Abstract
Let be a CSL subalgebra of a von Neumann algebra acting on a Hilbert space . It is shown that any Jordan -derivation on is an -derivation, where are any automorphisms on . Moreover, the th power -maps on are investigated.
1. Introduction and Preliminaries
Throughout the paper, let be a complex Hilbert space. Denote by the algebra of all bounded linear operators on and by the identity operator on . The terms “projection" and “subspace" will mean “orthogonal projection" and “norm closed linear manifold." For convenience, a subspace will be identified with the projection onto it. If is a collection of subspaces of , then the smallest subspace containing each will be denoted by and the largest subspace contained in each will be denoted by . A subspace lattice is a strongly closed lattice of subspaces (or projections) that is closed under the usual lattice operations and , containing and . If is a subspace lattice, denote by the algebra of all bounded operators in which leave every subspace in invariant; that is, . Dually, for a subalgebra of , denote by the lattice of all closed subspaces left invariant under every operator in ; that is, . A subspace lattice is reflexive if ; an algebra is reflexive if . If all projections in a subspace lattice commutate pairwise, then the subspace lattice is called a commutative subspace lattice, or a CSL for short, and is called a CSL algebra. If is a CSL, whose projections are contained in a von Neumann algebra acting on a Hilbert space , then is called a CSL subalgebra of the von Neumann algebra .
Let be an algebra. A linear map is called a derivation if for all . Let us review some generalizations of the notion of derivations. Suppose that and are two automorphisms on . Recall that a linear map is an -derivation if for all ; and is a Jordan -derivation if for all . For the identity isomorphism and an automorphism on , we refer to a -derivation as a -derivation and a Jordan -derivation as a Jordan -derivation. Note that an -derivation is just a derivation. A linear map on is called a generalized -derivation if for all , where is an -derivation on . The concept of generalized -derivations covers both the concepts of -derivations and of generalized derivations. The notions of -derivation as a generalization of the concept of derivation have been studied in ([1–5] and references therein).
It is straightforward to show that if is an -derivation of and is an integer, then for any in , where for any element in . This equation is called the th power -property. When , the th power -property makes a Jordan -derivation. The natural question is that if is a linear map on and satisfies the th power -property, must be an -derivation?
The purpose of the present article is to answer the above question if is a CSL subalgebra of a von Neumann algebra acting on a Hilbert space. It can be proved that is a Jordan -derivation if is a linear map on and satisfies the th power -property. So we need to prove that a Jordan -derivation on is an -derivation.
The characterization of Jordan -derivations on algebras or rings is a subject in various areas. Clearly, any -derivation is a Jordan -derivation. Ashraf et al. [6] and Lanski [7] provided us with beautiful counterexamples to illustrate that the converse statements are not true in general. However, a classical result of Herstein ([8], Theorem3.3) shows that a Jordan derivation of a 2-torsion free prime ring is a derivation. This result was extended to 2-torsion free semiprime rings [9]. Brešar and Vukman proved [10] that a Jordan -derivation on a 2-torsion free prime ring is an -derivation. Then it was proved [7] that a Jordan -derivation on a 2-torsion free semiprime ring is an -derivation. It was proved [11] that a Jordan derivation on a CSL algebra is a derivation. And it was extended [12] that a Jordan derivation on a CSL subalgebra of von Neumann algebra is a derivation.
Motivated by these results, we are concerned with a Jordan -derivation and a Jordan -derivation on , a CSL subalgebra of a von Neumann algebra, which is not a semiprime ring.
The main results are as follows.
Theorem 1. Let be a von Neumann algebra acting on a Hilbert space , let be a CSL, whose projections are contained in , and let be the CSL subalgebra of the von Neumann algebra . Suppose that is any automorphism on ; then a Jordan -derivation on is a -derivation.
Theorem 2. Let be a von Neumann algebra acting on a Hilbert space , let be a CSL, whose projections are contained in , and let be the CSL subalgebra of the von Neumann algebra . Suppose that are any automorphisms on ; then a Jordan -derivation on is an -derivation.
Then we extend the Jordan -derivation case to the th power -property case for arbitrary , and we extend the derivation case to the generalized derivation case.
Corollary 3. Let be an integer, let be a CSL subalgebra of a von Neumann algebra acting on a Hilbert space , and let be any automorphisms on . If is a linear map on such that for any , where , then is an -derivation.
Corollary 4. Let be an integer, let be a CSL subalgebra of a von Neumann algebra acting on a Hilbert space , and let be any automorphisms on . If and are two linear maps on such that for any , where , then is an -derivation and is a generalized -derivation on with respect to .
2. Proof of Theorems 1 and 2
Throughout the paper, we suppose that is a von Neumann algebra acting on a Hilbert space , is a CSL whose projections are contained in , and is the CSL subalgebra of the von Neumann algebra .
Lemma 5 (see [12]). Suppose that is a CSL subalgebra of the von Neumann algebra . Denote , or simply, by the orthogonal projection onto the linear span of the set and , or simply, by the orthogonal projection onto the linear span of the set , and . Then, one has the following:(i). And , , , where is the commutant of . commutes with . Furthermore, for any , so that .(ii)If , then is a von Neumann algebra on .
Proposition 6. Let be a CSL subalgebra of a von Neumann algebra . Suppose that is an automorphism on with the property that, for any projection , there are , depending on , such that In such case, a Jordan -derivation on is a -derivation.
Proof. Let be a CSL subalgebra of a von Neumann algebra . Suppose that is a Jordan -derivation on . For any in , Since , Replacing by in (4), we have that In the sequel of proof, we choose an arbitrary nontrivial projection in the set . Let . Then and for any .
From , we have that . By the condition of , For any , by (4), we have that Since and , It follows that . Since by the condition of , By the fact that we get that Similarly,It follows that By (3), and Let . Then by (11) and (13), we have that Let , . Then we have that It follows that . The arbitrariness of and gives that Similarly, for any , we have that Let , . Then we have that It follows that So by involution. The arbitrariness of and gives that We now define a new linear map by Then is also a Jordan -derivation. Since by (6), it follows that and . Note that and for each . ThenClaim. If , then is a -derivation.
For this, it suffices obviously to prove that is a -derivation. In fact, let . Then equalities (16) and (22) hold if is replaced by . By (24), we have that By (16) and (26), we have that By (22), Since , and By (25), we get that It follows that Equation (14) gives that By (17), we have that By (24), (25), (26), (27), (31), (32), and (33), It follows that is a -derivation, and so is . The proof of Claim is complete.
For the general case, let . By Lemma 5, the algebra can be written as the direct sum We now define a new linear map by Then is a Jordan -derivation and . For any , by (5) and the assumption of , Then is of form . By Lemma 5, . It follows that for any . Then maps , onto themselves, respectively. So can be written as the direct sum , where , are the restrictions of on , , respectively. And , are both Jordan -derivations. Since we have that is a CSL subalgebra of the von Neumann algebra . For any , , and , we have that and Since we have that and similarly . It follows that is the identity element of . By the Claim, we have that is a -derivation. Notice that is von Neumann algebra and von Neumann algebra is a semiprime ring. By Theorem2 of [7], we know that is also a -derivation. Consequently, is a -derivation and so is , as required.
Lemma 7. Suppose that is CSL subalgebra of the von Neumann algebra . Denote , or simply, by the orthogonal projection onto the linear span of the set ; , or simply, by the orthogonal projection onto the linear span of the set , and . Then , the double commutant of .
Proof. Let .
Since , we have that and . It follows that . Hence .
Similarly, for any , , . It follows that and by involution. Noting that . So . Hence .
It follows that .
Lemma 8 (Theorem2.1, [13]). Let , be commutative subspace lattices on Hilbert spaces and , respectively, and be an isomorphism. Let be a masa (maximal abelian self-adjoint subalgebra) contained in . Then there is a bounded invertible operator and an automorphism such that(i) for all .(ii) for any .
Remark 9 (Remark1, [13]). Lemma 8 holds for an isomorphism , where and are assumed to contain masas.
Proof of Theorem 1. Suppose that is a Jordan -derivation on . We first note that any CSL subalgebra of a von Neumann algebra contains a masa which contains , the double commutant of .
In fact, for a CSL , , which gives that is a von Neumann algebra. It follows that is an abelian von Neumann algebra. Since , is an abelian von Neumann algebra.
As in the proof of Proposition 6, we can choose an arbitrary nontrivial projection in . Lemma 7 gives that , the masa (maximal abelian self-adjoint subalgebra). Lemma 8 with Remark 9 following it gives that the automorphism and the projection meet the property of Proposition 6. Using the proof of Proposition 6, we can similarly prove that a Jordan -derivation on is a -derivation.
Proof of Theorem 2. Suppose that is a Jordan -derivation on . For any , Let and ; then the above equality gives that is a Jordan -derivation. It follows that is a -derivation by Theorem 1. So for any , which gives that is an -derivation on .
Example 10. Let be CSL subalgebra of von Neumann algebra. If there is an invertible operator such that is not the identity operator on . Consider the mapping on by . Then is an automorphism on and meets the property in Proposition 6. Let , where is the identity function on . It is easy to see that is a -derivation but not a derivation on .
For example, let be an orthonormal basis for . We use the notation for the span of the first, second, and fourth of these vectors and so on. is the collection of matrices of the form where are complex numbers. The algebra is reflexive and its lattice consists of the subspaces . is a CSL subalgebra of a von Neumann algebra. Let . Then is invertible and . Consider the mapping on by ; that is, It is obvious that is an automorphism on and meets the property in Proposition 6. Define a mapping on by It is easy to see that is a -derivation and but not a derivation on .
It shows that Proposition 6 and Theorems 1 and 2 in the present paper generalize Theorem1.2 of [12].
3. Generalized th Power -Maps on CSL Subalgebras of Von Neumann Algebras
The following proposition is essentially due to [7]. For the sake of completeness, we outline the proof.
Proposition 11. Let be an integer, let be a unital operator algebra on a Hilbert space with unit , and let be automorphisms on . If is a linear map on such that for any , where , then is a Jordan -derivation.
Proof. When , our assumption is , and we reduce the general situation to this case. Substituting for in our assumed relation (46) gives that . Next, replace with for any integer . The result of this substitution, using the additivity of and , is Expanding (47), now using the additivity of and , we collect terms containing to write Note that is the relation assumed in the proposition, so it is zero, which leads to We replace by in turn in equality (48). Expressing the resulting system of homogeneous equations, we see that the coefficient matrix of the system is a Vandermonde matrix Since the determinant of the matrix is different from zero, it follows that the system has only trivial solution, so each In particular, We get that It follows that . The proof is complete.
The proof of Corollary 3 is as follows.
Proof. It follows from Proposition 11 and Theorem 2.
The proof of Corollary 4 is as follows.
Proof. It is similar to the proof of Theorem3 in [7].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Part of the work was done during the first author’s visit to Queen’s University, Canada. The first author gratefully acknowledges the kind hospitality from the institute, especially Professor Jamie Mingo. The authors would like to thank Professor Jamie Mingo for his many very helpful comments and suggestions that helped to improve the presentation of this paper in the preparation of this paper. This work was supported by the National Nature Science Foundation of China (nos. 11401273 and 11371279), the Natural Science Research Foundation of Jiangxi Province Education Department (no. GJJ160915), and the Special Fund Project of the Young and Middle-Aged Teachers’ Development Plan of the Ordinary Undergraduate Colleges and Universities in Jiangxi Province.