# Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Authors: Xinfeng Liang
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetA be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S:A⟶A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J. We show that, under mild conditions, every generalized Jordan n-derivation S:A⟶A is of the form Sx=λx+Jx in the current work. As an application, we give a description of generalized Jordan derivations for the condition n=2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.


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## Body

## 1. Introduction

Throughout this paper, letℛ be a commutative ring with an identity and let A be a unital algebras over ℛ. Let us assume that A has an idempotent e≠0,1 and let f=1−e. In this case, A can be represented in the so called Peirce decomposition form(1)A=eAe+eAf+fAe+fAf,where eAe and fAf are subalgebras with unitary elements e and f, respectively, eAf is an eAe,fAf-bimodule and fAe is an fAf,eAe-bimodule. It is worth to mention that A is isomorphic to a generalized matrix algebra [1]. We assume that A satisfies(2)exe.eAf=0=fAe.exe implies exe=0,eAf.fxf=0=fxf.fAe impliesfxf=0,for all x∈A. Some special examples of unital algebras with a nontrivial idempotents having the property (♣) are triangular algebras, matrix algebras, and prime (and hence in particular simple) algebras with nontrivial idempotent, nest algebras, standard operator algebras (see [2] for more details). It follows from (♣) that at least one of the bimodules eAf and fAe is nonzero.Letℛ be an associate algebra or ring. x,y=xy+yx is the Jordan product of elements x,y∈ℛ. For any integer n≥1 and any x1,x2,…,xn∈ℛ. Set p1x1=x1 and(3)pnx1,x2,…,xn=pn−1x1,x2,…,xn−1,xn=pn−1x1,x2,…,xn−1xn+xnpn−1x1,x2,…,xn−1,for n≥2, which is called the Jordan n-product of x1,x2,…,xn. Denote by φ:ℛ⟶ℛ a linear mapping; we call φ a Jordan n-derivation if(4)φpnx1,x2,…,xn=∑k=1npnx1,x2,…,…xk,…,xn,for all x1,x2,…,xn∈ℛ. It is obvious that Jordan n-derivations are usual Jordan derivations for n=2; moreover, it is also easily checked that the definition of Jordan 3-derivations are equivalent to the conception of Jordan triple derivations.A linear mappingS:A⟶A is said to be a generalized Jordan n-derivation if there exists a Jordan n-derivation J:A⟶A such that(5)Spnx1,x2,…,xn=pnSx1,x2,…,xn+∑k=2npnx1,x2,…,Jxk,…,xn,for all x1,x2,…,xn∈A. It is clear that any Jordan 2-derivations are usual generalized Jordan derivations.Note that any Jordann-derivation is an example of a generalized Jordan n-derivation. On the other hand, any multiplier function Fx=λx for all x∈A and λ∈ZA is a Jordan example of a generalized Jordan n-derivation.The motivation for this study comes from the results of [2–6]. Benkovič and Sčirovnik considered the structure of Jordan derivations on unital algebra with nonzero idempotent e and introduced the notation of singular Jordan derivations which comes out to be very important in the study of mappings on unital algebra with nonzero idempotent e. It turns out that, under some mild conditions, every Jordan derivation is the sum of a derivation and a singular Jordan derivation. By introducing the concept of Jordan n-derivation (n is any positive integer), Qi, Guo and Zhang [5] showed that every multiplicative Jordan n-derivation on A is additive; and furthermore, it turns out that a mapping on A is a multiplicative Jordan n-derivation if and only if it is an additive Jordan derivation. In Ref. [3], Benkovič present a new approach and show that every generalized Lie n-derivation F:A⟶A is of the form(6)Fx=λx+Δxfor allx∈A,where some λ∈ZA and Δ is a Lie n-derivation of A. Inspired by the structures of Jordan derivations [3], multiplicative Jordan n-derivations [5], and genelized Lie n-derivations [2] of unital algebras with a nonzero idempotent, the main purpose of this article is to study generalized Jordan n-derivation of unital algebras with the property (♣). In the main theorem of the paper, Theorem 1, we show that, under certain mild assumptions, every generalized Jordan n-derivation is of the form(7)Sx=λx+Jxfor allx∈A,where λ∈ZA, J is a Jordan n-derivation of A. We shall use some known results about the form of Jordan derivations on unital algebras with nonzero idempotent that were obtained in [2–4]. Let us mention that in different papers [2, 3, 7–11] Jordan derivations on triangular algebras and related algebras were studied from different perspectives.In this paper, we present a new approach which is analogue to Benkovič’s article [3], which elegantly reduces the problem of describing a generalized Jordan n-derivation to the describing a Jordan n-derivation. It turns out that if S:A⟶A is a generalized Jordan n-derivation associated with a Jordan n-derivation J, then a linear mapping H=S−J:A⟶A satisfies(8)Hpnx1,…,xn=pnHx1,…,xn,for all x1,…,xn∈A. Therefore, it suffices to consider linear mappings with property (♣). Under suitable assumptions on unital algebra A with a nonzero idempotent (Proposition 1), any such generalized Jordan n-derivation is of the form Hx=λx for all x∈A and some λ∈ZA.


## 2. Preliminaries and the Main Theorem

LetA be a unital algebra with idempotent e and f=1−e, which satisfies (♣). For convenience, we shall use the following notations a=eae∈eAe,m=emf∈eAf,t=fte∈fAe, and b=fbf∈fAf. Thus, every element x∈A can be represented in the form(9)x=eae+emf+fte+fbf=a+m+t+b,where eAf is an eAe,fAf-bimodule and fAe is an fAf,eAe-bimodule.Let us list some classical example of unital algebras with idempotente and f=1−e, which satisfies (♣). Since these examples have already been presented in many papers, see [2, 12–14], we just state their title without any details.(a)
Matrix algebraA=MnA,n≥2, where A is a unital algebra.(b)
Every simple unital algebraA with nontrivial idempotent, which satisfies (♣).(c)
Unital prime algebras with nontrivial idempotent.(d)
Triangular algebraT=AMB such that the bimodule M is faithful as a left A and also as a right B-module. The most important examples of triangular algebras are upper triangular matrix algebras TnA and block upper triangular matrix algebras BnA over a unital algebra A and also nest algebras TN, where N is a nest in a Hilbert space ℍ.The first very useful observation refers to the form of the center ofA, which is identical to ([15], Proposition 3) and ([16], Lemma 3.1, Lemma 3.2). Throughout the paper, A denotes a unital algebra with a nonzero idempotents e satisfying (♣).By [12], [Proposition 2.1], it follows that the center of A is equal to(10)ZA=a+b∈eAe⊕fAfam=mb,na=bn,∀m∈eAf,∀n∈fAe.Furthermore, we know that the mapτ:ZAe⟶ZAf is an algebraic isomorphism such that am=mτa and na=τan for all a∈ZAe,m∈eAf,n∈fAe.Remark 1.
LetA be a unital algebra with nontrivial idempotent e and f=1−e. For any x∈A and for any integer n≥2, we have(11)pnx,e,…,e=2n−1exe+exf+fxe,pnx,f,…,f=2n−1fxf+exf+fxe.
In particular,x,e=xe+ex=2exe+exf+fxe and x,f=xf+fx=2fxf+exf+fxe.
In this section, we will prove the main result, Theorem1. As we mentioned in the introduction, the problem of a description of a generalized Jordan n-derivations can be reduced to a description of a map satisfying (8). Let us begin with the solution of this problem.Proposition 1.
LetA be a unital algebra with a nontrivial idempotent e satisfying (♣) over a 2-trosionfree commutative ring ℛ. Let us assume that ZeAe=ZAe and ZfAf=ZAf. Let a linear mapping H:A⟶A satisfies(12)Hpnx1,…,xn=pnHx1,x2,…,xnfor all x1,…,xn∈A. Then(13)Hx=λxfor allx∈A,where λ∈ZAProof.
To prove this proposition, we will need the following Claims:(i)
Claim 1.H0=0. By taking xi=0 for all i=1,2,…,n, one can show H0=0.(ii)
Claim 2. With notations as above, we haveeHee⊕fHff∈ZA, Hm=eHeem=mHff and Ht=teHee=fHfft for all m∈eAf,t∈fAe.By the definitionH, we have(14)0=Hpne,f,…,f=pnHe,f,…,f=2n−1fHef+eHef+fHee,and(15)0=Hpnf,e,…,e=pnHf,e,…,e=2n−1eHfe+eHff+fHfe,which implyeHef=fHee=eHff=fHfe=0 and fHef=eHfe=0 as char ℛ≠2. Therefore He=eHee∈eAe and Hf=fHff∈fAf.For allm∈eAf, we have(16)H2n−2m=Hpnf,…,f,m=pnHf,…,f,m=pn−1Hf,f,…,f,m=2n−2mHff,and(17)H2n−2m=Hpne,…,e,m=pnHe,…,e,m=pn−1He,e,…,e,m=2n−2eHeem,for allm∈eAf. Hence,(18)Hm=eHeem=mHffas charℛ≠2. On the other hand, for all t∈fAe,(19)H2n−2t=Hpnf,…,f,t=pnHf,…,f,t=pn−1Hf,f,…,f,t=2n−2fHfft,and(20)H2n−2t=Hpne,…,e,t=pnHe,…,e,t=pn−1He,e,…,e,t=2n−2teHee.Therefore,(21)Ht=teHee=fHfftfor allt∈fAe. Combining (18) and (21) with the definition of the center ZA, we have(22)eHee⊕fHff∈ZA.(iii)
Claim 3. With notations as above, we haveHa=eHea and Hb=fHfb for all a∈eAe and b∈fAf
For alla11∈eAe, we have(23)2n−1Ha=Hpne,…,e,a=pnHe,e,…,e,a=pn−1He,e,…,e,a=2n−2eHea+2n−2aHee.Since the characteristic ℛ is not 2, this implies(24)2Ha=eHea+aHee=2eHea,and then(25)Ha=eHea=aeHeefor all a∈eAe.
Similarly, one can check that(26)Hb=bHff=bfHffor all b∈fAf.
For allx=a+m+t+b∈A, by Claims 1−3, we have(27)Hx=Ha+Hm+Ht+Hb=eHea+eHem+fHft+fHfb=λx,where λ=eHee+fHff∈ZA
The main result of the the paper state:Theorem 1.
LetA be a unital algebra with a nontrivial idempotent e satisfying (♣). Let us assume that(i)
ZeAe=ZAe(ii)
ZfAf=ZAf
Then any generalized Jordann-derivation S:A⟶A is of the from Sx=λx+Jx for all x∈A, where λ∈ZA and J:A⟶A is a Jordan n-derivation.Proof.
LetS:A⟶A be a generalized Jordan n-derivation associated with a Jordan n-derivation J. According to the definition(28)Spnx1,…,xn=pnSx1,…,xn+∑k=2npnx1,…,Jxk,…,xn,Jpnx1,…,xn=pnJx1,…,xn+∑k=2npnx1,…,Jxk,…,xn,for all x1,…,xn∈A. Let us denote H=S−J. If we subtract upper equalities, we see that a linear mapping H:A⟶A satisfies(29)Hpnx1,…,xn=pnHx1,…,xnfor all x1,…,xn∈A. Since all assumption from Proposition 1 are fulfilled there exists λ∈ZA such that Hx=λx+Jx.
According to Theorem1 and ([2], Theorem 4.1), Jordan n-derivation is usual Jordan derivation if n=2, the following result is immediate.Corollary 1.
LetA be a unital algebra with a nontrivial idempotent e satisfying (♣). Let us assume that(i)
ZeAe=ZAe(ii)
ZfAf=ZAf
Then any generalized Jordan derivationF:A⟶A is of the from Fx=λx+J1x+J2x for all x∈A, where λ∈ZA, J1:A⟶A is a derivation and J2:A⟶A is a singular Jordan derivation.Corollary 2.
([4], Theorem 3.11) Let A be a unital algebra with a nontrivial idempotent e satisfying (♣) and a bimodule eAf is faithful as a left eAe-module and as a right fAf-module. Then every Jordan derivation of A can be expressed as the sum of a derivation and an antiderivation.We apply Theorem1 to the classical examples of unital algebras: triangular algebras (upper triangular matrix algebras, nest algebras), matrix algebras, and algebras of bounded linear operators. Our main result reduces the description of a generalized Jordan n-derivation to the description of a Jordan n-derivation.It is well-known that all Jordan derivations of matrix algebras and prime algebras are derivations [17, 18]. Using the results from ([2], Section 3), one could prove that there are no nonzero singular Jordan derivations of the matrix algebra MnA over a unital algebra A. One can also obtain that there are no nonzero singular Jordan derivations of a unital prime algebra A with a nontrivial idempotent e. Therefore, Theorem 1 implies the following corollaries.Corollary 3.
LetA=MsA, s≥3, where A is a unital s−1-torsionfree algebra. Then every generalized Jordan derivation F:A⟶A is of the form Fx=λx+Jx, where λ∈ZA, J:A⟶A is a derivation.We conclude this article with some applications of the main theorem. In case A is a unital algebra with a nontrivial idempotente such that fAe=0, and that the bimodule eAf is faithful as a left eAe-module and also as a right fAf-module, the algebra A is a triangular algebra. The triangular algebra A satisfies (♣) and by the definition A has no nonzero singular Jordan derivations. Therefore, combining with [7], we obtain the following result.Corollary 4.
LetA be a triangular algebra. Let us assume that(i)
ZeAe=ZAe(ii)
ZfAf=ZAf
Then, any generalized Jordan derivationF:A⟶A is of the from Fx=λx+Jx for all x∈A, where λ∈ZA and J:A⟶A is a derivation.Corollary 5.
Every Jordan derivation of a unital prime algebra A with a nontrivial idempotente is a derivation.Algebras of all bounded linear operators.LetX be a Banach space over ℂ of dimension greater than 1. By B=BX, we denote the algebra of all bounded linear operators on X. B contains nontrivial idempotent e and hence can be presented in the form B=eBe+eBf+fBe+fBf, where f=1−e. Since B is a prime algebra, B satisfies (♣). Note that eBe and fBf are algebras of all bounded linear operators and all B,eBe,fBf are central algebras over ℂ. Therefore, eZBe=ZeBe=ℂe and fZBf=ZfBf=ℂf. Hence B meets assumptions of Theorem 1 and we have.Corollary 6.
LetX be a Banach space over ℂ, dim X≥2. Then every generalized Jordan n-derivation S of B is of the form Sx=λx+Jx for all x∈B, where λ∈ℂ and J:B⟶B is a Jordan n-derivation.We know thatB is a prime algebra and all B,eBe,fBf are central algebras over ℂ. Thus, the assumptions of the Wang’result ([2], Corollary 4.5) are fulfilled. Hence, any Jordan derivation J:B⟶B is a derivation.Corollary 7.
LetX be a Banach space over ℂ, dim X≥2. Then every generalized Jordan derivation S of B is of the form Sx=λx+Jx for all x∈B, where λ∈ℂ and J:B⟶B is a derivation.



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