Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Suppose that T � Tri ( A , M , B ) is a 2-torsion free triangular ring, and S � ( A, B )| AB � { 0 ,A, B ∈ T } 􏽓 ( A, X ) | A ∈ T , X ∈ P, Q { } { } , where P is the standard idempotent of T and Q � I − P . Let δ : T ⟶ T be a mapping (not necessarily additive) satisfying, ( A, B ) ∈ S ⇒ δ ( A ∘ B ) � A ∘ δ ( B ) + δ ( A ) ∘ B , where A ∘ B � AB + BA is the Jordan product of T . We obtain various equivalent conditions for δ , specifcally, we show that δ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, δ on nest algebras are determined.


Introduction
Let R be a ring and δ: R ⟶ R be a mapping (not necessarily additive). δ is called a derivable map if δ(ab) � aδ(b) + δ(a)b for all a, b ∈ R. Moreover, δ is called a Jordan derivable map if δ(a ∘ b) � a ∘ δ(b) + δ(a) ∘ b for all a, b ∈ R, where a ∘ b � ab + ba is the Jordan product of R. An additive derivable mapping δ is called additive derivation. If δ is an additive Jordan derivable mapping, then it is called an additive Jordan derivation. Tese defned maps are important classes of maps on the rings and there has been many studies on them from diferent 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, frst Martinadle [1] considered some conditions on a ring R, proved that any multiplicative bijection map of R is additive. Ten the question, what maps on a ring R are automatically additive was considered and diferent 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][5][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? Te frst 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][11][12][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 R like special maps. More precisely, assume that S ⊆ R × R, in this line of investigation, considering those maps that for all (a, b) ∈ S, operate like derivable maps (additive derivations) or Jordan derivable maps (additive Jordan derivations). First Brešar in [14] proved that if δ is an additive map on a unital ring R, that contains a nontrivial idempotent and is an additive derivation on R and c belongs to the center of R. 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][16][17][18][19][20][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 A and B be unital rings and M be a unital (A, B)-bimodule, which is faithful as a left A-module and also as a right B -module. Te triangular ring Tri (A, M, B) is as follows:  [22] proved that any linear Jordan derivation on a 2torsion free triangular algebra Tri (A, M, B) 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 δ on T n (C) (all n × n upper triangular matrices over the complex feld C), satisfying the following equation: is a linear derivation. In [21] it has been shown that a mapping δ (not necessarily additive) on T n (F) (F is a feld and n ≥ 3) satisfying the following equation: is an additive derivation. Let T � Tri(A, M, B) be a triangular ring and consider the subset S of T × T as follows: whereP � 1 A 0 0 0 is the standard idempotent in T and Q � I − P. In this paper we consider a mapping δ (not necessarily additive) on T which satisfes the following condition: and prove that if T is 2-torsion free, then δ is an additive derivation. Note that if the mapping δ on T is derivable, Jordan derivable, additive Jordan derivation or δ is an additive map on T satisfying then δ satisfes (49) (see proof of Teorem 1). So our result generalizes various results in these directions for triangular rings, especially each of the results of [21,23], Teorem 4.2 (for G � 0), [22]. Next theorem is the main result of our paper.

Theorem 1. Suppose that T � Tri(A, M, B) is a 2-torsion free triangular ring and δ: T ⟶ T is a mapping (not necessarily additive). Let S ⊆ T × T be as follows:
where P ∈ T is the standard idempotent and Q � I − P. Ten the following are equivalent:

is an additive Jordan derivation; (v) δ is a derivable map; (vi) δ is an additive derivation.
Te proof of this theorem will be given in Section 3. In the above theorem, we have considered the 2-torsion free condition. Te necessity of this condition can be a question of interest.
Let x be a fxed element of the ring R. Te mapping I x : R ⟶ R defned by I x (a) � ax − xa(a ∈ R) 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, Teorem 1 can be obtained in more specifc on nest algebras. We present this result in Section 2 as an application of Teorem 1 on nest algebras.

Application to Nest Algebras
Let X be a (real or complex) Banach space, let B(X) be the Banach algebra of all bounded linear operators on X. A nest N on X is a chain of closed (under norm topology) subspaces of X with 0 { } and X in N such that for every family N α of elements of N, both ∩ N α and ∨N α (closed linear span of N α ) belong to N. Te nest algebra associated to the nest N, denoted by alg N is as follows: We say that N is nontrivial whenever { } and N ≠ X, is complemented. Ten there exists an idempotent P ∈ algN such that ranP � N, and the nest algebra alg N has a representation as the following triangular algebra.
Since any closed (under norm topology) subspace of a Hilbert space is complemented,it follows that for any nontrivial nest N on a Hilbert space H, each N ∈ N with N ≠ 0 { } and N ≠ H, is complemented. Tus, every nontrivial nest algebra on a Hilbert space satisfes the conclusion in Remark 1.
We have the following result on nest algebras.

Theorem 2.
Let N be a nest on a Banach space X, and there exists a nontrivial element N in N which is complemented in X. Suppose that δ: algN ⟶ algN is a mapping (not necessarily additive), and S ⊆ algN × algN is as follows: where P ∈ algN is the idempotent with ranP � N and Q � I − P. Ten the following are equivalent:

iv) δ is an additive Jordan derivation; (v) δ is a derivable map; (vi) δ is an additive derivation.
Suppose, further, that X is an infnite-dimensional Banach space. Ten the above conditions are also equivalent to: (vii) δ is an inner derivation.
Proof. From Remark 1, alg N is a triangular algebra, and all the assumptions of Teorem 1 hold. So all cases (i) to (vi) are equal. If X is an infnite-dimensional Banach space, then by [25] every additive derivation of alg N is linear. From [26], Teorem 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], Teorem 2.3). Given this, it is proved that if X is an infnite-dimensional Banach space, condition (vii) is equivalent to condition (vi). Te proof is complete.
□ By Teorem 2, we have the following corollary.

Suppose that δ: algN ⟶ algN is a mapping (not necessarily additive), and S ⊆ algN × algN is as follows
where P ∈ algN is the is the orthogonal projection on a nontrivial element N ∈ N, and Q � I − P. Ten the following are equivalent:

Suppose, further, that H is an infnite-dimensional Banach space. Ten the above conditions are also equivalent to: (vii) δ is an inner derivation.
Note that if H is a Hilbert space, and dimH < ∞, then there exist additive derivations of the nest algebra which are not inner (see [29]).

Proof of Theorem 1
In this section weassume that T � Tri (A, M, B) is a 2torsion free triangular ring, and P � 1 A 0 0 0 is the standard idempotent of T and Q � I − P � 0 0 0 1 B which is also an idempotent. Also we put It is obvious that T � T 11 ⊕ T 12 ⊕ T 22 and for any A ∈ T we have Proof of Teorem 1: Te following statements are clear: (vi) ⇒ (i), (vi) ⇒ (ii) , (vi) ⇒ (iii), (vi) ⇒ (iv), (vi) ⇒ (v), (ii) ⇒ (i) and (iv) ⇒ (i). We just prove the next items and so that the proof is complete.
(iii) ⇒ (i): It is enough to prove that for all A ∈ T we have
For all A ∈ T we have (PAP)Q � Q(PAP) � 0, and hence (20) for all A ∈ T. Now according to the results we have for all A ∈ T. By a similar argument as given above, we can prove that for all A ∈ T. Since I δ(P) is an additive derivation, the condition (i) is obtained for δ.

Journal of Mathematics
For all A ∈ T, we have so Qτ(PAQ)Q � 0. Also thus Pτ(PAQ)P � 0. Terefore for all A ∈ T. Further

(29)
By adding two recent statements, we arrive at the following equation: for all A ∈ T. So for all A, B ∈ T, we have From the above equality and (28) we fnd that for all A, B ∈ T.
We have so that for all A ∈ T. It follows from (32) that Comparing recent two statements we get for all A, B, C ∈ T. From the above statement, (34) and faithfulness of M as a left A-module, we conclude that for all A, B ∈ T. By (38), we have for all A ∈ T. Beside Journal of Mathematics By adding two recent statements, we obtain for all A ∈ T. Now by the above identity and the fact that τ is a derivable map, we arrive at for all A ∈ T. According to this statement and that I δ(P) is a derivation, we conclude that for all A ∈ T. By the simillar argument as above, we can prove that for all A ∈ T. Terefore δ satisfes (i).
Also Pτ(P)Q � 0. We prove that τ is an additive derivation through the following steps.
Proof. Since 0 ∘ 0 � 0 then Proof. For any A ∈ T we have (PAQ, P) ∈ S. So that If we multiplying both sides of (48) by P, we arrive at Pτ(PAQ)P � 0 and if we multiplying both sides of (48) by Q, we obtain Qτ(PAQ)Q � 0. Terefore τ(PAQ) � Pτ(PAQ)Q for all A ∈ T.
By multiplying both sides of (50) by Q, we have Qτ(P)Q � 0. Now multiplying (50) from the left by P and from the right by Q, we see that Pτ(P)PAQ � 0 for all A ∈ T . Faithfulness of M implies that Pτ(P)P � 0. By the results obtained and that Pτ(P)Q � 0, we conclude that τ(P) � 0. From this result and (50), we have τ(Q)P + Pτ(Q) � 0 and so Pτ(Q)P � 0 and Pτ(Q)Q � 0. Since for all A ∈ T, (Q, PAQ) ∈ S, so we have Multiplying both sides of the above identity, we arrive at PAQτ(Q)Q � 0, for all A ∈ T. By faithfulness of M, we have Proof. Since (PAP, Q) ∈ S for all A ∈ T, from Step 3, we have Hence Qτ(PAP)Q � 0 and Pτ(PAP)Q � 0. So for all A ∈ T. Given that (QAQ, P) ∈ S for all A ∈ T, using Step 3 and same argument as above, we can prove that for all A ∈ T.

□
Step 5. For all A, B ∈ T Qτ(PAP + PBQ)Q � 0, In the above, multiplying both sides of frst statement by Q and multiplying both sides of second statement by P and according to Step 2, the desired result is obtained.
for all A, B ∈ T.
Since (QBQ, PAQ) ∈ S for all A, B ∈ T, from Steps 2, 4 and similar argument as above, we conclude that for all A, B ∈ T.
for all A, B ∈ T. By adding two recent statements we arrive at for all A, B ∈ T. Using Steps 4, 6 and similar arguments as above, we can show that for all A, B ∈ T. (74) Comparing two recent statements and multiplying outcome relation from the left by P and from the right by Q, and using Step 5, we arrive at