Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Abstract

Let be a unital algebra with idempotent over a 2-torsionfree unital commutative ring and be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation . We show that, under mild conditions, every generalized Jordan n-derivation is of the form in the current work. As an application, we give a description of generalized Jordan derivations for the condition 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.

1. Introduction

Throughout this paper, let be a commutative ring with an identity and let be a unital algebras over . Let us assume that has an idempotent and let . In this case, can be represented in the so called Peirce decomposition formwhere and are subalgebras with unitary elements and , respectively, is an -bimodule and is an -bimodule. It is worth to mention that is isomorphic to a generalized matrix algebra [1]. We assume that satisfiesfor all . 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 and is nonzero.

Let be an associate algebra or ring. is the Jordan product of elements . For any integer and any . Set andfor , which is called the Jordan -product of . Denote by a linear mapping; we call a Jordan -derivation iffor all . It is obvious that Jordan -derivations are usual Jordan derivations for ; moreover, it is also easily checked that the definition of Jordan 3-derivations are equivalent to the conception of Jordan triple derivations.

A linear mapping is said to be a generalized Jordan -derivation if there exists a Jordan -derivation such thatfor all . It is clear that any Jordan 2-derivations are usual generalized Jordan derivations.

Note that any Jordan -derivation is an example of a generalized Jordan -derivation. On the other hand, any multiplier function for all and 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 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 . 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 -derivation on is additive; and furthermore, it turns out that a mapping on is a multiplicative Jordan -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 -derivation is of the formwhere some and is a Lie n-derivation of . 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 -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 -derivation is of the formwhere , is a Jordan -derivation of . 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 -derivation to the describing a Jordan -derivation. It turns out that if is a generalized Jordan -derivation associated with a Jordan -derivation , then a linear mapping satisfiesfor all . Therefore, it suffices to consider linear mappings with property (♣). Under suitable assumptions on unital algebra with a nonzero idempotent (Proposition 1), any such generalized Jordan -derivation is of the form for all and some .

2. Preliminaries and the Main Theorem

Let be a unital algebra with idempotent and , which satisfies (♣). For convenience, we shall use the following notations , and . Thus, every element can be represented in the formwhere is an -bimodule and is an -bimodule.

Let us list some classical example of unital algebras with idempotent and , 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 algebra , where is a unital algebra.(b)Every simple unital algebra with nontrivial idempotent, which satisfies (♣).(c)Unital prime algebras with nontrivial idempotent.(d)Triangular algebra such that the bimodule is faithful as a left and also as a right -module. The most important examples of triangular algebras are upper triangular matrix algebras and block upper triangular matrix algebras over a unital algebra A and also nest algebras , where is a nest in a Hilbert space .

The first very useful observation refers to the form of the center of , which is identical to ([15], Proposition 3) and ([16], Lemma 3.1, Lemma 3.2). Throughout the paper, denotes a unital algebra with a nonzero idempotents satisfying (♣).

By [12], [Proposition 2.1], it follows that the center of is equal to

Furthermore, we know that the map is an algebraic isomorphism such that and for all .

Remark 1. Let be a unital algebra with nontrivial idempotent and . For any and for any integer , we haveIn particular, and .
In this section, we will prove the main result, Theorem 1. As we mentioned in the introduction, the problem of a description of a generalized Jordan -derivations can be reduced to a description of a map satisfying (8). Let us begin with the solution of this problem.

Proposition 1. Let be a unital algebra with a nontrivial idempotent satisfying (♣) over a 2-trosionfree commutative ring . Let us assume that and . Let a linear mapping satisfiesfor all . Thenwhere

Proof. To prove this proposition, we will need the following Claims:(i)Claim 1. . By taking for all , one can show .(ii)Claim 2. With notations as above, we have , and for all . By the definition , we have and which imply and as char . Therefore and . For all , we have and for all . Hence, as char . On the other hand, for all , and Therefore, for all . Combining (18) and (21) with the definition of the center , we have(iii)Claim 3. With notations as above, we have and for all and For all , we haveSince the characteristic is not 2, this impliesand thenfor all .
Similarly, one can check thatfor all .
For all , by Claims , we havewhere
The main result of the the paper state:

Theorem 1. Let be a unital algebra with a nontrivial idempotent satisfying (♣). Let us assume that(i)(ii)Then any generalized Jordan -derivation is of the from for all , where and is a Jordan -derivation.

Proof. Let be a generalized Jordan -derivation associated with a Jordan n-derivation . According to the definitionfor all . Let us denote . If we subtract upper equalities, we see that a linear mapping satisfiesfor all . Since all assumption from Proposition 1 are fulfilled there exists such that .
According to Theorem 1 and ([2], Theorem 4.1), Jordan -derivation is usual Jordan derivation if , the following result is immediate.

Corollary 1. Let be a unital algebra with a nontrivial idempotent satisfying (♣). Let us assume that(i)(ii)Then any generalized Jordan derivation is of the from for all , where , is a derivation and is a singular Jordan derivation.

Corollary 2. ([4], Theorem 3.11) Let be a unital algebra with a nontrivial idempotent satisfying (♣) and a bimodule is faithful as a left -module and as a right -module. Then every Jordan derivation of A can be expressed as the sum of a derivation and an antiderivation.

We apply Theorem 1 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 -derivation to the description of a Jordan -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 over a unital algebra . One can also obtain that there are no nonzero singular Jordan derivations of a unital prime algebra with a nontrivial idempotent . Therefore, Theorem 1 implies the following corollaries.

Corollary 3. Let , , where is a unital -torsionfree algebra. Then every generalized Jordan derivation is of the form , where , is a derivation.

We conclude this article with some applications of the main theorem. In case A is a unital algebra with a nontrivial idempotent e such that , and that the bimodule is faithful as a left -module and also as a right -module, the algebra is a triangular algebra. The triangular algebra satisfies (♣) and by the definition has no nonzero singular Jordan derivations. Therefore, combining with [7], we obtain the following result.

Corollary 4. Let be a triangular algebra. Let us assume that(i)(ii)Then, any generalized Jordan derivation is of the from for all , where and is a derivation.

Corollary 5. Every Jordan derivation of a unital prime algebra A with a nontrivial idempotent e is a derivation.

Algebras of all bounded linear operators.

Let be a Banach space over of dimension greater than 1. By , we denote the algebra of all bounded linear operators on . contains nontrivial idempotent and hence can be presented in the form , where . Since is a prime algebra, satisfies (♣). Note that and are algebras of all bounded linear operators and all are central algebras over . Therefore, and . Hence meets assumptions of Theorem 1 and we have.

Corollary 6. Let be a Banach space over , dim . Then every generalized Jordan -derivation of is of the form for all , where and is a Jordan -derivation.

We know that is a prime algebra and all are central algebras over . Thus, the assumptions of the Wang’result ([2], Corollary 4.5) are fulfilled. Hence, any Jordan derivation is a derivation.

Corollary 7. Let be a Banach space over , dim . Then every generalized Jordan derivation of is of the form for all , where and is a derivation.

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Key Projects of Natural Science Research in Anhui Province (Grant No. 2008085QA01), Youth fund of Anhui Natural Science Foundation (Grant No. KJ2019A0107), National Natural Science Foundation of China (No. 11901030), and Natural Science Foundation of Beijing Municipality (No. 1204034).