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Generalized Derivations and Bilocal Jordan Derivations of Nest Algebras
Let be a complex Hilbert space and the collection of all linear bounded operators, is the closed subspace lattice including 0 an , then is a nest, accordingly alg is a nest algebra. It will be shown that of nest algebra, generalized derivations are generalized inner derivations, and bilocal Jordan derivations are inner derivations.
The concept of local derivations was introduced by Kadison  who showed that on a Von-nemann algebra all norm-continuous local derivations are derivations. Larson and Sourour  proved that on the algebra local derivations are derivations. M. Brešar and P. Šemrl [3, 4] generalized the results of the three authors above under a weaker condition. Shulman  showed that all local derivations on -algebra are derivations.
Based on a great deal of research works of many mathematicians, some scholars paid more interests in similar kind of problems under more generalized conditions, such as considering local derivations on nest algebras and generalized derivation. Zhu and Xiong [6, 7] proved that local derivations of nest algebra and standard operator algebra are derivations, Zhang  considered the Jordan derivations of nest algebras, Lee  discussed generalized derivations of left faithful rings. Recently, some scholars discussed some new types of derivations, as Li and Zhou  and Majieed and Zhou  investigated some new types of generalized derivations associated with Hochschild 2 cocycles, other examples are in [12–15]. In fact, under appropriate conditions, local derivations are derivations.
In this paper, we will show that of nest algebra, a generalized derivation is a generalized inner derivation, and bilocal Jordan derivations are inner derivations.
2. Some Notations and Definitions
In what follows, some notations and basic definitions are introduced.
Let be a complex Hilbert space and the collection of all linear bounded operators on , is the closed subspace lattice including 0 an , then is a nest, correspondingly the Nest algebra is .
If , we denote and if , denote , where is real inclusion, and we define , .
For all , represent the project operator from to , and .
Let be a Banach algebra and a subalgebra of , we call the linear map a generalized inner derivation if and only if for all , there exist operators and in such that ; if for all , we have , then is called a Jordan derivation; if for all , there is a Jordan derivation , such that , then is said to be a local Jordan derivation.
Definition 2.1. Let be an additive mapping, if there exists a derivation that , for all , then is called a generalized derivation.
Definition 2.2. We call the linear mapping a bilocal Jordan derivation, if for every , there is a Jordan derivation , such that .
3. Main Results
Next to give out the main conclusions.
Theorem 3.1. If is a generalized derivation, then there are operators and in , such that , for all .
Proof. From the definition of generalized derivation, we can find a derivation: , such that , for all , so when , we have , for all , denote , apparently and , for all .
Since is a derivation, by , it is an inner derivation, namely, there exists , such that , consequently Denote , , then , for all .
Theorem 3.2. If is a local Jordan derivation, then is an inner derivation.
Proof. Since is a local Jordan derivation, there exists a Jordan derivation , such that , from Theorem 2.12 in , we know that the Jordan derivation of nest algebra is an inner derivation, so there exists , such that , by imitating the proof in , we can conclude that , so , namely, is an inner derivation.
The following is the main result.
Theorem 3.3. If is a bilocal Jordan derivation, then it is an inner derivation.
Proof. We will prove this proposition by the following three steps.
(1) , where and are the kernal of and range of , respectively. In fact, since for all , for all , and is a Jordan derivation, by Theorem 2.12 in , there is an , such that , so if , we have .
(2) For all , , there exists and , such that , for all , .
For arbitrary fixed , , and for all , we know that , from step (1), we have , so there exists a linear function , such that , for all , in succession we will prove that is independent of . Take a which is linear independent of , we have , then On the other hand, , so Since is linear independent of , we know that , that is, is independent of , so can be denoted by , and Let be the linear continuous span on of , we define as follows: Obviously, is linear and , , .
Next is independent of , which reduce to show (i) , ; (ii) , where , .
In fact, (i) is evident. For (ii), since for all , and , we have , namely, , on account of , so , that is, is independent of , so we can mark by , as a result, we have Consequently
Define , now we will show that is a linear bounded operator. Because is a continuous linear function, so is bounded, consequently is bounded, according to (3.4), we know so ; on the other hand, so , this is enough to show that , , so when , That is to say is linear, so (3.7) has the form of In succession we will prove that is independent of , arbitrarily choose , where is linear independent of , then and On the other hand, So , then , that is, is independent of , which can be marked by , so (3.11) has the form of We proved that is bounded, because .
On account of the boundary of and , we know that is bounded, namely, , .
(3) For arbitrary , there is .
For all , , select , , then , and , from the result of step (2), it is easy to know that Consequently , so there exists a scalar , such that By imitating Lemma 2 mentioned in , we can prove that
Since the collection of all rank one operators is dense in , so for every , we have , let , then , considering to be a bilocal Jordan derivation, namely, , we can conclude that , so , thereby and , which shows that is an inner derivation.
The authors wish to thank the anonymous reviewers for their valuable suggestions. This work was supported by the Natural Science Foundation Project of CQ CSTC under Contract no. 2010BB2240.
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Copyright © 2011 Dangui Yan and Chengchang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.