Let be the set of nonnegative integers and be a 2-torsion free triangular algebra over a commutative ring . In the present paper, under some lenient assumptions on , it is proved that if is a sequence of -linear mappings satisfying for all with (resp. , where p is a nontrivial idempotent of ), then for each , ; where is -linear mapping satisfying for all , i.e. is a higher derivation on and (where is the center of ) is an -linear map vanishing at every second commutator with xy=0 (resp. ).
Let ℛ be a commutative ring with unity, 𝒜 be an algebra over ℛ and Z(𝒜) be the center of 𝒜. Recall that an ℛ-linear map d:𝒜⇒𝒜 is called a derivation on 𝒜 if d(xy)=d(x)y+xd(y) holds for all x,y∈𝒜. An ℛ-linear map δ:𝒜⇒𝒜 is called a Lie derivation on 𝒜 if δ([x,y])=[δ(x),y]+[x,δ(y)] holds for all x,y∈𝒜, where [x,y]=xy-yx is the usual Lie product. An ℛ-linear map δ:𝒜⇒𝒜 is called a Lie triple derivation on 𝒜 if δ([[x,y],z])=[[δ(x),y],z]+[[x,δ(y)],z]+[[x,y],δ(z)] holds for all x,y,z∈𝒜. It is easy to check that every derivation on 𝒜 is a Lie derivation on 𝒜 and that every Lie derivation on 𝒜 is a Lie triple derivation on 𝒜. However, the converse need not be true in general. For example if we consider the algebra 𝒜 of all 3 × 3 strictly upper triangular matrices over , the ring of integers, and define a map such that
Then it can easily be seen that L is a Lie triple derivation on 𝒜 which is neither a derivation nor a Lie derivation on 𝒜. Now, let be the set of nonnegative integers and be a sequence of ℛ-linear mappings such that , the identity map on 𝒜. Then Δ is said to be
(i) a higher derivation on 𝒜 if
(ii) a Lie higher derivation on 𝒜 if
(iii) a Lie triple higher derivation on 𝒜 if
It is also easy to observe that there exists Lie triple higher derivation on an algebra 𝒜 which is not a Lie higher derivation on 𝒜. For example consider the algebra 𝒜 of all 3 × 3 strictly upper triangular matrices over the field 𝒬 of rational numbers, and consider the sequence of linear mappings such that , where L is Lie triple derivation on 𝒜 which is not a Lie derivation on 𝒜. Then by using induction on n, it can be easily verified that ℒ is a Lie triple higher derivation on 𝒜 but not a Lie higher derivation on 𝒜.
The ℛ-algebra under the usual matrix operations is called a triangular algebra, where 𝒜 and 𝓑 are unital algebras over ℛ and ℳ is an (𝒜,𝓑)-bimodule. Recall that a left (resp. right) 𝒜-module ℳ is faithful if aℳ=0 (resp. ℳa=0) implies that a=0 for every a∈ 𝒜. The notion of triangular ring was first introduced by Chase [5] in 1960. Further, in the year 2000, Cheung [7] initiated the study of linear maps on triangular algebras. He described Lie derivations, commuting maps and automorphisms of triangular algebras (see for reference [8, 9]).
In the recent years, derivation and Lie derivation have been studied by several authors (see [1, 2, 3, 4, 9, 10, 12, 14, 16, 19, 20]) in various directions. One direction of investigation is to study the conditions under which derivations and Lie derivations can be completely determined by the action on some subsets of 𝒜. We say that an ℛ-linear map is derivable at a given point if for every with xy=c and such c is called a derivable point of 𝒜. This kind of maps were discussed by several authors (see [6, 15, 22]). Similarly, an ℛ-linear map is said to be a Lie derivable at a given point if for all with xy=c. Lu and Jing [18] discussed such maps on B(X) where X is a Banach space with and B(X) is the algebra of all bounded linear operators acting on X and proved that if δ is Lie derivable at c=0 (resp. c=p, where p is a fixed nontrivial idempotent of B(X)), then , where d is a derivation of B(X) and is a linear map vanishing at every commutator [x,y] with xy=0 (resp. xy=p). Ji and Qi [13] investigated this problem on triangular algebras and obtained that under some mild conditions on , if is an -linear map satisfying for any with xy = 0 (resp. xy = p, where p is a fixed nontrivial idempotent of ), then where d is a derivation of and (where is the center of ) is an ℛ-linear map vanishing at commutators [x, y] with xy = 0 (resp. xy = p). Furthermore, in [17] Liu analysed Lie triple derivation on factor von Neumann algebra 𝒜 of dimension greater than one and found that if a linear map satisfies for any with xy = 0 (resp. xy = p, where p is a fixed nontrivial projection of 𝒜), then there exist an operator r ∈ 𝒜 and a linear map (where is the center of 𝒜) vanishing at every second commutator [[x, y], z] with xy = 0 (resp. xy = p) such that for any
An ℛ-linear map δ:𝒜⇒ 𝒜 is Lie triple derivable at a given point c ∈ 𝒜 if for all with xy=c. It is obvious that the condition of being a Lie triple derivable map at some point is much weaker than the condition of being a Lie triple derivation. So far, there has been no result on the study of the local actions of Lie triple derivations on triangular algebras. Motivated by these observations, the purpose of the present paper is to characterize the additive mapping δn on triangular algebra satisfying for any x,y,z ∈ with xy=0 (resp. xy=p, where p is a fixed nontrivial idempotent).
Throughout the present paper will denote a triangular algebra which is 2-torsion free. Define two natural projections and by and . The center of coincides with
Moreover, and , and there exists a unique algebra isomorphism such that for all .
Let (resp.) be the identity of the algebra (resp.) and let be the identity of triangular algebra . Throughout this paper, we shall use the following notations: and . Set , and . Then we can write , where is a subalgebra of isomorphic to , is a subalgebra of isomorphic to and is a -bimodule isomorphic to the bimodule . To simplify the notation we will use the following convention: , and . Then each element can be represented in the form where and
In what follows, we write , it indicates and the corresponding element in or . Note that if .
The proof of the following lemma can be seen in [8, Propositon 3].
Lemma 1.1.
Letbe a triangular algebra. If and , then there is a unique algebra isomorphismsuch thatfor any.
2. Characterizations of Lie Triple Higher Derivations by Action on Zero Product
The main result of the present paper states as follows:
Theorem 2.1.Letbe a 2-torsion free triangular algebra consisting of algebrasandover a commutative ringwith unityand respectively and be a faithful-bimodule, which is faithful as a left-module and also as a right-module. Suppose that
(i) and .
(ii) For any , if , then or for any , if , then .
Ifis a sequence of-linear mapssuch that
for all with xy=0, then for each,
; where is -linear mapping satisfyingfor all, i.e., is a higher derivation of and is an-linear map vanishing at every second commutator [[x,y],z] withxy=0.
The proof of Theorem 2.1 is based on the induction on n. We provide the proof, for n=1, through several claims. Indeed, we show that under the given assumptions of our theorem every Lie triple derivation on there exists a derivation d on and a linear mapping vanishing on second commutators such that for all .
Proof of Theorem 2.1.Claim 1.; for any .
Since for any , we have
On multiplying the above equality from left by p1 and right by p2, we get
This implies that that is, and hence By putting , in (2.1), we get and so we have for ,
Similarly we can get the following result:
Claim 2.
Claim 3.
Since , we have
This yields that . By Claims 1 & 2, we have .
In the sequel, we define
where is the inner derivation determined by , that is,
One can verify that
for all with xy=0. Moreover by Claim 1, we have
By Claim 3, we get . Consequently
Clearly for any , . By Claim 1, we have , and hence
Claim 4. For any
Claim 5.. There exists an -linear map such that for all
First we show for i=1. Since a11p2=0 for any and , it follows that
Multiplying by p1 from the left we get and hence .
Similarly, we can prove the result for i=2. Thus, .
Now we can write . Moreover, since , for any and , it is easy to check that
Multiplying by p2 on both the sides of the above equation, we get
This implies that . Hence by hypothesis (ii), we find that
Define such that , where η is the map defined in Lemma 1.1. Thus, we get
Since f is -linear, one can verify that τ1 is -linear. Similarly, we can define -linear map by . Then
Now, for any, we define two -linear maps and by
Then, .
Claim 6.d is a derivation.
Since f & τ are ℛ-linear and d(x)=f(x)-τ(x), d is ℛ-linear. It remains to show that , for all . Now, we divide the proof into the following three steps:
Step 1. Since a12a11=0 for any , and τ(x) is in , we have
The proof of the following theorem shares the same outline as that of Theorem 2.1 but requires different technique.
Theorem 3.1.
Let be a 2-torsion free triangular algebra consisting of algebras and over a commutative ring with unity and respectively and be a faithful -bimodule. Suppose that
(i) and .
(ii) For any , if , then or for any , if , then .
(iii) For every , there exists an integer t such that is invertible.
If is a sequence of -linear mappings such that for all with xy=p, then for each , ; where is -linear mapping satisfying for all , i.e., is a higher derivation of and is an -linear map vanishing at every second commutator [[x,y],z] with xy=p.
For the proof of Theorem 3.1, we proceed by induction on n. We provide the proof, for n=1, through several claims. Indeed, we show that for every Lie triple derivation on there exist a derivation d on and a linear mapping vanishing on second commutators such that for all .
Proof of Theorem 3.1.Claim 1.; for any .
Since for any , we have
On multiplying the above equality by p1 and p2 from the left and the right respectively, we get This gives that that is, It follows that . By putting , in (3.1), we get and so we have for . Hence, Now, define
where is the inner derivation determined by . Then, we have
and for all with xy=p. Moreover, for any , by Claim 1, we have
Claim 2. and
Since Ip1=p1, we have
This yields that and hence we find that
So, we get For any , since , we have
On multiplying above equality by p1 and p2 from the left and the right respectively, we get and
his implies that . Consequently, .
Claim 3.. There exists an -linear map such that for all .
First we show for i=1. Suppose that a11 is invertible in , that is, there exists an element such that . From and , we have
and hence by Claim 2,
On multiplying by p1 from the left and by p2 from the right, we get , and hence we find that
If a11 is not invertible in , by the hypothesis (iii), there exists an integer t such that is invertible in . It follows from the preceding case that . Therefore, we have
Similarly, we can prove the result for i=2. Now we can write . First suppose that a11 is invertible in with inverse element . Note that and , we get
and hence
Multiplying by p2 on both the sides, we get
This implies that . Hence by hypothesis (ii), we get .
If a11 is not invertible in , by the hypothesis (iii), there exists an integer t such that is invertible in . It follows from the preceding case that
Multiplying by p2 on both the sides, we get
This implies that . Hence by hypothesis (ii), we get .
Define by , where η is the map defined in Lemma 1.1. Thus, we get
Since f is an ℛ-linear, one can verify that τ1 is ℛ-linear.
Similarly, we can define ℛ-linear map by . Then
Now, for any , we define two -linear mappings and by and respectively. Then, .
Claim 4.d is a derivation.
Since f & τ are ℛ-linear and , d is an ℛ-linear. It remains to show that , for all . We divide the proof into the following three Steps:
Step 1. If a11 is invertible in with inverse element , then for any , , we have
Since , we have
Hence, Replacing a12 by , we arrive at
For any , let be invertible in . Then
Since , we have
Step 2. Let and . Observe that and . Since , we have
Thus, for any .
With the same approach as used in the proof of Claim 6 of Theorem 2.1, we can get:
Step 3. For any and
(i) ,
(ii) .
Step 4. for all
Claim 5. τ vanishes at second commutator [[x,y],z] with xy=p for all .
Since xy=p, we find that
for all The proof for n=1 is now complete.
Now, suppose that the conclusion holds for all . That is, there exist linear maps and such that , with xy=p and for all .
Moreover, has the following properties:
We shall show that also satisfies the similar properties. We prove this through the following claims:
Claim 6.; for any .
Since for any , by induction hypothesis, we have
On multiplying by p1 from the left and by p2 from the right in the above equation, we get . This gives that that is, It follows that . By putting , in (3.2), we get and so, we have for . Hence,
Now, define
where is the inner derivation determined by . Then, we have
and
for all with xy=p. Moreover, for any , by Claim 6, we have
Claim 7. and .
Since , using induction hypothesis, we have
On multiplying by p1 and by p2 from the left and the right respectively, we get Hence, we find that For any , since , using induction hypothesis, we have
Further, multiply by p1 from the left and by p2 from the right, we find that
This implies that . Consequently, .
Claim 8.. There exists an ℛ-linear map such that for all .
First we show the result for i=1. Suppose that a11 is invertible in , that is, there exists an element such that . From and , by induction hypothesis, we have
and hence by Claim 7,
This yields that and hence we find that . From this we get .
On the other hand if a11 is not invertible in , by the hypothesis (iii), there exists an integer t such that tp1-a11 is invertible in . It follows from the preceding case that . Therefore, we have Similarly, we can prove that for i=2.
Now we can write . First, suppose that a11 is invertible in with inverse element . Note that and , using induction hypothesis, we get
and hence
Multiplying by p2 on both the sides, we get
This implies that . Hence by hypothesis (ii), we get .
If a11 is not invertible in , by the hypothesis (iii), there exists an integer t such that is invertible in . It follows from the preceding case that
Multiplying by p2 on both the sides, we get
This implies that . Hence by hypothesis (ii), we get .
Define by , where η is the map defined in Lemma 1.1. Thus, we get
Since fn is ℛ-linear, one can verify that is ℛ-linear. Similarly, we can define ℛ-linear map by . Then
Now, for any , we define two ℛ-linear maps and by
Then, .
Claim 9. for all .
Since fn & τn are ℛ-linear and is an ℛ-linear. It remains to show that , for all .
We divide the proof into the following three Steps:
Step 1. If a11 is invertible in with inverse element , then for any , , we have
Since,
we find that
Hence, Replacing a12 by , we arrive at
For any , let be invertible in $\mathfrak{A}_{11}$. Then
Since , we have
Step 2. Let and . Observe that and . Since , we have
Thus, for any .
Using the same approach as used in the proof of Claim 13 of Theorem 2.1, we find that
Step 3. For any and
(i)
(ii)
Step 4. for all
Claim 10.τn vanishes at second commutator[[x,y],z] with xy=p for all
As an application of Theorems 2.1 & 3.1, we consider the nest algebra case. We know that every nontrivial nest algebra is a triangular algebra (see [11]), which satisfies the conditions of Theorems 2.1 & 3.1 and hence we have the following results.
Theorem 4.1.
Let 𝒩 be an arbitrary nontrivial nest on a Hilbert space T of dimension greater than 2, Alg𝒩 be the associated nest algebra. If is a sequence of ℛ-linear maps satisfying for all with xy=0. Then for each , where is an inner higher derivation on and (where is the center of ) is an -linear map vanishing at the second commutator [[x,y],z] with xy=0.
Proof. Since is nontrivial nest, the associated nest algebra is a triangular algebra which satisfies the conditions of Theorem 2.1. Then there exists a higher derivation of and a linear map vanishing at the second commutator [[x,y],z] with xy=0 such that for each , Since every higher derivation on is inner (see [10, 21]), there is an inner higher derivation on . This implies that for each
Theorem 4.2.
Let 𝒩 be a nontrivial nest on a Hilbert space T of dimension greater than 2, Alg𝒩 be the associated nest algebra and p be a nontrivial projection in 𝒩. If is a sequence of ℛ-linear maps satisfying for all with xy=p. Then for each , where is an inner higher derivation on and (where is the center of ) is an -linear map vanishing at the second commutator [[x,y],z] with xy=p.
Proof. Let , and . Then and are unital algebras with unit element p and I-p respectively and is a triangular algebra. Also satisfies the conditions of Theorem 3.1, then there exists a higher derivation of and a linear map vanishing at the second commutator [[x,y],z] with xy=p such that for each , Since every higher derivation on is inner (see [10, 21]) there exists an inner higher derivation on such that for each ,
If Hilbert space T is finite dimensional, then nest algebras are upper block triangular matrices algebras [7].
Theorem 4.3.
Let be a proper block upper triangular matrix algebra over a commutative ring ℛ. If is a sequence of ℛ-linear maps satisfying for all with xy=0 (resp. xy=p, p be a nontrivial projection in . Then for each , where is an inner higher derivation on and (where is the center of ) is an ℛ-linear map vanishing at the second commutator [[x,y],z] with xy=0 (resp. xy=p).
Proof. It can be easily seen that conditions of Theorems 2.1 & 3.1 hold for block upper triangular matrix algebra and from [21, Proposition 2.6] all higher derivations of are inner. Hence δn is the sum of an inner higher derivation and a functional that vanishes on all second commutators of
M. Ashraf and A. Jabeen, Nonlinear generalized Lie triple derivation on triangular algebras, Comm. Algebra, 45(2017), 4380-4395.
M. Ashraf and N. Parveen, Lie triple higher derivable maps on rings, Comm. Algebra, 45(2017), 2256-2275.
D. Benkovivč, Biderivations of triangular algebras, Linear Algebra Appl., 431(2009), 1587-1602.
D. Benkovič and D. Eremita, Multiplicative Lie n-derivations of triangular rings, Linear Algebra Appl., 436(2012), 4223-4240.
S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J., 18(1961), 13-25.
M. A. Chebotar, W. F. Ke and P. H. Lee, Maps characterized by action on zero products, Pacific J. Math., 216(2004), 217-228.
W. S. Cheung, Maps on triangular algebras, Ph. D. Dissertation, University of Vic- toria, 2000.
W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc., 63(2001), 117-127.
W. S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra, 51(2003), 299-310.
E. Christensen, Derivations of nest algebras, Math. Ann., 229(1977), 155-161.
K. R. Davidson, Nest algebras. Triangular forms for operator algebras on Hilbert space, Pitman Research Notes in Mathematics Series 191, Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK, 1988.
Y. Q. Du and Y. Wang, Lie derivations of generalized matrix algebras, Linear Algebra Appl., 437(2012), 2719-2726.
P. Ji and W. Qi, Characterizations of Lie derivations of triangular algebras, Linear Algebra Appl., 435(2011), 1137-1146.
P. Ji, W. Qi and X. Sun, Characterizations of Lie derivations of factor von Neumann algebras, Linear Multilinear Algebra, 61(2013), 417-428.
W. Jing, S. Lu and P. Li, Characterisations of derivations on some operator algebras, Bull. Austral. Math. Soc., 66(2002), 227-232.
J. Li and Q. Shen, Characterizations of Lie higher and Lie triple derivations on triangular algebras, J. Korean Math. Soc., 49(2012), 419-433.
L. Liu, Lie triple derivations on factor von neumann algebras, Bull. Korean Math. Soc., 52(2015), 581-591.
F. Y. Lu and W. Jing, Characterizations of Lie derivations of B(X), Linear Algebra Appl., 432(2010), 89-99.
M. Mathieu and A. R. Villena, The structure of Lie derivations on C∗-algebras, J. Funct. Anal., 202(2003), 504-525.
X. F. Qi, Characterization of Lie higher derivations on triangular algebras, Acta Math. Sinica, 29(2013), 1007-1018.
F. Wei and Z. Xiao, Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl., 435(2011), 1034-1054.
J. Zhu and S. Zhao, Characterizations of all-derivable points in nest algebras, Proc. Amer. Math. Soc., 141(2013), 2343-2350.