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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 407427, 11 pages
http://dx.doi.org/10.1155/2013/407427
Research Article

Characterizing Derivations on Von Neumann Algebras by Local Actions

Department of Mathematics, Shanxi University, Taiyuan 030006, China

Received 25 August 2013; Accepted 25 November 2013

Academic Editor: P. Veeramani

Copyright © 2013 Xiaofei Qi and Jia Ji. 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.

Abstract

Let be any von Neumann algebra without central summands of type and a core-free projection with the central carrier . For any scalar , it is shown that every additive map on satisfies whenever if and only if (1) , , where is an additive derivation and is a central valued additive map vanishing on with ; (2) , is a derivation with for each .

1. Introduction

Let be an algebra over a field . For a scalar and for , we say that commutes with up to a factor if . The notion of commutativity up to a factor for pairs of operators is an important concept and has been studied in the context of operator algebras and quantum groups [1, 2]. Motivated by this, a binary operation , called -Lie product of and , was introduced in [3]. An additive map is called an additive -Lie derivation if holds for all . It is clear that a -Lie derivation is a derivation if ; is a Lie derivation if ; is a Jordan derivation if . The structure of -Lie derivations on various operator algebras was discussed by several authors (see [35] and the references therein).

Recently, the question of under what conditions that a map becomes a (Jordan) derivation attracted much attention of many researchers (see [69] and the references therein). For -Lie derivations, an additive (a linear) map on is said to be -Lie derivable at a point if for any with . Clearly, this definition is only valid for -Lie commutators, that is, the elements of the form . For instance, if and , as the unit may not be a commutator in general, there is no sense to define that is Lie derivable at . Since zero is a -Lie commutator for any and any algebra, as a start, Qi and Hou [10] characterized the linear maps Lie derivable at zero between -subspace lattice algebras. In [11] Qi et al. considered further the additive maps -Lie derivable at zero on unital prime algebras over a field containing a non-trivial idempotent . Since factor von Neumann algebras are prime, as a consequence of the result for prime algebras, all additive maps -Lie derivable at zero on factor von Neumann algebras are characterized. Recently, Qi et al. [12] gave a characterization of additive maps -Lie derivable at zero on general von Neumann algebras for all possible .

Lu and Jing [13] considered the question of characterizing Lie derivations from another direction. Let be a Banach space with and the algebra of all bounded linear operators acting on . They [13] showed that, if is a linear map satisfying for any with (resp., , where is a fixed nontrivial idempotent), then , where is a derivation of and is a linear map vanishing at commutators with (resp., ). Later, this result was generalized to the additive maps on triangular algebras and prime rings in [14, 15], respectively. Let be a von Neumann algebra without central summands of type and an additive map. For -Lie derivations, Qi and Hou [16] showed that satisfies for any with if and only if there exists an additive derivation such that (1) , , where is an additive map vanishing each commutator whenever ; (2) , ; (3) , for all ; (4) , , and for all .

Let be any von Neumann algebra and let . Recall that the central carrier of , denoted by , is the intersection of all central projections such that . If is self-adjoint, then the core of , denoted by , is sup. Particularly, if is a projection, it is clear that is the largest central projection . A projection is called core free if . It is easy to see that if and only if . By [17], if is a von Neumann algebra without central summands of type , there exists a nonzero core-free projection with . So, it is easily seen that is a von Neumann algebra without central summands of type if and only if it has a projection with and . Fixed such . The purpose of the present paper is to give a complete characterization of additive maps satisfying for any with on von Neumann algebras without central summands of type for all possible .

Let be a von Neumann algebra without central summands of type and a nonzero core-free projection with . Assume that is an additive map. In this paper, we prove that satisfies for any with if and only if (1) , for all , where is an additive derivation, is an additive map vanishing each commutator whenever , and denotes the center of (Theorem 1); (2) , is an additive derivation satisfying for all (Theorem 4).

2. Characterizing Lie Derivations by Local Action

In this section, we consider the question of characterizing Lie derivations by local action at a core-free projection with on general von Neumann algebras having no central summands of type .

Theorem 1. Let be a von Neumann algebra without central summands of type and a nonzero core-free projection with . Suppose that is an additive map. Then satisfies for any with if and only if there exists an additive derivation and an additive map vanishing each commutator whenever such that for all , where denotes the center of .

We first give two lemmas, which are needed.

Lemma 2 (see [17]). Let be a von Neumann algebra. For projections , if and , then commutes with and for all implies .

Lemma 3 (see [16]). Let be a von Neumann algebra. Assume that is a projection with and . Then .

Proof of Theorem 1. By the definitions of core and central carrier, is also core free and . For convenience, denote , , where and . Then .
The “if” part is obvious. We will prove the “only if” part by checking several claims.
Claim  1. Consider . Since , we have , and so
For any , by , we get ; that is, Let . Combining (2) with (1), one gets It follows that , which implies for all .
Similarly, for any , by using the equation , one can show that holds for all . It follows from (4)-(5) and Lemma 2 that , as desired.
Let and define a map by for every . It is easily checked that, for any , Moreover, by Claim 1, we have
Claim  2. Consider , . Here we only give the proof for . For the case , the proof is similar.
For any , let . Since , we have Note that . Then, by (1) and (7), the above equation reduces to , which implies . Hence .
Claim  3. Consider  . Take any . Since , we have By (7) and Claim 2, the above equation yields ; that is,
Similarly, by the equation , one can show that Equations (10)-(11) and Lemma 2 ensure that .
Note that, by (7) and Claim 3, one shows
Claim  4. For any , there exists some map such that , .
Firstly, take any invertible and let . Since , we have
For any , let . Since , we have By (7), the above equation yields
Since , one gets which and (13) imply Let in (17), and by (12), we get , and so Thus, combining (15) and (18), (17) reduces to . It follows that holds for all invertible and all .
Now taking any , there is a scalar such that is invertible. Then (19) implies for all , and so . Note that, for , there must be some such that . So
On the other hand, by (19) and the arbitrariness of , one can show that , and so for all invertible . Now consider any . There is a scalar such that is invertible. Also note that is invertible in . Hence for all . Again, for , there is some such that . It follows that Claim 4 is true.
Now define two maps and , respectively, by for all . Then by Claims 2 and 4, we have
Claim  5.   and are additive. By the definitions of and , we only need to verify that is additive on . In fact, for any , we have That is, Since by Lemma 3, one sees , and consequently, .
Similarly, one can prove that is additive on .
Claim  6.   is a derivation; that is, for all .
We will complete the proof of the claim by three steps.
Step  1. For any , , and , we have and , .
In fact, for any invertible and any , since , by (23) and (13), we have That is, for any invertible and all .
Take any . There is a scalar such that is invertible. Note that is also invertible in . So by (27) one gets Hence, holds for all and all .
Similarly, by , one can show that holds for all and ; by , one can show that holds for all and ; and by using the relation , one can obtain that holds for all and .
Step  2. For any , we have , .
Let . For any and any , by Step 1, on the one hand, we have on the other hand, Comparing the above two equations gets that holds for all ; that is, holds for all . Note that . It follows from the definition of the central carrier that span is dense in . It follows from (23) that holds for all , as desired.
Note that, by using Step 2 and the fact that (), one can get
Step  3. Consider for all and , .
Take any and . Since , by the definition of , Claim 5, (23), and (32), we have It follows from Step 1 that Multiplying by from the left side and the right side, respectively, in (34) and applying (23), we get These two equations, together with Step 1, yield Comparing the above two equations, one achieves
Similarly, multiplying by from the left side and the right side, respectively, in (34), one can verify
Next, we will prove . To do this, for any , let be its polar decomposition. Then (37) implies , and so . It follows that Similarly, one can show Multiplying by from the right side in (34), by using (39)-(40), we get Note that, by Step 2, (23), and (39)-(40), we have Hence, (41) implies and so . Thus, (34) reduces to which implies that and hold for all and , as desired.
Now combining Steps  1–3, it is easily checked that is an additive derivation.
Claim  7. Consider for all with .
In fact, for any with , we have
Claim  8. The theorem holds.
Indeed, let for all ; then, by the definitions of and , we have for all . It is easy to check that is an additive derivation on .
The proof is finished.

3. Characterizing -Lie Derivations by Local Action

In this section, we will give a complete characterization of additive -Lie derivations for by local action at some core-free projection with on general von Neumann algebras having no central summands of type .

The following is the main result of this section.

Theorem 4. Let be a von Neumann algebra without central summands of type and a nonzero core-free projection with . Suppose that is an additive map and is a scalar with . Then satisfies for any with if and only if is an additive derivation and for all .

Proof. Still, the “if” part is clear. For the “only if” part, we use the same symbols as that in the proof of Theorem 1. In the sequel, we always assume that and is an additive map satisfying for with . We will prove the “only if” part by several claims.
Claim  1.  Consider and . Since , we have ; that is, Multiplying by and from the left and the right sides, respectively, in (47), one gets
As , , and , that is, Multiplying by and from the left and the right sides, respectively, in (49) and combining with (48), one gets ; multiplying by and from the left and the right sides, respectively, in (50) and combining with (48), one gets .
On the other hand, since , we have , which and (47) yield This implies and . It follows from the fact that that and . Claim 1 is true.
Define a map by for each , where . It is easily verified that is also an additive map satisfying Moreover, by Claim 1. Thus we obtain Note that (47) is also true for the map . So, by (53), one can also get
Claim  2. Consider .  For any , since , we have By (53) and (55), the above equation reduces to Multiplying by from the right side in (57), one gets ; multiplying by from both sides in (57), one gets .
Similarly, by , one can show . Thus, we have proved that for all .
Next, take any invertible . By and , we get Comparing the two above equations, and by (54), one has , which implies as .
Similarly, by and , one can prove .
Thus, we have proved that if is invertible. Now for any , we can find a scalar such that is invertible in . It follows from the preceding case that , . Therefore, for each . Claim 2 holds.
Claim  3. For any , , we have . Moreover, the following statements hold.(1)If , then .(2)If , then .
First, consider the case . Since , we have By (53) and (55), we get Multiplying by and from the left and the right sides, respectively, in (60), we get . Since is arbitrary, replacing by if , one achieves
Since , we have That is,
Since , we have That is, Comparing (63)-(65) and by (53)-(54), one can obtain Then, (60) and (66) yield Multiplying by from both sides in (67), and by (53)-(54), we have multiplying by and and from the left and the right sides, respectively, in (67), we have Now combining (61) and (68)-(69), we achieve that the claim holds for any .
For any , by the relations , , and , and by using a similar argument to that of the above, one can show that and the claim also holds for .
Claim  4.   is an additive derivation with for each .
We will prove the claim by considering three cases.
Case  1. Consider . In this case, for any , , by Claim 3(1), (60), and (70), we get
We will complete the proof of Claim 5 by the following several steps.
Step    1. For any , , , , the following statements hold:(i);(ii);(iii);(iv).
For any , that is, invertible, and any , since , by Claim 2, Claim 3(1), and (58), we get . Now for any , note that there exists some such that is invertible in . It is easy to check that Also note that, by (71), . As , this and (72) yield
For any and , one can similarly check that and ; for any , , and , since and , by the definition of , Claim 3(1), (55), and (71), one can check that Step 1 holds.
Step  2. For any , we have , .
In fact, let . For any and any , by (71) and Step 1, on the one hand, we have on the other hand, The above two equations yield that holds for all and all , which implies that