The Characterization of Generalized Jordan Centralizers on Triangular Algebras

Abstract

In this paper, it is shown that if is a triangular algebra and is an additive operator on such that for any , then is a centralizer. It follows that an Jordan centralizer on a triangular algebra is a centralizer.

1. Introduction and Preliminaries

The triangular algebras are firstly introduced in [1]. Let and be two unital algebras over a commutative ring , where is a unital commutative ring, and let be a unital bimodule. Suppose that is a faithful bimodule; that is, is an bimodule satisfying : and implies that , and : implies that . Consider the setThe usual matrix addition and matrix multiplication on are defined as follows: for any ,It is easy to see that is an algebra under the usual matrix addition and matrix multiplication and is called a triangular algebra. We denote by for simplicity.

Let be a ring or an algebra and be an additive mapping on . If (resp., ) for any , then is called a left centralizer (resp., a right centralizer). A centralizer of is an additive mapping which is both a left centralizer and a right centralizer. An additive mapping is called a left (resp., right) Jordan centralizer, if (resp., ) for any . A Jordan centralizer of is an additive mapping which is both a left Jordan and a right Jordan centralizer. An Jordan centralizer is defined in ([2]) as follows: let be a ring or an algebra and be an additive mapping on . If there are such that for any , then is called an Jordan centralizer.

The characterization of centralizers on algebras or rings is a subject in various areas. Bresar and Zalar ([3]) proved that if is a prime ring and is an additive mapping on such that (resp., ) for any , then is a left (resp., a right) centralizer. Zalar ([4]) generalized the result to 2-torsion free semiprime rings as follows: if is a 2-torsion free semiprime ring and is an additive mapping on such that (resp., ) for any , then is a left (resp., a right) centralizer. Vukman ([5]) proved that if is a 2-torsion free semiprime ring and is an additive mapping on such that for any , then is a centralizer. Benkovic and Eremita ([6]) proved that if is a prime ring with or , where is a fixed positive integer such that , and is an additive mapping on such that for any , then is a centralizer. Vukman and Kosi-Ulbl ([7]) proved that if is a Banach space over the field , and is a standard subalgebra of and is an additive mapping such that for any , where , then is a centralizer. Qi et al. ([8]) proved that if is a standard subalgebra of with the identity and is an additive mapping such that for any , where is a Banach space over the field and , then is a centralizer. Yang and Zhang ([9]) proved that, if is an additive mapping on a nest algebra , such that or for any , where is a nontrivial nest on , is the corresponding nest algebra, and , then is a centralizer. J. Vukman ([2]) proved that an Jordan centralizer on a prime ring with is a centralizer. Li et al. ([10]) proved that a Jordan centralizer on a CSL subalgebra of a von Neumann algebra is a centralizer. Chen et al. ([11]) proved that an additive mapping on a CSL subalgebra of a von Neumann algebra such that for any is a centralizer. Lately several authors investigated Jordan centralizers or their related mappings on rings and algebras. Some of the results can be found in ([5, 12–20]).

Motivated by these results, we are concerned with an additive mapping such that for any . It is shown that if is a triangular algebra and is an additive operator on such that for any , then is a centralizer. And it follows that an Jordan centralizer on a triangular algebra is a centralizer. As applications, Jordan centralizers on (block) upper triangular matrix algebras and nest algebras are centralizers.

2. Generalized Jordan Centralizers on Triangular Algebras

If is a unital commutative ring, and are unital algebras over with units and , respectively, and is a unital, faithful bimodule, thenis a triangular algebra.

Let be a triangular algebra. And let , , where and are the identity elements of algebras and , respectively. And let (). Then and .

The main result is as follows.

Theorem 1. Let be a triangular algebra and be an additive mapping. If there exist numbers such thatfor any , then is a centralizer; that is, and for any , where .

The proof of Theorem 1 can be divided into several lemmas.

Lemma 2. Suppose that is a unital algebra, and is an additive mapping on such thatThat is, for any , there is (depending on ) such thatwhere . Then(1)(2)for any .

Proof. For any ,On the other hand,Comparing the above two equalities, we obtain thatPutting in (11) , it follows from that

Lemma 3. Let be a unital algebra and be an additive mapping on satisfying (4). If with such that , then and .

Proof. Since , . By (12), we have thatandHenceOn the other hand,Comparing the two equalities, we have that . Since and , and .

Lemma 4. Let be a unital algebra and be an additive mapping on satisfying (4). If with , then .

Proof. If or , the result is trivial.
Let be a nontrivial idempotent; that is, and . By (4),By (12),Multiplying (17) by from the left and the right sides givesMultiplying (18) by from the left and the right sides, we have thatBy comparing the above two equalities,Multiplying (17) by from the left side gives thatThat is,It follows from (20) thatThusMultiplying (18) by from the left side yields thatand comparing and (25), we obtain thatIt follows from (21) that andIt follows from (24) thatSimilarly,andEquations (27) and (29) yield that . And by (28) and (30). By Lemma 3 and , it follows that and . And by (21), and . Identity (17) yields that

Lemma 5. Let be a unital algebra and be an additive mapping on satisfying (4). If with , then(1),(2), where .

Proof. By (12), Similarly, .

Lemma 6. Let be a unital algebra and be an additive mapping on satisfying (4). If with , then .

Proof. If or , the result is trivial.
Let be a nontrivial idempotent; that is, and . It follows from Lemma 5 that Comparing the above two equalities, we have that It follows from Lemma 2(2) that By Lemma 4, we have that , , and . Putting for and for in (11), we have that By (36) with (37), It follows from (36) that Comparing (38) and (39), we have that Multiplying (36) by from the left and the right sides gives that Multiplying (37) by from both sides yields that It follows that Comparing (41) and (43), we have that It follows from the above equality and (40) that Since , and By (37) and (45), By (36) and (45), Comparing it with (46), we have that and

Lemma 7. Let be a triangular algebra. Suppose that .(1)If , then ;(2)If , then .

Proof. (1) If and , then for any . So for any . It follows from the definition of a triangular algebra that . And
(2) It is similar to the proof of (1).

Lemma 8. Let be a triangular algebra, and be an additive mapping satisfying (4). Then .

Proof. It follows that and from Lemma 6.
For any , by Lemma 5, we have that and It follows that andBy (12), we have that Multiplying the above equality by from the right side gives that It follows that and Similarly, we have that It follows that and

Lemma 9. Let be a triangular algebra, and be an additive mapping satisfying (4). Let . Then(1);(2);(3);(4).

Proof. (1) By (11), Since , , and , it follows that Multiplying (60) by from the right side, using the fact that , yields that and . Since , it follows from Lemma 7(2) that Since , Combining it with (60), we have that . Since , (2) By (11), Using the fact that yields that Multiplying (65) by from the right side, we have that and . It follows from Lemma 7(2) that and Combining it with (65), we have that . Since , (3) By (1), for any , and by Lemma 7(1).
(4) By (2), for any , and by Lemma 7(2).

Proof of Theorem 1. Let . Then and . By Lemmas 8 and 9, we have that So that is a centralizer.

Corollary 10. Let be a triangular algebra, and be an additive mapping. If there are positive numbers , such that for any , then is a centralizer.

Theorem 11. Let be a triangular algebra, and be an additive mapping. If there are positive numbers and a positive integer , such that for any , then is a centralizer.

Proof. Putting for in (73), where is any rational number, we get that for any , so that is a centralizer by Corollary 10.

Theorem 12. Let be a triangular algebra and be an additive mapping. If there are positive integers , such that for any , then is a centralizer.

Proof. Putting for in (74), where is any rational number, we can also get that for any , so that is a centralizer by Corollary 10.

3. Applications

Upper triangular matrix algebras, block upper triangular matrix algebras, and nest algebras over a Hilbert space are important examples of triangular algebras.

Upper triangular matrix algebras. Let be the set of all matrices and be the algebra of all upper triangular matrices over a ring . For and each , the algebra can be represented as a triangular algebra of the form .

Block upper triangular matrix algebras. Let be the set of all positive integers and let . For every positive integer , , we denote by an ordered vector of positive integers such that . The block upper triangular matrix algebra with corresponding vector is a subalgebra of which contains all block upper triangular matrices, where diagonal blocks have sizes , respectively. Note that and are two special cases of block upper triangular matrix algebras. If we have that and , then is a triangular algebra and can be represented as , where and and are block upper triangular algebras with suitable vectors ,.

Corollary 13. Every Jordan centralizer of a block upper triangular matrix algebra ) is a centralizer, where is the complex field.

Corollary 14. Let be a block upper triangular matrix algebra and be an additive mapping on ). If there are positive numbers and a positive integer , such that for any , then is a centralizer.

Corollary 15. Let be a block upper triangular matrix algebra and be an additive mapping on ). If there are positive numbers and a positive integer , such that for any , then is a centralizer.

Nest algebras. A nest is a chain of closed subspaces of a real or complex Hilbert space containing and , which is closed under arbitrary intersections and closed liner span. The nest algebra associated with is the algebra A nest is called trivial if . A nontrivial nest algebra over a Hilbert space is a triangular algebra.

In fact, by Peirce decomposition, a nontrivial nest algebra is a triangular algebra. Suppose that is a nontrivial nest algebra, and is the corresponding nest. Choose any such that , . Set , , , and . Then ; that is, Since is a faithful bimodule, the nest algebra is a triangular algebra.

Corollary 16. Let be a nontrivial nest algebra over a Hilbert space. Then every Jordan centralizer on the nest algebra is a centralizer.

Corollary 17 (see [9]). Let be a nontrivial nest algebra over a Hilbert space and let be an additive mapping on . If there are positive numbers and a positive integer , such that for any , then is a centralizer.

Corollary 18 (see [9]). Let be a nontrivial nest algebra over a Hilbert space and let be an additive mapping on . If there are positive numbers and a positive integer , such that for any , then is a centralizer.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors are supported by the National Science Foundation of China (No. 11401273, No. 11871375) and the Natural Science Research Foundation of Jiangxi Provincial Education Department (No. GJJ170759).