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Approximate *-derivations and approximate quadratic *-derivations on C*-algebras
Journal of Inequalities and Applications volume 2011, Article number: 55 (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.
1 Introduction and preliminaries
Suppose that is a complex Banach *-algebra. A ℂ-linear mapping is said to be a derivation on if δ(ab) = δ(a)+ b + aδ(b) for all , where D(δ) is a domain of δ and D(δ) is dense in . If δ satisfies the additional condition δ(a*) = δ(a)* for all , then δ is called a *-derivation on . It is well known that if is a C*-algebra and D(δ) is , then the derivation δ is bounded.
A C*-dynamical system is a triple (, G, α) consisting of a C*-algebra , a locally compact group G, and a pointwise norm continuous homomorphism α of G into the group Aut() of *-automorphisms of . Every bounded *-derivation δ arises as an infinitesimal generator of a dynamical system for ℝ. In fact, if δ is a bounded *-derivation of on a Hilbert space , then there exists an element h in the enveloping von Neumann algebra such that
for all .
If, for each t ∈ ℝ, α t is defined by α t (x) = ei th xe-ith for all , then α t is a *-automorphism of induced by unitaries U t = ei th for each t ∈ ℝ. The action , t → α t , is a strongly continuous one-parameter group of *-automorphisms of . 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.
2 Stability of *-derivations on Banach *-algebras
In this section, let be a Banach *-algebra. We prove the stability of *-derivations on .
Theorem 2.1 Suppose thatis a mapping with f(0) = 0 for which there exists a functionsuch that
for all and all . Then there exists a unique *-derivation δ on satisfying
for all .
Proof. Setting a = b, c = d = 0 and λ = 1 in (2.2), we have
for all . One can use induction to show that
for all n > m ≥ 0 and all . It follows from (2.5) and (2.1) that the sequence is Cauchy. Due to the completeness of , this sequence is convergent. Define
for all . 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 and all . 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 .
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 such that . From (2.7) and (2.8), it follows that
for all . Hence, δ is ℂ-linear, and so it is a derivation on . Moreover, it follows from (2.5) with m = 0 and (2.6) that for all . 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 . So δ is a *-derivation on , as desired. □
Corollary 2.2 Let ε, p be positive real numbers with p < 1. Suppose thatis a mapping satisfying
for alland all. 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 2.1, we get the desired result. □
Similarly, we can obtain the following. We will omit the proof.
Theorem 2.3 Suppose thatis a mapping with f (0) = 0 for which there exists a functionsatisfying (2.2), (2.3) and
for all. Then there exists a unique *-derivation δ onsatisfying
for all, where
Corollary 2.4 Let ε, p be positive real numbers with p > 2. Suppose thatis a mapping satisfying (2.10) and (2.11). 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 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 .
Theorem 3.1 Letbe a Banach *-algebra. Suppose thatis a mapping with f (0) = 0 for which there exists a functionsuch that
for alland all. Then there exists a unique *-derivation δ onsatisfying
for all.
Proof. Letting λ = 1 and b = -a in (3.2), we get
for all . Letting λ = 1 and replacing a and b by -a and 3a, respectively, in (3.2), we get
for all . Thus,
for all . So
for all nonnegative integers n, m with n > m and all . It follows from (3.6) that the sequence is a Cauchy sequence for all . Since is complete, the sequence is convergent. So one can define the mapping by
for all . By (3.2),
for all . Thus
for all . Since f(0) = 0, we have δ(0) = 0. Putting b = 0 in (3.7), we get for all and therefore for all . 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 . 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 . Hence, δ is a *-derivation on .
Corollary 3.2 Let ε, p be positive real numbers with p < 1. Suppose thatis a mapping satisfying
for alland all. Then there exists a unique *-derivation δ onsatisfying
for all.
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 Letbe a Banach *-algebra. Suppose thatis a mapping with f(0) = 0 for which there exists a functionsatisfying (3.2), (3.3), (3.4) and
for all. Then there exists a unique *-derivation δ onsatisfying
for all, where
Corollary 3.4 Let ε, p be positive real numbers with p > 2. Suppose thatis a mapping satisfying (3.8), (3.9) and (3.10). Then there exists a unique *-derivation δ onsatisfying
for all.
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 .
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
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
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
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
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. □
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 thatis a *-normed algebra and s ∈{1, -1}. Letbe a mapping for which there exist a mapping , and a function satisfying
such that
for all. Then δ is called a (ψ, ε)-approximate *-derivation on.
Theorem 5.2 Letbe a C*-algebra. Then any (ψ, ε)-approximate *-derivation δ onis a *-derivation.
Proof. We assume that (5.1) holds. Let and λ ∈ ℂ. We have
which tends to zero as n → ∞, and so b(δ(λa) - λδ(a)) = 0 for all . Let {e i }i∈Ibe an approximate unit of . If we replace b with {e i }, then we have
for all i ∈ I. So we conclude that δ(λa) = λδ(a) for all and λ ∈ ℂ.
The additivity of δ follows from
By the same process, using the approximate unit of , we have that δ(a + b) - δ(a) -δ(b) for all .
The following computation
yields that δ(ab) = δ(a)b + aδ(b) for all .
Finally, on the involution, we have that
Thus, δ(a)* = δ(a) * for all . □
Therefore, δ is a *-derivation on .
Corollary 5.3 Suppose thatis a C*-algebra and thatis 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 . Then δ is a *-derivation of .
Next, we prove the superstability of quadratic *-derivations on C*-algebras.
Definition 5.4 Suppose thatis a *-normed algebra and s ∈{-1, 1}. Letbe a mapping for which there exist a functionand a mappingsatisfying
such that
for all a,b,c,d ∈ A. Then δ is called a (ψ, ε)-approximate quadratic *-derivation on.
Theorem 5.5 Suppose thatis a C*-algebra and s ∈{-1, 1}. Letbe a (ψ, ε)-approximate quadratic *-derivation on. Then δ is a quadratic *-derivation on.
Proof. We assume that (5.2) holds. We first show that δ is quadratic homogeneous. To do this, pick λ ∈ ℂ and . Then, we have
So
which tends to 0 as n → ∞. Let {e i }i∈Ibe an approximate unit of . Then, {f(e i )|i ∈ I} is also an approximate unit of 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
for all . Thus, we have δ(a + b) + δ(a - b) -2δ(a) - 2δ(b) = 0 for all .
for all . 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 . □
Corollary 5.6 Suppose thatis a C*-algebra and thatis a mapping for which there exist a nonnegative real number α and a positive real number p with p < 2 such that
for all a, b, c, d ∈ A. Then δ is a quadratic *-derivation on .
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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.
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Jang, S.Y., Park, C. Approximate *-derivations and approximate quadratic *-derivations on C*-algebras. J Inequal Appl 2011, 55 (2011). https://doi.org/10.1186/1029-242X-2011-55
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DOI: https://doi.org/10.1186/1029-242X-2011-55