Abstract
We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.
1. Introduction and Preliminaries
Let be an algebra over the real or complex field . An additive mapping is called a ring derivation if the functional equation is valid for all . In addition, if the identity holds for all and all , then is said to be a linear derivation.
Singer and Wermer [1] obtained a fundamental result which started investigation into the ranges of linear derivations on Banach algebras. The result states that every continuous linear derivation on a commutative Banach algebra maps into the radical. They also made a very insightful conjecture that the assumption of continuity is unnecessary. This conjecture was proved by Thomas [2]. So, in this paper, we will take into account the problems in [1, 2] for the derivations on noncommutative Banach algebras.
On the other hand, the stability problem for ring derivations on Banach algebras was considered by Miura et al. in [3]: under suitable conditions every approximate ring derivation on Banach algebra is an exact ring derivation. Šemrl [4] obtained the first stability result concerning derivations between operator algebras. As just mentioned, the study of stability problem has originally been formulated by Ulam [5]: under what condition does there exist a homomorphism near an approximate homomorphism? Hyers [6] had answered affirmatively the question of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [7] and for approximately linear mappings was presented by Rassias [8] by considering an unbounded Cauchy difference. Since then, many interesting results of the stability problems to a number of functional equations and inequalities (or involving derivations) have been investigated (see, e.g., [9–16]). The reader is referred to the book [17] for more information on stability problem with a large variety of applications.
In this paper, we will establish the stability of functional inequalities with ring derivations and will deal with the problem for the radical range of these functional inequalities on Banach algebra by considering the base of noncommutative versions for the result of Singer and Wermer: Mathieu and Murphy [18] verify that every continuous centralizing linear derivation on a Banach algebra maps into the radical and Brešar [19] proved that every centralizing linear derivation on a semiprime Banach algebra maps into the intersection of the center and the radical. Moreover, Chaudhry and Thaheem [20] showed that two ring derivations on semiprime ring with suitable property map into the center.
2. Functional Inequalities for a Derivation and Its Applications
Theorem 1. Let be a normed algebra. Assume that mappings and satisfy the assumptions, , where , , and are fixed positive real numbers with and . Suppose that is a mapping subjected to the inequalities for all . Then is a ring derivation.
Proof. By letting in (1), we get . And by putting , , and in (1), we obtain
for all . Also, by letting , , and in (1), it follows that
for all . Replacing by and setting and in (1), we have
for all . From (3), (4), and (5), we arrive at
for all . So the relations (3), (4), (5), and (6) give
for all . Replacing by and letting and in (1), the following yields
for all . Due to (7) and (8), we conclude that
for all , , .
Next we are in the position to show that is a ring derivation. We will consider two different cases for the second assumption of
Case I. Assume that
for all . We get by (7)
for all positive integers and all . Therefore, one can obtain that
for all . Due to (6) and (9), we see that
for all , . Thus .
By (2), we find that
for all , . Thus .
Case II. Assume that
for all , . We get by (7)
for all positive integers and all . Therefore, one can obtain that
for all . The remainder of the proof is similar to the proof of Case I.
Corollary 2. Let be a Banach algebra and let , , and be fixed positive real numbers with and . Suppose that mapping satisfies the assumption (1) and the first case of assumption (2) in Theorem 1. Assume that is a continuous mapping subjected to the inequality for all , , and all , , where . In addition, if a mapping is fulfilled with the following inequalities: for some and and all , where , then maps into its radical .
Proof. We first consider in (18) and in (19). It follows from Theorem 1 that is a ring derivation. In this case, is a mapping defined by for all and satisfies (7). Since is continuous, is continuous in for each fixed . Thus is -linear as in [8]. Replacing by and putting , , and in (18) with (7), we see that and so we see that for all and all . Hence is -linear.
In view of (20), we have
for all , which means that is a centralizing mapping. With the help of Mathieu and Murphy’s result [18], we arrive at the conclusion.
Corollary 3. Let be a Banach algebra with identity and let , , and be fixed positive real numbers with and . Suppose that mapping satisfies the assumption (1) and the second case of assumption (2) in Theorem 1 and suppose that is a continuous centralizing mapping subjected to (18) and (19). Then maps into its radical .
Proof. We let in (18). In the proof of Theorem 1, we find that is an additive mapping. In this case, is a mapping defined by for all and satisfies (7). As in the proof of the Corollary 2, the mapping is linear. Since contains the identity, Badora’s result [10] implies that for all . So is a centralizing linear derivation. Based on the result of Mathieu and Murphy [18], we conclude that .
3. Approximate Derivations and Their Applications
Theorem 4. Let be a Banach algebra. Assume that mappings and satisfy the following assumptions: + + < , ,, , = , ,where , , and are fixed positive real numbers with and . Suppose that is a mapping subjected to the inequalities (1) and (2). Then, there exists a unique ring derivation such that the inequality for all , where In addition, the equation holds for all .
Proof. By letting in (1), we get . Replacing by and letting and in (1), we also have
for all , which implies that
Next, by letting , and in (1), we obtain
for all . Again, replacing by and setting and in (1), we find that
for all . It follows from (25), (27), and (28) that
for all . So the relation (29) can be rewritten as
for all . Therefore, by (26) and (30), we see that
for all nonnegative integers , with and all . This means that is a Cauchy sequence. Hence the sequence converges. So one can define a mapping by for all . Letting and taking the limit , we arrive at (22).
Now we claim that the mapping is additive. By (30), one notes
for all . So we have . By virtue of (25), we get
for all , which means that . Using the similar way with (27), we feel that . Therefore, we see that
which yields that
for all . By (1), we obtain
for all , . So we know that
Due to (34) and (37), we conclude that .
In particular, by (2), we note that
for all , . Thus we get
The conditions (35) and (39) guarantee that
which implies that
Therefore, we obtain (24) and .
Now, to show uniqueness of the mapping , let us assume that is another ring derivation satisfying (22). Then, we have by (22) and (35)
for all , which means that .
From now on, we suppose that .
Corollary 5. Let be a semiprime Banach algebra. Assume that the mappings and satisfy the assumptions of Theorem 4. Suppose that is a mapping such that the inequality (18) holds for all and all . Moreover, if a mapping satisfies the conditions (2) and (20), then maps into the intersection of its center and its radical .
Proof. Now we take in (18). Replacing instead of in (24), we see that for all . Since is semiprime, we conclude that . So Theorem 4 provides that is a ring derivation and one can obtain that for all . Moreover, . Replacing by and setting and in (18), then we get This implies that Thus for all and all . Clearly, . In addition, we prove that holds for all and all (see [21]). So is linear. As in the proof of Corollary 2, we find that is a centralizing linear derivation. Therefore Brešar’s result [19] yields our claim.
Theorem 6. Let be a Banach algebra. Assume that mappings and satisfy the assumptions + + + < , , = , , = ,where , , and are fixed positive real numbers with and . Suppose that is a mapping subjected to inequalities (1) and (2). Then there exists a unique ring derivation such that the inequality for all . Moreover, (24) holds.
Proof. Letting in (1), we get . By assumption of , we should have . Thus . Replacing by in (25) and then dividing in the resulting inequality, for all . Starting from (25) and employing the same argument in the proof of the previous theorem, we obtain for all . The rest of proof can be carried out similarly as the corresponding part of Theorem 4.
Corollary 7. Let be a semiprime Banach algebra. Assume that a mapping satisfies the assumptions of Theorem 6 and that is a centralizing mapping such that the inequality (18) holds for all , , and all . Suppose that a mapping is fulfilled with the inequality (19). Then maps into the intersection of its center and its radical .
Proof. Employing the same method in the proof of Corollary 5, we find that is linear derivation. According to Brešar’s result [19], we get the result.
Corollary 8. Let be a semiprime Banach algebra with identity. Suppose that mappings and fulfilled the assumptions of Theorem 4 (resp., Theorem 6). Assume that and are mappings such that, for each , for all , , and all . If for all , then and map into the intersection of its center and its radical .
Proof. Employing the same argument in the proof of the previous corollaries, we see that and are linear derivations. Chaudhry and Thaheem’s result [20] guarantees that and . This implies that and are centralizing mappings. Hence, by Brešar’s result [19], we see that and map into .
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referees for giving useful suggestions and for the improvement of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2010-002338 and no. 2013R1A1A2A10004419). This research was also supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (no. 2012R1A1A2021721).