Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Ghahramani, Hoger; Nader Ghosseiri, Mohammad; Rezaei, Tahereh

doi:https://doi.org/10.1155/2022/9941760

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Research Article | Open Access

Volume 2022 | Article ID 9941760 | https://doi.org/10.1155/2022/9941760

Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Hoger Ghahramani,¹Mohammad Nader Ghosseiri,¹and Tahereh Rezaei¹

Academic Editor: Ji Gao

Received09 Jul 2022

Revised07 Nov 2022

Accepted06 Dec 2022

Published20 Dec 2022

Abstract

Suppose that is a 2-torsion free triangular ring, and , where is the standard idempotent of and . Let be a mapping (not necessarily additive) satisfying, , where is the Jordan product of . We obtain various equivalent conditions for , specifically, we show that is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, on nest algebras are determined.

1. Introduction

Let be a ring and be a mapping (not necessarily additive). is called a derivable map if for all . Moreover, is called a Jordan derivable map if for all , where is the Jordan product of . An additive derivable mapping is called additive derivation. If is an additive Jordan derivable mapping, then it is called an additive Jordan derivation. These defined maps are important classes of maps on the rings and there has been many studies on them from different directions, and here we mention some of these study routes which is interesting for us.

One of the interesting issues is the study of relationship between the additive and multiplicative structure of maps on the rings. In this line of investigation, first Martinadle [1] considered some conditions on a ring , proved that any multiplicative bijection map of is additive. Then the question, what maps on a ring are automatically additive was considered and different results were obtained in this line, we refer the reader to [2, 3] and references therein for more details. Especially, it has been proved on special rings that every derivable map or Jordan derivable map is additive, for instance, see [4–6].

Obviously, any additive derivation is an additive Jordan derivation, but the converse may not hold in general (see [7]). Another interesting study routes on derivations and Jordan derivations is: on what rings (algebras) is any (linear) additive Jordan derivation is (linear) additive derivation? The first result in this way was obtained by Herstein [8], which proved that on 2-torsion-free prime rings, any additive Jordan derivation is an additive derivation. Later in [9], this result was generalized for 2-torsion free semiprime rings, and after that, this result was proved for other various rings (algebras) or the structure of additive (linear) Jordan derivations on some rings characterized in terms of additive (linear) derivations (see [7, 10–13] and references therein).

Another way to study derivable maps (additive derivations) and Jordan derivable maps (additive Jordan derivations) is to study them according to local conditions. One of these local conditions is studying the maps which operate on special pairs of elements of a ring like special maps. More precisely, assume that , in this line of investigation, considering those maps that for all , operate like derivable maps (additive derivations) or Jordan derivable maps (additive Jordan derivations). First Brešar in [14] proved that if is an additive map on a unital ring , that contains a nontrivial idempotent and for all with , then , where is an additive derivation on and belongs to the center of . Following this line of investigation, derivations and Jordan derivations (additive or nonadditive) at zero products or another special pairs of several rings or algebras has been studied and considerable results has been achieved. For instance, see [12, 15–21] and the references therein.

In the research lines mentioned above, some considerable results on prime rings or semiprime rings have been achieved. Of course this study routes have been established on some non-semiprime rings or operator algebras (especially nest algebras), of which we can mention triangular rings as one of the most important ones. In the following we introduce this ring and hint at some results on it. Let and be unital rings and be a unital -bimodule, which is faithful as a left -module and also as a right -module. The triangular ring Tri is as follows:under the usual matrix operations. Tri is a unital ring with identity , where , are identities of and , respectively. This ring contains important class of rings like upper triangular block matrices over a unital ring , especially the ring of upper triangular matrices over a unital ring , some nest algebras on Banach spaces, especially nest algebras on Hilbert spaces. Note that, if and be unital algebras over a commutative ring and be a unital faithful -bimodule, then Tri is an algebra over . Zhang in [22] proved that any linear Jordan derivation on a 2-torsion free triangular algebra Tri is a linear derivation and in [18] this result is obtained for additive Jordan derivations on 2-torsion free triangular rings. In [23] has been shown that any Jordan drivable map on a 2-torsion free triangular algebra is an additive derivation, which is a generalization of result of [22]. In [24] it has been proved that a linear map on (all upper triangular matrices over the complex field ), satisfying the following equation:is a linear derivation. In [21] it has been shown that a mapping (not necessarily additive) on ( is a field and ) satisfying the following equation:is an additive derivation. Let be a triangular ring and consider the subset of as follows:where is the standard idempotent in and . In this paper we consider a mapping (not necessarily additive) on which satisfies the following condition:and prove that if is 2-torsion free, then is an additive derivation. Note that if the mapping on is derivable, Jordan derivable, additive Jordan derivation or is an additive map on satisfyingthen satisfies (49) (see proof of Theorem 1). So our result generalizes various results in these directions for triangular rings, especially each of the results of [21, 23], Theorem 4.2 (for ), [22]. Next theorem is the main result of our paper.

Theorem 1. Suppose that is a 2-torsion free triangular ring and is a mapping (not necessarily additive). Let be as follows:where is the standard idempotent and . Then the following are equivalent:(i);(ii);(iii) is additive and ;(iv) is an additive Jordan derivation;(v) is a derivable map;(vi) is an additive derivation.

The proof of this theorem will be given in Section 3. In the above theorem, we have considered the 2-torsion free condition. The necessity of this condition can be a question of interest.

Let be a fixed element of the ring . The mapping defined by is an additive derivation which is called inner derivation. On nest algebras, derivations can be characterized in terms of inner derivations. According to this point, Theorem 1 can be obtained in more specific on nest algebras. We present this result in Section 2 as an application of Theorem 1 on nest algebras.

2. Application to Nest Algebras

Let be a (real or complex) Banach space, let be the Banach algebra of all bounded linear operators on . A nest on is a chain of closed (under norm topology) subspaces of with and in such that for every family of elements of , both and (closed linear span of ) belong to . The nest algebra associated to the nest , denoted by alg is as follows:

We say that is nontrivial whenever . If is trivial, then alg .

Remark 1. Let is a nontrivial nest on and with and , is complemented. Then there exists an idempotent such that , and the nest algebra alg has a representation as the following triangular algebra.where is the identity operator. and are unital algebras with unities and , respectively, and is a faithful unital -bimodule.

Since any closed (under norm topology) subspace of a Hilbert space is complemented,it follows that for any nontrivial nest on a Hilbert space , each with and , is complemented. Thus, every nontrivial nest algebra on a Hilbert space satisfies the conclusion in Remark 1.

We have the following result on nest algebras.

Theorem 2. Let be a nest on a Banach space , and there exists a nontrivial element in which is complemented in . Suppose that is a mapping (not necessarily additive), and is as follows:where is the idempotent with and . Then the following are equivalent:(i);(ii);(iii) is additive and ;(iv) is an additive Jordan derivation;(v) is a derivable map;(vi) is an additive derivation. Suppose, further, that is an infinite-dimensional Banach space. Then the above conditions are also equivalent to:(vii) is an inner derivation.

Proof. From Remark 1, alg is a triangular algebra, and all the assumptions of Theorem 1 hold. So all cases (i) to (vi) are equal. If is an infinite-dimensional Banach space, then by [25] every additive derivation of alg is linear. From [26], Theorem 2 any linear derivation of a nest algebra on a Banach space is continuous and by [27] all continuous linear derivations of a nest algebra on a Banach space are inner derivations (see also [28], Theorem 2.3). Given this, it is proved that if is an infinite-dimensional Banach space, condition (vii) is equivalent to condition (vi). The proof is complete.

By Theorem 2, we have the following corollary.

Corollary 1. Let be a nontrivial nest on a Hilbert space . Suppose that is a mapping (not necessarily additive), and is as followswhere is the is the orthogonal projection on a nontrivial element , and . Then the following are equivalent:(i);(ii);(iii) is additive and ;(iv) is an additive Jordan derivation;(v) is a derivable map;(vi) is an additive derivation. Suppose, further, that is an infinite-dimensional Banach space. Then the above conditions are also equivalent to:(vii) is an inner derivation.

Note that if is a Hilbert space, and , then there exist additive derivations of the nest algebra which are not inner (see [29]).

3. Proof of Theorem 1

In this section weassume that is a 2-torsion free triangular ring, and is the standard idempotent of and which is also an idempotent. Also we put

It is obvious that and for any we have

Proof of Theorem 1:

The following statements are clear: , , , , , and . We just prove the next items and so that the proof is complete.

(iii)(i): It is enough to prove that for all we have

Define the additive mapping by . Given that is an additive derivation, it is easily checked that

Also, .

Since , thus

Multiplying both sides of (16) by , we arrive at , and hence . Since for all , it follows that

By (17) and the fact that we get

Multiplying both sides of (18) by , we conclude that . Now multiplying (18) from the left by , it yields for all . Since is faithful as a left -module, we conclude that . By using the results obtained, . In view of (16) and the fact that , we get , thus . Given that and with the same argument above, we can prove that and hence .

For all we have , and hence

It follows from that for all . Thusfor all . Now according to the results we havefor all . By a similar argument as given above, we can prove thatfor all . Since is an additive derivation, the condition is obtained for .

(v)(i): Since is derivable, soand therefore for all with , we have

Thus

Define by . So that is a derivable map and . It follows from that and . Hence . Since is derivable, then and . So .

For all , we haveso . Alsothus . Thereforefor all . Further

By adding two recent statements, we arrive at the following equation:for all .

So for all , we have

From the above equality and (28) we find thatfor all .

We haveso thatfor all .

It follows from (32) that

On the other hand

Comparing recent two statements we getfor all . From the above statement, (34) and faithfulness of as a left -module, we conclude thatfor all .

By (38), we havefor all . Beside

By adding two recent statements, we obtainfor all . Now by the above identity and the fact that is a derivable map, we arrive atfor all . According to this statement and that is a derivation, we conclude thatfor all . By the simillar argument as above, we can prove thatfor all . Therefore satisfies (i).

(i)(vi): Define the mapping by and easily can be seen that satisfies

Also .

We prove that is an additive derivation through the following steps.

Step 1. .

Proof. Since then

Step 2. For all

Proof. For any we have . So thatIf we multiplying both sides of (48) by , we arrive at and if we multiplying both sides of (48) by , we obtain . Therefore for all .

Step 3. and .

Proof. Since , it follows thatSoBy multiplying both sides of (50) by , we have . Now multiplying (50) from the left by and from the right by , we see that for all . Faithfulness of implies that . By the results obtained and that , we conclude that . From this result and (50), we have and so and . Since for all , , so we haveMultiplying both sides of the above identity, we arrive at , for all . By faithfulness of , we have , so .

Step 4. For all

Proof. Since for all , from Step 3, we haveHence and . Sofor all . Given that for all , using Step 3 and same argument as above, we can prove thatfor all .

Step 5. For all

Proof. Since and , for all , from Step 3 we obtainIn the above, multiplying both sides of first statement by and multiplying both sides of second statement by and according to Step 2, the desired result is obtained.

Step 6. For all

Proof. Since for all , so thatFrom Steps 2 and 4, we find that and , hencefor all .
Since for all , from Steps 2, 4 and similar argument as above, we conclude thatfor all .

Step 7. For all

Proof. According to Step 6, for all we haveOn the other handComparing above statements and according to Step 4, we getNow by the faithfulness of , we haveSofor all . It follows from Step 4 thatfor all . By adding two recent statements we arrive atfor all . Using Steps 4, 6 and similar arguments as above, we can show thatfor all .

Step 8. For all

Proof. For all we have . SoFrom Step 2, we getBeside by Step 6, we haveComparing two recent statements and multiplying outcome relation from the left by and from the right by , and using Step 5, we arrive atfor all . By faithfulness of , we conclude thatFrom the above statement, Steps 4 and 5, we obtainSo thatfor all . Since for all we haveComparing the above statements, we getfor all .
Since and for all , by the same computation as above, we arrive atfor all .

Step 9. on , , is additive.

Proof. For all , we have . So thatNow from Steps 2, 3 and 8, we getfor all . Therefore is additive on .
From Step 6, we haveOn the other hand, according to additivity of on and from Step 6, we haveBy comparing the above statements and faithfulness of , we conclude thatfor all . Now Step 4 implies that is additive on .
By the similar argument as above we conclude that is also additive on .

Step 10. For all .

Proof. For all we have . So thatOn the other handSince , and from Steps 3, 4 and 8, we conclude thatThusNow for all , since and by the similar argument as above we prove thatTherefore we get the required result.

Step 11. is an additive derivation.

Proof. From Steps 9, 10, for all we haveTherefore is additive on .
It follows from Steps 2, 4, 6 and 7 thatfor all . Hence is an additive derivation. Since is also an additive derivation, so is an additive derivation and the proof is complete.

3.1. Declarations

(i)The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.(ii)The authors have no relevant financial or non-financial interests to disclose.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Hoger Ghahramani et al. 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.

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