \infty

for all

a, b \in A

. Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{1}{3} (\tilde{φ} (a, - a) + \tilde{φ} (- a, 3 a))

for all

a \in A

, where

\tilde{φ} (a, b) : = \sum_{n = 1}^{\infty} 3^{n} φ (\frac{a}{3^{n}}, \frac{b}{3^{n}}) .

Corollary 3.4 Let ε, p be positive real numbers with p > 2. Suppose that

f : A \to A

is a mapping satisfying (3.8), (3.9) and (3.10). Then there exists a unique *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{3^{p} + 3}{3^{p} - 3} ε ∥ a ∥^{p}

for all $a \in A$ .

Proof. Putting φ(a, b) = ε(||a||^p+ ||b||^p) in Theorem 3.3, we get the desired result. □

4 Stability of quadratic -derivations on Banach -algebras

In this section, we prove the stability of quadratic *-derivations on a Banach *-algebra $A$ .

Definition 4.1 Let $A$ be a *-normed algebra. A mapping $δ : A \to A$ is a quadratic *-derivation on $A$ if δ satisfies the following properties:

(1) δ is a quadratic mapping,

(2) δ is quadratic homogeneous, that is, δ(λa) = λ²δ(a) for all $a \in A$ and all λ ∈ ℂ,

(3) δ(a b) = δ(a)b² + a²δ(b) for all $a, b \in A$ ,

(4) δ(a*) = δ(a)* for all $a \in A$ .

Theorem 4.2 Suppose that

f : A \to A

is a mapping with f(0) = 0 for which there exists a function

φ : A^{4} \to [0, \infty)

such that

\tilde{φ} (a, b, c, d) : = \sum_{k = 0}^{\infty} \frac{1}{4^{k}} φ (2^{k} a, 2^{k} b, 2^{k} c, 2^{k} d) < \infty,

\begin{gathered} ∥ f (λ a + λ b + c d) + f (λ a - λ b + c d) - 2 λ^{2} f (a) - 2 λ^{2} f (b) - 2 f (c) d^{2} - 2 c^{2} f (d) ∥ \\ \leq φ (a, b, c, d), \end{gathered}

(4.1)

∥ f (a^{*}) - f {(a)}^{*} ∥ \leq φ (a, a, a, a)

(4.2)

for all

a, b, c, d \in A

and all

λ \in T

. Also, if for each fixed

a \in A

the mapping t → f(ta) from ℝ to

A

is continuous, then there exists a unique quadratic *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{1}{4} \tilde{φ} (a, a, 0, 0)

for all $a \in A$ .

Proof. Putting a = b, c = d = 0, , and λ = 1 in (4.1), we have

∥ f (2 a) - 4 f (a) ∥ \leq φ (a, a, 0, 0)

for all

a \in A

. One can use induction to show that

‖ \frac{f {(2}^{n} a)}{4^{n}} - \frac{f {(2}^{m} a)}{4^{m}} ‱ \leq \frac{1}{4} \sum_{k = m}^{n - 1} \frac{μ {(2}^{k} a {,2}^{k} a,0,0)}{4^{k}}

(4.3)

for all n > m ≥ 0 and all

a \in A

. It follows from (4.3) that the sequence

{\frac{f {(2}^{n} a)}{4^{n}}}

is Cauchy. Since

A

is complete, this sequence is convergent. Define

δ (a) : = \lim_{n \to \infty} \frac{f {(2}^{n} a)}{4^{n}} .

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

‖ \frac{f {(2}^{n} (λ a + λ b))}{4^{n}} + \frac{f {(2}^{n} (λ a - λ b))}{4^{n}} - 2 λ^{2} \frac{f {(2}^{n} a)}{4^{n}} - 2 λ^{2} \frac{f {(2}^{n} b)}{4^{n}} ‱ \leq \frac{μ {(2}^{n} a {,2}^{n} b,0,0)}{4^{n}} .

Taking the limit as n → ∞, we obtain

δ (λ a + λ b) + δ (λ a - λ b) = 2 λ^{2} δ (a) + 2 λ^{2} δ (b)

(4.4)

for all

a, b \in A

and all

λ \in T

. Putting λ = 1 in (4.4), we obtain that δ is a quadratic mapping. Setting b: = a in (4.4), we get

δ (2 λ a) = 4 λ^{2} δ (a)

for all

a \in A

and all

λ \in T

. Hence,

δ (λ a) = λ^{2} δ (a)

for all

a \in A

and all

λ \in T

. Under the assumption that f(ta) is continuous in t ∈ ℝ for each fixed

a \in A

, by the same reasoning as in the proof of [10], we obtain that δ(λa) = λ²δ(a) for all

a \in A

and all λ ∈ ℝ. Hence,

δ (λ a) = δ (\frac{λ}{| λ |} | λ | a) = \frac{λ^{2}}{| λ |^{2}} δ (| λ | a) = \frac{λ^{2}}{| λ |^{2}} | λ |^{2} δ (a) = λ^{2} δ (a)

for all $a \in A$ 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

\begin{array}{l} ‖ \frac{f {(2}^{n} c \cdot 2^{n} d)}{4^{2 n}} + \frac{f {(2}^{n} c \cdot 2^{n} d)}{4^{2 n}} - 2 \frac{2^{2 n} c^{2} f {(2}^{n} d)}{4^{2 n}} - 2 \frac{f {(2}^{n} c {) 2}^{2 n} d^{2}}{4^{2 n}} ‱ \\ = ‖ \frac{f {(2}^{2 n} c d)}{4^{2 n}} + \frac{f {(2}^{2 n} c d)}{4^{2 n}} - 2 \frac{2^{2 n} c^{2}}{2^{2 n}} \frac{f {(2}^{n} d)}{4^{n}} - 2 \frac{f {(2}^{n} c)}{4^{n}} \frac{2^{2 n} d^{2}}{2^{2 n}} ‱ \\ \leq \frac{μ {(0,0,2}^{n} c {,2}^{n} d)}{4^{2 n}} \leq \frac{μ {(0,0,2}^{n} c {,2}^{n} d)}{4^{n}} \end{array}

for all $c, d \in A$ .

Hence, we have

∥ δ (c d) - c^{2} δ (d) - δ (c) d^{2} ∥ \leq \lim_{n \to \infty} \frac{μ {(0,0,2}^{n} c {,2}^{n} d)}{4^{n}} = 0.

Thus, δ is a quadratic *-derivation on $A$ .

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 that

f : A \to A

is a mapping such that

\begin{gathered} ∥ f (λ a + λ b + c d) + f (λ a - λ b + c d) - 2 λ^{2} f (a) - 2 λ^{2} f (b) - 2 c^{2} f (d) - 2 f (c) d^{2} ∥ \\ \leq ε (∥ a ∥^{p} + ∥ b ∥^{p} + ∥ c ∥^{p} + ∥ d ∥^{p}) \end{gathered}

(4.5)

for all

a, b, c, d \in A

and all

λ \in T

. Also, if for each fixed

a \in A

the mapping t → f(ta) is continuous, then there exists a unique derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{2 ε}{4 - 2^{p}} ∥ a ∥^{p}

for all $a \in A$ .

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 that

f : A \to A

is a mapping with f(0) = 0 for which there exists a function

φ : A^{4} \to [0, \infty)

satisfying (4.1), (4.2) and

\sum_{k = 1}^{\infty} 4^{2 k} φ (\frac{a}{2^{k}}, \frac{b}{2^{k}}, \frac{c}{2^{k}}, \frac{d}{2^{k}}) < \infty

for all

a, b, c, d \in A

. Also, if for each fixed

a \in A

the mapping t → f(ta) from ℝ to

A

is continuous, then there exists a unique quadratic *-derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{1}{4} \tilde{φ} (a, a, 0, 0)

for all

a \in A

, where

\tilde{φ} (a, b, c, d) : = \sum_{k = 1}^{\infty} 4^{k} φ (\frac{a}{2^{k}}, \frac{b}{2^{k}}, \frac{c}{2^{k}}, \frac{d}{2^{k}})

Corollary 4.5 Let ε, p be positive real numbers with p > 4. Suppose that

f : A \to A

is a mapping satisfying (4.5). Also, if for each fixed

a \in A

the mapping t → f(ta) is continuous, then there exists a unique derivation δ on

A

satisfying

∥ f (a) - δ (a) ∥ \leq \frac{2 ε}{2^{p} - 4} ∥ a ∥^{p}

for all $a \in A$ .

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. □

5 Superstability of -derivations and of quadratic -derivations On C*-algebras

We prove the superstability of *-derivations and of quadratic *-derivations on C*-algebras. More precisely, we introduce the concept of (ψ, ε) -approximate *-derivations and of (ψ, ε)-approximate quadratic *-derivations on C*-algebras and show that every (ψ, ε)-approximate *-derivation is a *-derivation and that every (ψ, ε)-approximate quadratic *-derivation is a quadratic *-derivation. Thus, we extend the results of [21].

Definition 5.1 Suppose that

A

is a *-normed algebra and s ∈{1, -1}. Let

δ : A \to A

be a mapping for which there exist a mapping

ε : A \to A

, and a function

ψ : A \times A \to ℝ

satisfying

lim_{n \to \infty} n^{- s} ψ (n^{s} a, b) = lim_{n \to \infty} n^{- s} ψ (a, n^{s} b) = 0 (a, b \in A)

(5.1)

such that

\begin{gathered} ∥ a δ (b) - ε (a) b ∥ \leq ψ (a, b) \\ ∥ ε (a) c d - a (δ (c) d - c δ (d)) ∥ \leq ψ (a, c d) \\ ∥ a δ {(b)}^{*} - ε (a) b^{*} ∥ \leq ψ (a, b) \end{gathered}

for all $a, b, c, d \in A$ . Then δ is called a (ψ, ε)-approximate *-derivation on $A$ .

Theorem 5.2 Let $A$ be a C*-algebra. Then any (ψ, ε)-approximate *-derivation δ on $A$ is a *-derivation.

Proof. We assume that (5.1) holds. Let

a, b \in A

and λ ∈ ℂ. We have

\begin{align} ∥ b (δ (λ a) - λ δ (a)) ∥ & \leq n^{- s} ∥ n^{s} b δ (λ a) - λ n^{s} b δ (a) ∥ & (1) \\ \leq n^{- s} ∥ n^{s} b δ (λ a) - ε (n^{s} b) λ a ∥ + n^{- s} ∥ ε (n^{s} b) λ a - λ n^{s} b δ (a) ∥ & (2) \\ \leq n^{- s} ψ (n^{s} b, λ a) + n^{- s} | λ | ψ (n^{s} b, a), & (3) \\ (4) \end{align}

which tends to zero as n → ∞, and so b(δ(λa) - λδ(a)) = 0 for all

b \in A

. Let {e_i}_i∈Ibe an approximate unit of

A

. If we replace b with {e_i }, then we have

∥ e_{i} (δ (λ a) - λ δ (a)) ∥ = 0

for all i ∈ I. So we conclude that δ(λa) = λδ(a) for all $a \in A$ and λ ∈ ℂ.

The additivity of δ follows from

\begin{array}{l} ∥ c (δ (a + b) - δ (a) - δ (b)) ∥ \\ \leq n^{- s} ∥ n^{s} c δ (a + b) - ε (n^{s} c) (a + b) ∥ + n^{- s} ∥ n^{s} c δ (a) - ε (n^{s} c) a) ∥ + n^{- s} ∥ n^{s} c δ (b) - ε (n^{s} c) b) \\ \leq n^{- s} ψ (n^{s} c, a + b) + n^{- s} ψ (n^{s} c, a) + n^{- s} ψ (n^{s} c, b). \end{array}

By the same process, using the approximate unit of $A$ , we have that δ(a + b) - δ(a) -δ(b) for all $a, b \in A$ .

The following computation

\begin{gathered} ∥ z (δ (a b) - δ (a) b - a δ (b)) ∥ \\ \leq n^{- s} ∥ n^{s} z δ (a b) - ε (n^{s} z) (a b) ∥ + n^{- s} ∥ ε (n^{s} z) a b - n^{s} z (δ (a) b + a δ (b)) ∥ \\ \leq n^{- s} ψ (n^{s} z, a b) + n^{- s} ψ (n^{s} z, a b) \end{gathered}

yields that δ(ab) = δ(a)b + aδ(b) for all $a, b \in A$ .

Finally, on the involution, we have that

\begin{align} ∥ z (δ (a^{*}) - δ {(a)}^{*}) ∥ & \leq n^{- s} ∥ n^{s} z δ (a^{*}) - ε (n^{s} z) a^{*} ∥ & (1) \\ + n^{- s} ∥ ε (n^{s} z) a^{*} - n^{s} z δ {(a)}^{*} ∥ & (2) \\ \leq n^{- s} ψ (n^{- s} z, a^{*}) + n^{- s} ψ (n^{s} z, a) . & (3) \\ (4) \end{align}

Thus, δ(a)* = δ(a) * for all $a \in A$ . □

Therefore, δ is a *-derivation on $A$ .

Corollary 5.3 Suppose that

A

is a C*-algebra and that

δ : A \to A

is a mapping for which there exist nonnegative real numbers α, β and positive real numbers p₁, p₂, q₁, q₂with p₁, p₂, q₁, q₂ < 1 such that

\begin{gathered} ∥ a δ (b) - ε (a) b ∥ \leq α (∥ a ∥^{p_{1}} + ∥ b ∥^{p_{2}}) + β ∥ a ∥^{q_{1}} ∥ b ∥^{q_{2}}, \\ ∥ ε (a) c d - a (δ (c) d - c δ (d)) ∥ \leq α (∥ a ∥^{p_{1}} + ∥ c d ∥^{p_{2}}) + β ∥ a ∥^{q_{1}} ∥ c d ∥^{q_{2}}, \\ ∥ a δ {(b)}^{*} - ε (a) b^{*} ∥ \leq α (∥ a ∥^{p_{1}} + ∥ b ∥^{p_{2}}) + β ∥ a ∥^{q_{1}} ∥ b ∥^{q_{2}} \end{gathered}

for all $a, b, c, d \in A$ . Then δ is a *-derivation of $A$ .

Next, we prove the superstability of quadratic *-derivations on C*-algebras.

Definition 5.4 Suppose that

A

is a *-normed algebra and s ∈{-1, 1}. Let

δ : A \to A

be a mapping for which there exist a function

ψ : A \times A \to [0, \infty)

and a mapping

ε : A \to A

satisfying

lim_{n \to \infty} n^{- 2 s} ψ (n^{s} a, b) = lim_{n \to \infty} n^{- 2 s} ψ (a, n^{s} b) = 0 (a, b \in A)

(5.2)

such that

\begin{array}{l} ∥ a^{2} δ (b) - ε (a) b^{2} ∥ \leq ψ (a, b) \\ ∥ ε (a) (c d)^{2} - a^{2} (δ (c) d^{2} - c^{2} δ (d)) ∥ \leq ψ (a, c d) \\ ∥ a^{2} δ (b^{*}) - ε (a) (b^{2})^{*} ∥ \leq ψ (a, b) \end{array}

for all a,b,c,d ∈ A. Then δ is called a (ψ, ε)-approximate quadratic *-derivation on $A$ .

Theorem 5.5 Suppose that $A$ is a C*-algebra and s ∈{-1, 1}. Let $δ : A \to A$ be a (ψ, ε)-approximate quadratic *-derivation on $A$ . Then δ is a quadratic *-derivation on $A$ .

Proof. We assume that (5.2) holds. We first show that δ is quadratic homogeneous. To do this, pick λ ∈ ℂ and

a, b \in A

. Then, we have

\begin{matrix} ∥ b^{2} (δ (λ a) - λ^{2} δ (a)) ∥ = n^{- 2 s} ∥ n^{2 s} b^{2} δ (λ a) - λ^{2} n^{2 s} b^{2} δ (a) ∥ \\ \leq n^{- 2 s} ∥ n^{2 s} b^{2} δ (λ a) - ε (n^{s} b) (λ a)^{2} ∥ + n^{- 2 s} ∥ λ^{2} ε (n^{s} b) a^{2} - λ^{2} n^{2 s} b^{2} δ (a) ∥ \\ \leq n^{- 2 s} ψ (n^{s} b, λ a) + n^{- 2 s} | λ |^{2} ψ (n^{s} b, a). \end{matrix}

So

∥ b^{2} (δ (λ a) - λ^{2} δ (a)) ∥ \leq n^{- 2 s} ψ (n^{s} b, λ a) + | λ |^{2} n^{- 2 s} ψ (n^{s} b, a),

which tends to 0 as n → ∞. Let {e_i }_i∈Ibe an approximate unit of $A$ . Then, {f(e_i )|i ∈ I} is also an approximate unit of $A$ for every polynomial f. Considering e_i instead of b in the above inequality, we conclude that δ(λa) = λ²δ(a) for all λ ∈ ℂ.

The quadraticity of δ follows from

\begin{array}{l} ∥ d^{2} (δ (a + b) + δ (a - b) - 2 δ (a) - 2 δ (b)) ∥ \\ = n^{- 2 s} ∥ n^{2 s} d^{2} δ (a + b) + n^{2 s} d^{2} δ (a - b) - 2 n^{2 s} d^{2} δ (a) - 2 n^{2 s} d^{2} δ (b) ∥ \\ \leq n^{- 2 s} [∥ n^{2 s} d^{2} δ (a + b) - ε (n^{s} d) (a + b)^{2} ∥ \\ + n^{- 2 s} ∥ n^{2 s} d^{2} δ (a - b) - ε (n^{s} d) (a - b)^{2} ∥ \\ + 2 n^{- 2 s} ∥ δ (n^{s} d) a^{2} - n^{2 s} d^{2} δ (a) ∥ + 2 n^{- 2 s} ∥ δ (n^{s} d) b^{2} - n^{2 s} d^{2} δ (b) ∥] \\ \leq n^{- 2 s} [ψ (n^{s} d, a + b) + ψ (n^{s} d, a - b) + 2 ψ (a, n^{s} d) + 2 ψ (b, n^{s} d)] \end{array}

for all

a, b, d \in A

. Thus, we have δ(a + b) + δ(a - b) -2δ(a) - 2δ(b) = 0 for all

a, b \in A

.

\begin{matrix} ‖ d^{2} (δ (a b) - (δ (a) b^{2} + a^{2} δ (b)) ‱ = n^{- 2 s} ∥ n^{2 s} d^{2} (δ (a b) - δ (a) b^{2} - a^{2} δ (b)) ∥ \\ \leq n^{- 2 s} [∥ n^{2 s} d^{2} δ (a b) - ε (n^{s} d) (a b)^{2} ∥ \\ + n^{- 2 s} ∥ ε (n^{s} d) (a b)^{2} - n^{2 s} d^{2} δ (a) b^{2} + n^{2 s} d^{2} a^{2} δ (b) ∥] \\ \leq n^{- 2 s} [ψ (n^{s} d, a b) + ψ (n^{s} d, a b)] \end{matrix}

for all $a, b, d \in A$ . So δ(ab) = δ(a)b² + a²δ(b).

The rest of the proof is similar to the proof of Theorem 5.2.

Therefore, δ is a quadratic *-derivation on $A$ . □

Corollary 5.6 Suppose that

A

is a C*-algebra and that

δ : A \to A

is a mapping for which there exist a nonnegative real number α and a positive real number p with p < 2 such that

\begin{array}{l} ∥ a^{2} δ (b) - δ (a) b^{2} ∥ \leq α ∥ a ∥^{p} ∥ b ∥^{p}, \\ ∥ ε (a) (c d)^{2} - a^{2} (δ (c) d^{2} - c^{2} δ (d)) ∥ \leq α ∥ a ∥^{p} ∥ c d ∥^{p}, \\ ∥ a^{2} δ (b^{*}) - ε (a) (b^{2})^{*} ∥ \leq α ∥ a ∥^{p} ∥ b ∥^{p} \end{array}

for all a, b, c, d ∈ A. Then δ is a quadratic *-derivation on $A$ .

Declarations

Acknowledgements

The first author and the second author were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0013211) and (NRF-2009-0070788), respectively.

Authors’ Affiliations

(1)

Department of Mathematics, University of Ulsan

(2)

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University

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Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Approximate *-derivations and approximate quadratic *-derivations on C*-algebras

Abstract

Keywords

1 Introduction and preliminaries

2 Stability of *-derivations on Banach *-algebras

3 Stability of *-derivations associated with the Jensen functional equation

4 Stability of quadratic *-derivations on Banach *-algebras

5 Superstability of *-derivations and of quadratic *-derivations On C*-algebras

Declarations

Acknowledgements

Authors’ Affiliations

References

Copyright

Approximate -derivations and approximate quadratic -derivations on C*-algebras

2 Stability of -derivations on Banach -algebras

4 Stability of quadratic -derivations on Banach -algebras

5 Superstability of -derivations and of quadratic -derivations On C*-algebras