Asymptotic aspect of derivations in Banach algebras.

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

Introduction and preliminaries

Let be an algebra. A linear mapping is called a left derivation (resp., derivation) if (resp., ) is fulfilled for all . A linear mapping is said to be a left Jordan derivation if holds for all . A linear mapping is called a generalized left derivation if there exists a linear left derivation such that for all . A linear mapping is said to be a generalized left Jordan derivation if there exists a linear left Jordan derivation such that for all .

Singer and Wermer [1] obtained a fundamental result which started the investigation of the ranges of linear derivations on Banach algebras. The result, which is called the Singer-Wermer theorem, states that every continuous linear derivation on a commutative Banach algebra maps into the radical. In the same paper, they made a very insightful conjecture: that the assumption of continuity is unnecessary. Thomas [2] proved this conjecture. Hence linear derivations on Banach algebras (if everywhere defined) genuinely belong to the noncommutative setting.

On the other hand, the study of stability problems had been formulated by Ulam [3]. Hyers [4] had answered affirmatively the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [5] for additive mappings and by Rassias [6] for linear mappings by considering an unbounded difference. In particular, the stability result concerning derivations between operator algebras was first obtained by Šemrl [7]. Badora gave a generalization of the Bourgin result and he also dealt with the stability and the superstability of Bourgin-type for derivations; see [8-10] and the references therein. Recently, the stability problems for derivations are considered by some authors in [11-13].

In this work, we first take into account the functional inequality which expands the functional inequality in [14]. It is well known that every ring left derivation (resp., ring left Jordan derivation) on a semiprime ring maps into its center; see [15, 16]. Considering the base of the previous result, we show that every approximate ring left derivation on a semiprime normed algebra maps into its center and then, by using this fact, we prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. We also establish the functional inequalities related to a linear derivation and their stability. In particular, mappings satisfying such functional inequalities on a semiprime Banach algebra are linear derivations which map into the intersection of the center and the radical. We finally investigate a linear generalized left Jordan derivation on a semisimple Banach algebra with application.

Approximate left derivations

We first demonstrate the following proposition quoted in this work.

Proposition 2.1

[15], Proposition 1.6

Let be a ring, be a left -module, and be a left derivation.

Suppose that with implies or . If , then is commutative.

Suppose that is a semiprime ring. Then δ is a derivation which maps into its center.

Let be a normed algebra. An additive mapping is said to be an approximate ring derivation (resp., approximate ring left derivation) if for some , for all . In addition, if for all and , then δ is called an approximate linear derivation (resp., approximate linear left derivation).

From now on, we suppose that . The commutator will be denoted by . We start our investigations for approximate ring left derivations with some results.

Theorem 2.2

Let be a semiprime normed algebra. Assume that is a fixed integer and are fixed positive real numbers, where () and . Suppose that is a mapping such that for all and for some , for all . Then δ is an approximate ring derivation which maps into its center .

Proof

By letting in (2.1), we get . And we put in (2.1) and then set to obtain It follows by the result of [14] that δ is additive. In particular, in view of (2.2), we find that δ is an approximate ring left derivation.

By virtue of (2.2), we see that for all . Combining (2.2) and (2.4), we get for all . It follows from (2.2) and (2.5) that for all . Replacing x by nx in (2.6) and then dividing on both sides by , we have for all and all positive integer n. Taking the limit as in the above relation, we see that Just proceeding as in the proof of Proposition 2.1, we get for all . That is, belong to its center . So δ is an approximate ring derivation. Therefore we arrive at the desired conclusion. □

Theorem 2.3

Let be a noncommutative prime normed algebra. Assume that is a fixed integer and are fixed positive real numbers, where () and . Suppose that is a mapping subject to the conditions (2.1) and (2.2). Then δ is identically zero.

Employing the same argument as the proof Theorem 2.2, we feel that δ satisfies equation (2.7). Since is noncommutative, choose a z that does not belong to the center of . Using the same method in the proof of Proposition 2.1, we see that , which completes the proof. □

Theorem 2.4

Let be a semisimple Banach algebra. Assume that is a fixed integer and are fixed positive real numbers, where (), and . Suppose that is a mapping subject to for all and all , where () and the inequality (2.2). Then δ is a continuous.

As we did in the proof of Theorem 2.2, we get . We take in (2.8) and then put to have for all and all . Now we consider in (2.9) and so δ satisfies the inequality (2.3). Hence we find that δ is additive [14].

Next, setting and in (2.3), we obtain . Letting , and in (2.9), we get for all and all and so we see that δ is linear [17].

Since semisimple algebras are semiprime [18], Theorem 2.2 guarantees that δ is an approximate linear derivation. Therefore δ is continuous [14]. The proof is complete. □

Inequalities related to a linear derivation

In this section, we write a unit element of algebra by e.

Theorem 3.1

Let be a semiprime unital Banach algebra. Suppose that is a mapping subject to the inequality (2.8) and for some , for all . Then δ is a linear derivation which maps into the intersection of its center and its radical .

Employing the same way in the proof Theorem 2.4, we find that δ is linear. By linearization of (3.1) and additivity of δ, we get for all . Substituting −x for x in (3.2), we have for all . Equations (3.2) and (3.3) yield for all . We have therefore

Putting for y in (3.4), we obtain for all . On the other hand, we have from (3.4) and the equation the result for all . By comparing (3.5) and (3.6), we arrive at for all . Applying equation (3.7) with (3.1) and (3.4), we have for all . Letting in (3.8) and then dividing the resulting inequality by , we get for all and all positive integers n. Taking the limit of (3.9), it is reduced to the equation Putting in (3.10), we get . Again, considering in (3.10), we easily prove that This means that δ is a linear left Jordan derivation.

On the other hand, from Vukman’s result [16], we see that δ is a linear derivation with . Since is a commutative Banach algebra, the Singer-Wermer theorem tells us that maps into and thus . Using the semiprimeness of as well as the identity we have . Therefore , which concludes the proof. □

As consequences of Theorem 3.1, we get the following.

Corollary 3.2

Let be a unital semisimple Banach algebra. Assume that a mapping satisfies the assumptions of Theorem 3.1. Then δ is identically zero.

Now we consider the result which is needed in the following theorems.

Lemma 3.3

Let be a Banach algebra. Suppose that is a bilinear mapping and that ξ and η are mappings satisfying for all . If is semiprime or unital, then ξ and η are linear mappings.

Note that, for all and all , Hence we see that, for all ,

If is unital, then we see that by letting in (3.11).

If is nonunital, then lies in the right annihilator of . If is semiprime, then , so that for all and all .

Observe that, for all , Hence for all . As above, we get for all , so that ξ is linear.

Similarly, one can prove that η is linear. □

Theorem 3.4

Let be a semiprime Banach algebra. Assume that is a fixed integer and are fixed positive real numbers, where (), and . Suppose that is a mapping with such that, for some , for all and all , where () and for some and all . Then δ is a linear derivation which maps into the intersection of its center and its radical .

We let in (3.12) and then put to have for all and all . Now we consider in (3.14). It follows from the result in [14] that there exists a unique additive mapping defined by Moreover, holds for all .

Letting , and in (3.14), we find that for all and all . This implies that Thus , so that for all and all . Thus we see that is linear [17].

By (3.13), we see that Hence we arrive at It follows from Lemma 3.3 that δ is linear. Then we have by (3.15) that . Therefore That is, δ is a linear left Jordan derivation.

The remainder of the proof can be carried out similarly to the corresponding part of Theorem 3.1. □

Theorem 3.5

Let be a unital Banach algebra. Assume that is a fixed integer and are fixed positive real numbers, where (), and . Suppose that is a mapping with such that, for some , for all and all , where () and (3.13). If for all irrational numbers p, then δ is a linear left Jordan derivation. In this case is a semiprime unital Banach algebra, δ is a linear derivation which maps into the intersection of its center and its radical .

We first consider in (3.16). We see by the result in [14] that there is a unique additive mapping defined by (3.15). In addition, for all .

Also we set in (3.16). And we take in (3.16) and then let to have for all . Putting and in (3.17), we obtain for all , which shows that Hence . So we have for all .

We have by (3.13) This implies that

Again, by virtue of (3.13), we see that This implies that Comparing (3.18) and (3.19), we arrive at for all . Since contains the unit element, we find that . Equation (3.19) can be written

Letting in (3.20), we have . Now we obtain by additivity of δ for all and all . So for all . This fact and the assumption of δ imply that for all . Considering in (3.20), we have for all and all . Thus δ is -linear. Hence we see that for all and all . So we see that δ is -linear. In view of (3.20), we get Thereby δ is a linear left Jordan derivation.

On the other hand, if is semiprime unital Banach algebra, then the rest of the proof is similar to the corresponding part of Theorem 3.1. □

Theorem 3.6

Let be a semisimple Banach algebra. Assume that is a fixed integer and are fixed positive real numbers, where (), and . Suppose that, for each , is a mapping with such that, for some , for all and all , where () and for some and all . Then is a linear generalized left Jordan derivation associated with a linear left Jordan derivation . In this case, is continuous.

It is well known that semisimple algebras are semiprime [18]. As we saw in the proof of Theorem 3.4, is a linear left Jordan derivation. In addition, we see that there exists a unique linear mapping defined by

According to (3.23) and (3.24), we see that which implies that So we obtain from (3.25) In particular, the left-side of equation (3.26) is a bilinear mapping. Lemma 3.3 guarantees that is linear. By (3.24), we have . Equation (3.25) gives Thus is a linear generalized left Jordan derivation.

Therefore, since is semisimple, we conclude that is continuous; see [19]. This completes the proof. □