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

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.

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

A C*-dynamical system is a triple (\mathcal{A}, G, α) consisting of a C*-algebra \mathcal{A}, a locally compact group G, and a pointwise norm continuous homomorphism α of G into the group Aut(\mathcal{A}) of *-automorphisms of \mathcal{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 \mathcal{A} on a Hilbert space \mathcal{H}, then there exists an element h in the enveloping von Neumann algebra {\mathcal{A}}^{{}^{″}} such that

for all x\in \mathcal{A}.

If, for each t ∈ ℝ, α _t is defined by α _t (x) = eⁱ th xe^-ith for all x\in \mathcal{A}, then α _t is a *-automorphism of \mathcal{A} induced by unitaries U _t = eⁱ th for each t ∈ ℝ. The action \alpha :ℝ\to Aut\left(\mathcal{A}\right), t → α _t , is a strongly continuous one-parameter group of *-automorphisms of \mathcal{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

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.

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

Theorem 2.1 Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping with f(0) = 0 for which there exists a function\phi :{\mathcal{A}}^4\to \left[0,\infty \right)such that

for all \lambda \in T:=\left\{\lambda \in ℂ:\left|\lambda \right|=1\right\} and all a,b,c,d\in \mathcal{A}. Then there exists a unique *-derivation δ on \mathcal{A} satisfying

for all a\in \mathcal{A}.

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

for all a\in \mathcal{A}. One can use induction to show that

for all n > m ≥ 0 and all a\in \mathcal{A}. It follows from (2.5) and (2.1) that the sequence \left\{\frac{f{(2}^{n}a)}{2^n}\right\} is Cauchy. Due to the completeness of \mathcal{A}, this sequence is convergent. Define

for all a\in \mathcal{A}. Then, we have

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

Taking the limit as n → ∞, we obtain

for all a,b\in \mathcal{A} and all \lambda \in T. Putting a = b = 0 and replacing c and d by 2 ⁿc and 2 ⁿd, respectively, in (2.2), we get

Taking the limit as n → ∞, we obtain

for all c,d\in \mathcal{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 {\gamma }_i=\frac{{\lambda }_{i,1}+{\lambda }_{i,2}}{2} such that {\lambda }_{i,j}\in T\left(1\le i,j\le 2\right). From (2.7) and (2.8), it follows that

for all a\in \mathcal{A}. Hence, δ is ℂ-linear, and so it is a derivation on \mathcal{A}. Moreover, it follows from (2.5) with m = 0 and (2.6) that \parallel \delta \left(a\right)-f\left(a\right)\parallel \le \tilde{\phi }\left(a,a,0,0\right) for all a\in \mathcal{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

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

Corollary 2.2 Let ε, p be positive real numbers with p < 1. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping satisfying

for all\lambda \in Tand alla,b,c,d\in \mathcal{A}. Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{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 thatf:\mathcal{A}\to \mathcal{A}is a mapping with f (0) = 0 for which there exists a function\phi :{\mathcal{A}}^4\to \left[0,\infty \right)satisfying (2.2), (2.3) and

for alla,b,c,d\in \mathcal{A}. Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}, where

Corollary 2.4 Let ε, p be positive real numbers with p > 2. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping satisfying (2.10) and (2.11). Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{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. □

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 \mathcal{A}.

Theorem 3.1 Let\mathcal{A}be a Banach *-algebra. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping with f (0) = 0 for which there exists a function\phi :\mathcal{A}\times \mathcal{A}\to \left[0,\infty \right)such that

for alla,b\in \mathcal{A}and all\lambda \in T. Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}.

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

for all a\in \mathcal{A}. Letting λ = 1 and replacing a and b by -a and 3a, respectively, in (3.2), we get

for all a\in \mathcal{A}. Thus,

for all a\in \mathcal{A}. So

for all nonnegative integers n, m with n > m and all a\in \mathcal{A}. It follows from (3.6) that the sequence \left\{\frac{1}{3^n}f\left(3^{n}a\right)\right\} is a Cauchy sequence for all a\in \mathcal{A}. Since \mathcal{A} is complete, the sequence \left\{\frac{1}{3^n}f\left(3^{n}a\right)\right\} is convergent. So one can define the mapping \delta :\mathcal{A}\to \mathcal{A} by

for all a\in \mathcal{A}. By (3.2),

for all a,b\in \mathcal{A}. Thus

for all a,b\in \mathcal{A}. Since f(0) = 0, we have δ(0) = 0. Putting b = 0 in (3.7), we get 2\delta \left(\frac{a}{2}\right)=\delta \left(a\right) for all a\in \mathcal{A} and therefore \delta \left(a\right)+\delta \left(b\right)=2\delta \left(\frac{a+b}{2}\right)=\delta \left(a+b\right) for all a,b\in \mathcal{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

Passing the limit as n → ∞, we get δ(λa) = λδ(a) for all \lambda \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

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

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

Corollary 3.2 Let ε, p be positive real numbers with p < 1. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping satisfying

for all\lambda \in Tand alla,b\in \mathcal{A}. Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{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\mathcal{A}be a Banach *-algebra. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping with f(0) = 0 for which there exists a function\parallel f\left(a\right)-\delta \left(a\right)\parallel \le \frac{2\epsilon }{2^p-2}\parallel a{\parallel }^psatisfying (3.2), (3.3), (3.4) and

for alla,b\in \mathcal{A}. Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}, where

Corollary 3.4 Let ε, p be positive real numbers with p > 2. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping satisfying (3.8), (3.9) and (3.10). Then there exists a unique *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}.

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

In this section, we prove the stability of quadratic *-derivations on a Banach *-algebra \mathcal{A}.

Definition 4.1 Let\mathcal{A}be a *-normed algebra. A mapping\delta :\mathcal{A}\to \mathcal{A}is a quadratic *-derivation on\mathcal{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 alla\in \mathcal{A}and all λ ∈ ℂ,

(3) δ(a b) = δ(a)b² + a²δ(b) for alla,b\in \mathcal{A},

(4) δ(a*) = δ(a)* for alla\in \mathcal{A}.

Theorem 4.2 Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping with f(0) = 0 for which there exists a function\phi :{\mathcal{A}}^4\to \left[0,\infty \right)such that

for alla,b,c,d\in \mathcal{A}and all\lambda \in T. Also, if for each fixeda\in \mathcal{A}the mapping t → f(ta) from ℝ to\mathcal{A}is continuous, then there exists a unique quadratic *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}.

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

for all a\in \mathcal{A}. One can use induction to show that

for all n > m ≥ 0 and all a\in \mathcal{A}. It follows from (4.3) that the sequence \left\{\frac{f{(2}^{n}a)}{4^n}\right\} is Cauchy. Since \mathcal{A} 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

for all a,b\in \mathcal{A} and all \lambda \in T. Putting λ = 1 in (4.4), we obtain that δ is a quadratic mapping. Setting b: = a in (4.4), we get

for all a\in \mathcal{A} and all \lambda \in T. Hence,

for all a\in \mathcal{A} and all \lambda \in T. Under the assumption that f(ta) is continuous in t ∈ ℝ for each fixed a\in \mathcal{A}, by the same reasoning as in the proof of [10], we obtain that δ(λa) = λ²δ(a) for all a\in \mathcal{A} and all λ ∈ ℝ. Hence,

for all a\in \mathcal{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

for all c,d\in \mathcal{A}.

Hence, we have

Thus, δ is a quadratic *-derivation on \mathcal{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 thatf:\mathcal{A}\to \mathcal{A}is a mapping such that

for alla,b,c,d\in \mathcal{A}and all\lambda \in T. Also, if for each fixeda\in \mathcal{A}the mapping t → f(ta) is continuous, then there exists a unique derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{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 thatf:\mathcal{A}\to \mathcal{A}is a mapping with f(0) = 0 for which there exists a function\phi :{\mathcal{A}}^4\to \left[0,\infty \right)satisfying (4.1), (4.2) and

for alla,b,c,d\in \mathcal{A}. Also, if for each fixeda\in \mathcal{A}the mapping t → f(ta) from ℝ to\mathcal{A}is continuous, then there exists a unique quadratic *-derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{A}, where

Corollary 4.5 Let ε, p be positive real numbers with p > 4. Suppose thatf:\mathcal{A}\to \mathcal{A}is a mapping satisfying (4.5). Also, if for each fixeda\in \mathcal{A}the mapping t → f(ta) is continuous, then there exists a unique derivation δ on\mathcal{A}satisfying

for alla\in \mathcal{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. □

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\mathcal{A}is a *-normed algebra and s ∈{1, -1}. Let\delta :\mathcal{A}\to \mathcal{A}be a mapping for which there exist a mapping \epsilon :\mathcal{A}\to \mathcal{A}, and a function \psi :\mathcal{A}\times \mathcal{A}\to ℝ satisfying

such that

for alla,b,c,d\in \mathcal{A}. Then δ is called a (ψ, ε)-approximate *-derivation on\mathcal{A}.

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

Proof. We assume that (5.1) holds. Let a,b\in \mathcal{A} and λ ∈ ℂ. We have

which tends to zero as n → ∞, and so b(δ(λa) - λδ(a)) = 0 for all b\in \mathcal{A}. Let {e _i }_i∈Ibe an approximate unit of \mathcal{A}. If we replace b with {e _i }, then we have

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

The additivity of δ follows from

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

The following computation

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

Finally, on the involution, we have that

Thus, δ(a)* = δ(a) * for all a\in \mathcal{A}. □

Therefore, δ is a *-derivation on \mathcal{A}.

Corollary 5.3 Suppose that\mathcal{A}is a C*-algebra and that\delta :\mathcal{A}\to \mathcal{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

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

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

Definition 5.4 Suppose that\mathcal{A}is a *-normed algebra and s ∈{-1, 1}. Let\delta :\mathcal{A}\to \mathcal{A}be a mapping for which there exist a function\psi :\mathcal{A}\times \mathcal{A}\to \left[0,\infty \right)and a mapping\epsilon :\mathcal{A}\to \mathcal{A}satisfying

such that

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

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

So