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

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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 A is a complex Banach *-algebra. A -linear mapping δ:D ( δ ) A is said to be a derivation on A if δ(ab) = δ(a)+ b + (b) for all a,bA, 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 aA, 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 . 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 xA.

If, for each t , α t is defined by α t (x) = ei th xe-ith for all xA, then α t is a *-automorphism of A induced by unitaries U t = ei th for each t . The action α: A u t ( A ) , tα 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 [13]).

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 [819]). In particular, those of the important functional equations are the following functional equations

f ( x + y ) = f ( x ) + f ( y ) ,
(1.1)
2 f 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 thatf:AAis a mapping with f(0) = 0 for which there exists a functionφ: A 4 [ 0 , ) such that

φ ̃ ( a , b , c , d ) : = n = 0 1 2 n + 1 φ ( 2 n a , 2 n b , 2 n c , 2 n d ) < ,
(2.1)
f ( λ a + b + c d ) - λ f ( a ) - f ( b ) - f ( c ) d - c f ( d ) φ ( a , b , c , d ) ,
(2.2)
f ( a * ) - f ( a ) * φ ( a , a , a , a )
(2.3)

for all λ T : = { λ : | λ | = 1 } and all a,b,c,dA. Then there exists a unique *-derivation δ on A satisfying

f ( a ) - δ ( a ) φ ̃ ( a , a , 0 , 0 ) ,
(2.4)

for all aA.

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

f ( 2 a ) - 2 f ( a ) φ ( a , a , 0 , 0 )

for all aA. One can use induction to show that

f ( 2 n a ) 2 n f ( 2 m a ) 2 m = k = m n 1 1 2 k + 1 μ ( 2 k a ,2 k a ,0,0 )
(2.5)

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

δ ( a ) : = lim n f ( 2 n a ) 2 n
(2.6)

for all aA. Then, we have

δ ( 1 2 k a ) = lim n 1 2 k f ( 2 n k a ) 2 n k = 1 2 k δ ( a )
(2.7)

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

1 2 n f ( 2 n ( λ a + b ) ) λ 1 2 n f ( 2 n a ) 1 2 n f ( 2 n b ) 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,bA and all λT. Putting a = b = 0 and replacing c and d by 2 nc and 2 nd, respectively, in (2.2), we get

1 2 2 n f ( 2 2 n c d ) 1 2 2 n f ( 2 n c ) ( 2 n d ) 1 2 2 n ( 2 n c ) f ( 2 n d ) 1 2 2 n μ ( 0,0,2 n c ,2 n d ) 1 2 n μ ( 0,0,2 n c ,2 n d ).

Taking the limit as n → ∞, we obtain

δ ( c d ) = δ ( c ) d + c δ ( d )
(2.9)

for all c,dA.

Next, let λ = λ1 +iλ2 where λ1, λ2, . 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 = λ i , 1 + λ i , 2 2 such that λ i , j T ( 1 i , j 2 ) . From (2.7) and (2.8), it follows that

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 ) + 1 2 δ ( λ 1 , 1 a + λ 1 , 2 a ) + i [ λ 2 ] δ ( a ) + 1 2 δ ( λ 2 , 1 a + λ 2 , 2 a ) (3) = [ λ 1 ] δ ( a ) + 1 2 λ 1 , 1 δ ( a ) + 1 2 λ 1 , 2 δ ( a ) + i [ λ 2 ] δ ( a ) + 1 2 λ 2 , 1 δ ( a ) + 1 2 λ 2 , 2 δ ( a ) (4) = λ 1 δ ( a ) + i λ 2 δ ( a ) = λ δ ( a ) (5) (6) 

for all aA. Hence, δ is -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 ) φ ̃ ( a , a , 0 , 0 ) for all aA. 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

1 2 n f ( 2 n a * ) 1 2 n f ( 2 n a ) * 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 aA. So δ is a *-derivation on A, as desired. □

Corollary 2.2 Let ε, p be positive real numbers with p < 1. Suppose thatf:AAis a mapping satisfying

f ( λ a + b + c d ) - λ f ( a ) - f ( b ) - c f ( d ) - f ( c ) d ε ( a p + b p + c p + d p ) ,
(2.10)
f ( a * ) - f ( a ) * 4 ε a p
(2.11)

for allλTand alla,b,c,dA. Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 2 ε 2 - 2 p a p

for allaA.

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:AAis a mapping with f (0) = 0 for which there exists a functionφ: A 4 [ 0 , ) satisfying (2.2), (2.3) and

n = 1 2 2 n - 1 φ a 2 n , b 2 n , c 2 n , d 2 n <

for alla,b,c,dA. Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) φ ̃ ( a , a , 0 , 0 ) ,

for allaA, where

φ ̃ ( a , b , c , d ) : = n = 1 2 n - 1 φ a 2 n , b 2 n , c 2 n , d 2 n .

Corollary 2.4 Let ε, p be positive real numbers with p > 2. Suppose thatf:AAis a mapping satisfying (2.10) and (2.11). Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 2 ε 2 p - 2 a p

for allaA.

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 LetAbe a Banach *-algebra. Suppose thatf:AAis a mapping with f (0) = 0 for which there exists a functionφ:A×A [ 0 , ) such that

φ ̃ ( a , b ) : = n = 0 1 3 n φ ( 3 n a , 3 n b ) < ,
(3.1)
2 f λ a + λ b 2 - λ f ( a ) - λ f ( b ) φ ( a , b ) ,
(3.2)
f ( a * ) - f ( a ) * φ ( a , a ) ,
(3.3)
f ( a b ) - a f ( b ) - f ( a ) b φ ( a , b )
(3.4)

for alla,bAand allλT. Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 1 3 ( φ ̃ ( a , - a ) + φ ̃ ( - a , 3 a ) )
(3.5)

for allaA.

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

- f ( a ) - f ( - a ) φ ( a , - a )

for all aA. 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 ) φ ( - a , 3 a )

for all aA. Thus,

f ( a ) - 1 3 f ( 3 a ) 1 3 f ( a ) + f ( - a ) + 2 f ( a ) - f ( - a ) - f ( 3 a ) (1) 1 3 φ ( a , - a ) + φ ( - a , 3 a ) (2) (3)

for all aA. So

1 3 n f ( 3 n a ) 1 3 m f ( 3 m a ) j = m n 1 1 3 j f ( 3 j a ) 1 3 j + 1 f ( 3 j + 1 a ) 1 3 j = m n 1 1 3 j ( μ ( 3 j a , 3 j a ) + μ ( 3 j a ,3 j + 1 a ) )
(3.6)

for all nonnegative integers n, m with n > m and all aA. It follows from (3.6) that the sequence { 1 3 n f ( 3 n a ) } is a Cauchy sequence for all aA. Since A is complete, the sequence { 1 3 n f ( 3 n a ) } is convergent. So one can define the mapping δ:AA by

δ ( a ) = lim n 1 3 n f ( 3 n a )

for all aA. By (3.2),

2 δ ( a + b 2 ) δ ( a ) δ ( b ) = lim n 1 3 n 2 f ( 3 n a + b 2 ) f ( 3 n a ) f ( 3 n b ) lim n 1 3 n μ ( 3 n a ,3 n b ) = 0

for all a,bA. Thus

2 δ a + b 2 = δ ( a ) + δ ( b )
(3.7)

for all a,bA. Since f(0) = 0, we have δ(0) = 0. Putting b = 0 in (3.7), we get 2δ ( a 2 ) =δ ( a ) for all aA and therefore δ ( a ) +δ ( b ) =2δ a + b 2 =δ ( a + b ) for all a,bA. 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

1 3 n f ( λ 3 n a ) λ 3 n f ( 3 n a ) 1 3 n μ ( 3 n a ,3 n a ).

Passing the limit as n → ∞, we get δa) = λδ(a) for all λT. Thus we can get δa) = λδ(a) for all λ 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

1 3 n f ( 3 n a * ) 1 3 n f ( 3 n a ) * 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

f ( 3 2 n a b ) 3 2 n 3 n a f ( 3 n b ) 3 2 n f ( 3 n a ) ( 3 n b ) 3 2 n 1 3 2 n μ ( 3 n a ,3 n b ) 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 + (b) for all a,bA. Hence, δ is a *-derivation on A.

Corollary 3.2 Let ε, p be positive real numbers with p < 1. Suppose thatf:AAis a mapping satisfying

2 f λ a + λ b 2 - λ f ( a ) - λ f ( b ) ε ( a p + b p ) ,
(3.8)
f ( a * ) - f ( a ) * 2 ε a p ,
(3.9)
f ( a b ) - a f ( b ) - f ( a ) b ε ( a p + b p )
(3.10)

for allλTand alla,bA. Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 3 + 3 p 3 - 3 p ε a p

for allaA.

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 LetAbe a Banach *-algebra. Suppose thatf:AAis a mapping with f(0) = 0 for which there exists a function f ( a ) - δ ( a ) 2 ε 2 p - 2 a p satisfying (3.2), (3.3), (3.4) and

n = 1 3 2 n φ a 3 n , b 3 n <

for alla,bA. Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 1 3 ( φ ̃ ( a , - a ) + φ ̃ ( - a , 3 a ) )

for allaA, where

φ ̃ ( a , b ) : = n = 1 3 n φ a 3 n , b 3 n .

Corollary 3.4 Let ε, p be positive real numbers with p > 2. Suppose thatf:AAis a mapping satisfying (3.8), (3.9) and (3.10). Then there exists a unique *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 3 p + 3 3 p - 3 ε a p

for allaA.

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 LetAbe a *-normed algebra. A mappingδ:AAis a quadratic *-derivation onAif δ satisfies the following properties:

(1) δ is a quadratic mapping,

(2) δ is quadratic homogeneous, that is, δ(λa) = λ2δ(a) for allaAand all λ ,

(3) δ(a b) = δ(a)b2 + a2δ(b) for alla,bA,

(4) δ(a*) = δ(a)* for allaA.

Theorem 4.2 Suppose thatf:AAis a mapping with f(0) = 0 for which there exists a functionφ: A 4 [ 0 , ) such that

φ ̃ ( a , b , c , d ) : = k = 0 1 4 k φ ( 2 k a , 2 k b , 2 k c , 2 k d ) < ,
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 ) φ ( a , b , c , d ) ,
(4.1)
f ( a * ) - f ( a ) * φ ( a , a , a , a )
(4.2)

for alla,b,c,dAand allλT. Also, if for each fixedaAthe mapping t → f(ta) from toAis continuous, then there exists a unique quadratic *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 1 4 φ ̃ ( a , a , 0 , 0 )

for allaA.

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

f ( 2 a ) - 4 f ( a ) φ ( a , a , 0 , 0 )

for all aA. One can use induction to show that

f ( 2 n a ) 4 n f ( 2 m a ) 4 m 1 4 k = m n 1 μ ( 2 k a ,2 k a ,0,0 ) 4 k
(4.3)

for all n > m ≥ 0 and all aA. It follows from (4.3) that the sequence { f ( 2 n a ) 4 n } is Cauchy. Since A is complete, this sequence is convergent. Define

δ ( a ) : = lim n 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

f ( 2 n ( λ a + λ b ) ) 4 n + f ( 2 n ( λ a λ b ) ) 4 n 2 λ 2 f ( 2 n a ) 4 n 2 λ 2 f ( 2 n b ) 4 n 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,bA and all λT. 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 aA and all λT. Hence,

δ ( λ a ) = λ 2 δ ( a )

for all aA and all λT. Under the assumption that f(ta) is continuous in t for each fixed aA, by the same reasoning as in the proof of [10], we obtain that δ(λa) = λ2δ(a) for all aA and all λ . Hence,

δ ( λ a ) = δ λ | λ | | λ | a = λ 2 | λ | 2 δ ( | λ | a ) = λ 2 | λ | 2 | λ | 2 δ ( a ) = λ 2 δ ( a )

for all aA and all λ (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

f ( 2 n c 2 n d ) 4 2 n + f ( 2 n c 2 n d ) 4 2 n 2 2 2 n c 2 f ( 2 n d ) 4 2 n 2 f ( 2 n c ) 2 2 n d 2 4 2 n = f ( 2 2 n c d ) 4 2 n + f ( 2 2 n c d ) 4 2 n 2 2 2 n c 2 2 2 n f ( 2 n d ) 4 n 2 f ( 2 n c ) 4 n 2 2 n d 2 2 2 n μ ( 0,0,2 n c ,2 n d ) 4 2 n μ ( 0,0,2 n c ,2 n d ) 4 n

for all c,dA.

Hence, we have

δ ( c d ) c 2 δ ( d ) δ ( c ) d 2 lim n 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 thatf:AAis a mapping such that

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 ε ( a p + b p + c p + d p )
(4.5)

for alla,b,c,dAand allλT. Also, if for each fixedaAthe mapping t → f(ta) is continuous, then there exists a unique derivation δ onAsatisfying

f ( a ) - δ ( a ) 2 ε 4 - 2 p a p

for allaA.

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:AAis a mapping with f(0) = 0 for which there exists a functionφ: A 4 [ 0 , ) satisfying (4.1), (4.2) and

k = 1 4 2 k φ a 2 k , b 2 k , c 2 k , d 2 k <

for alla,b,c,dA. Also, if for each fixedaAthe mapping t → f(ta) from toAis continuous, then there exists a unique quadratic *-derivation δ onAsatisfying

f ( a ) - δ ( a ) 1 4 φ ̃ ( a , a , 0 , 0 )

for allaA, where

φ ̃ ( a , b , c , d ) : = k = 1 4 k φ a 2 k , b 2 k , c 2 k , d 2 k

Corollary 4.5 Let ε, p be positive real numbers with p > 4. Suppose thatf:AAis a mapping satisfying (4.5). Also, if for each fixedaAthe mapping tf(ta) is continuous, then there exists a unique derivation δ onAsatisfying

f ( a ) - δ ( a ) 2 ε 2 p - 4 a p

for allaA.

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 thatAis a *-normed algebra and s {1, -1}. Letδ:AAbe a mapping for which there exist a mapping ε:AA, and a function ψ:A×A satisfying

lim n n - s ψ ( n s a , b ) = lim n n - s ψ ( a , n s b ) = 0 ( a , b A )
(5.1)

such that

a δ ( b ) - ε ( a ) b ψ ( a , b ) ε ( a ) c d - a ( δ ( c ) d - c δ ( d ) ) ψ ( a , c d ) a δ ( b ) * - ε ( a ) b * ψ ( a , b )

for alla,b,c,dA. Then δ is called a (ψ, ε)-approximate *-derivation onA.

Theorem 5.2 LetAbe a C*-algebra. Then any (ψ, ε)-approximate *-derivation δ onAis a *-derivation.

Proof. We assume that (5.1) holds. Let a,bA and λ . We have

b ( δ ( λ a ) - λ δ ( a ) ) n - s n s b δ ( λ a ) - λ n s b δ ( a ) (1) 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) n - s ψ ( n s b , λ a ) + n - s | λ | ψ ( n s b , a ) , (3) (4) 

which tends to zero as n → ∞, and so b(δ(λa) - λδ(a)) = 0 for all bA. Let {e i }iIbe 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 aA and λ .

The additivity of δ follows from

c ( δ ( a + b ) δ ( a ) δ ( b ) ) 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 ) n s ψ ( n s c , a + b ) + n s ψ ( n s c , a ) + n s ψ ( n s c , b ).

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

The following computation

z ( δ ( a b ) - δ ( a ) b - a δ ( b ) ) 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 ) ) n - s ψ ( n s z , a b ) + n - s ψ ( n s z , a b )

yields that δ(ab) = δ(a)b + (b) for all a,bA.

Finally, on the involution, we have that

z ( δ ( a * ) - δ ( a ) * ) 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) n - s ψ ( n - s z , a * ) + n - s ψ ( n s z , a ) . (3) (4)

Thus, δ(a)* = δ(a) * for all aA. □

Therefore, δ is a *-derivation on A.

Corollary 5.3 Suppose thatAis a C*-algebra and thatδ:AAis 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

a δ ( b ) - ε ( a ) b 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 ) ) α ( a p 1 + c d p 2 ) + β a q 1 c d q 2 , a δ ( b ) * - ε ( a ) b * α ( a p 1 + b p 2 ) + β a q 1 b q 2

for all a,b,c,dA. Then δ is a *-derivation of A.

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

Definition 5.4 Suppose thatAis a *-normed algebra and s {-1, 1}. Letδ:AAbe a mapping for which there exist a functionψ:A×A [ 0 , ) and a mappingε:AAsatisfying

lim n n - 2 s ψ ( n s a , b ) = lim n n - 2 s ψ ( a , n s b ) = 0 ( a , b A )
(5.2)

such that

a 2 δ ( b ) ε ( a ) b 2 ψ ( a , b ) ε ( a ) ( c d ) 2 a 2 ( δ ( c ) d 2 c 2 δ ( d ) ) ψ ( a , c d ) a 2 δ ( b * ) ε ( a ) ( b 2 ) * ψ ( a , b )

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

Theorem 5.5 Suppose thatAis a C*-algebra and s {-1, 1}. Letδ:AAbe a (ψ, ε)-approximate quadratic *-derivation onA. Then δ is a quadratic *-derivation onA.

Proof. We assume that (5.2) holds. We first show that δ is quadratic homogeneous. To do this, pick λ and a,bA. Then, we have

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 ) 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 ) n 2 s ψ ( n s b , λ a ) + n 2 s | λ | 2 ψ ( n s b , a ).

So

b 2 ( δ ( λ a ) - λ 2 δ ( a ) ) 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 }iIbe 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 .

The quadraticity of δ follows from

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 ) 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 ) ] 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 ) ]

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

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 ) ) 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 ) ] n 2 s [ ψ ( n s d , a b ) + ψ ( n s d , a b ) ]

for all a,b,dA. 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 thatAis a C*-algebra and thatδ:AAis a mapping for which there exist a nonnegative real number α and a positive real number p with p < 2 such that

a 2 δ ( b ) δ ( a ) b 2 α a p b p , ε ( a ) ( c d ) 2 a 2 ( δ ( c ) d 2 c 2 δ ( d ) ) α a p c d p , a 2 δ ( b * ) ε ( a ) ( b 2 ) * α a p b p

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

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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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Correspondence to Choonkil Park.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Keywords

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