Local Generalized (𝛼,𝛽)-Derivations

Abstract

We study local generalized ()-derivations on algebras generated by their idempotents and give some important applications of our results.

1. Introduction

A linear mapping on an algebra is called a local derivation if for every there exists a derivation depending on such that . This notion was introduced independently by Kadison [1] and Larson and Sourour [2]. In [1] Kadison investigated continuous local derivations on von Neumann algebras. He proved that if is a von Neumann algebra and a dual -module, then all norm-continuous local derivations from into are derivations. This research was motivated by problems concerning the Hochschild cohomology of operator algebras. On the other hand, Larson and Sourour [2] proved that every local derivation on the algebra of all bounded linear operators on a Banach space is a derivation.

Of course, every derivation on an algebra is a local derivation. But the converse is in general not true. Kadison [1] constructed an example (due to C. Jensen) of an algebra (not an operator algebra) which has nontrivial local derivations. Moreover, there are examples of local derivations on operator algebras in the literature, for example, using the subalgebra of complex matrices consisting of constant multiples of the identity plus strictly upper triangular matrices.

In the last few decades a lot of work has been done on local mappings on some algebras. The results show that, in many important cases, local mappings of some certain class of transformations on a given algebra are global (see [3]). In the context of derivations, the relation between local derivations and derivations was widely studied by several authors (see, e.g., [1, 2, 4–10] and the references therein). Recently, Hadwin and Li proved that every local derivation from an algebra which is generated by its idempotents into any -bimodule is a derivation. This motivated us to study local generalized -derivations on algebras generated by their idempotents.

Before continuing, let us fix the notation and write some basic definitions which we will need in our further investigation. Throughout the paper, will be an algebra and will be an -bimodule. A linear mapping is called a derivation if In [11], Brešar defined the notion of generalized derivation as follows. A linear mapping is a generalized derivation if there exists a derivation such that In this case we say that is a generalized derivation associated with a derivation (or simply, is a generalized -derivation). On the other hand, Nakajima [12] defined generalized derivations without using the corresponding derivations as follows. Let be a linear mapping and an element of . A pair is called a generalized derivation if In particular, if has a unit , then and is a generalized derivation if it satisfies Since we will deal just with unital algebras, we will say that a linear mapping is a generalized derivation if and only if it satisfies the above condition. We refer the readers to [12], where they can find more information about generalized derivations.

Let be endomorphisms of a unital algebra and the identity map on . Motivated by the above notions, we define -derivations as linear mappings satisfying Further, a linear mapping is called a generalized -derivation if Of course, -derivation (generalized -derivation, resp.) is just a derivation (generalized derivation, resp.). The next example will show that there exist -derivations which are not derivations.

Example 1. Let be an algebra with a nontrivial central idempotent . Let us define , for all , and . Then is an -derivation which is not a derivation since . Moreover, if is a semiprime algebra and is an endomorphism of , then is an -derivation but not a derivation.

The definition of local generalized -derivations (local -derivations, resp.) can be self-explanatory, A linear mapping is called a local generalized -derivation (local -derivation, resp.) if for every there exists a generalized -derivation (-derivation, resp.) depending on such that .

In the following, we assume that all algebras are unital topological algebras and all bimodules are unital topological bimodules. Recall that a topological algebra is an associative algebra equipped with a vector space topology compatible with its ring structure, in the sense that the ring multiplication is separately continuous. Let be an -bimodule. If is a topological vector space and a topological algebra such that the module multiplications are separately continuous, then we say that is a topological -bimodule.

2. Local Generalized -Derivations

Let be an -bimodule and an ideal of . We say that is a separating set of if, for all , implies and implies . In the following, will be unital homomorphisms. For an idempotent , we write .

Lemma 2. Let be a linear mapping from an algebra into an -bimodule . Then for every and all idempotents , the following are equivalent: (i), (ii).

Proof. Obviously, (ii) implies (i). So, assume that (i) holds for every and all idempotents . Then On the other hand, This yields that for every and all idempotents , . Therefore, for every and all idempotents , .

Lemma 3. If is a linear mapping from an algebra into an -bimodule such that for every and all idempotents , , then for every and all idempotents .

Proof. Using the induction on , we first prove that for every and all idempotents , The case is clear. Assume now that (12) holds true for . Then To show our lemma, we use the induction on . We have just proved the case . So, assume that our assertion holds true for . Then for every and all idempotents ,

Our first theorem is a generalization of Theorem  2.7 in [8] (see also [7, Theorem 22]).

Theorem 4. Let be a separating set of an -bimodule contained in the algebra generated by all idempotents in , , endomorphisms of such that , , and let be a linear mapping. If for every and all idempotents , , then is a generalized -derivation. In particular, if , then is an -derivation.

The idea of the proof comes from [8, Proof of Theorem 2.7]. For the sake of completeness, we write the main steps.

Proof. Let , be arbitrary elements. Then, according to (15) and Lemma 3, we have This yields that for all since is an ideal of . On the other hand, by (15), we have and, consequently, Since is a separating set of and , it follows that for all . Furthermore, if and , then On the other hand, Therefore, and, as above, for all , . If , then is, obviously, an -derivation.

Let be a local generalized -derivation from an algebra into an -bimodule . Suppose that is an arbitrary element and suppose that , are idempotents. Then there exists a generalized -derivation such that . It is also easy to see that Hence, for every and all idempotents , . Thus, by Lemma 2, satisfies the condition (15), and, using Theorem 4, we have the next result.

Theorem 5. Let be a separating set of an -bimodule contained in the algebra generated by all idempotents in and let , be endomorphisms of such that , . Then every local generalized -derivation (local -derivation, resp.) from an algebra into an -bimodule is a generalized -derivation (-derivation, resp.).

Taking , , we have the next direct consequence of Theorem 5.

Corollary 6. Let be a separating set of an -bimodule contained in the algebra generated by all idempotents in . Then every local generalized derivation (local derivation, resp.) from an algebra into an -bimodule is a generalized derivation (derivation, resp.).

At the end, if , then we have the next result for local generalized skew derivations, that is; linear mappings with the property

Corollary 7. Let be a separating set of an -bimodule contained in the algebra generated by all idempotents in and let be an endomorphism of such that . Then every local generalized skew derivation (local skew derivation, resp.) from an algebra into an -bimodule is a generalized skew derivation (skew derivation, resp.).

3. Applications

Our results in Section 2 hold for unital algebras that can be generated (as algebras) by their idempotents. This class of algebras contains many important algebras. For example, if is a unital algebra and , a positive integer, then , that is, the algebra of all matrices over , belongs to this class (see [8, 13]). Thus, we have the next result.

Corollary 8. Let , be automorphisms of . Then every local generalized -derivation (local -derivation, resp.) from an algebra into any -bimodule is a generalized -derivation (-derivation, resp.).

The next result involves local matrix algebras: an algebra is called a local matrix algebra if any finite subset of can be embedded in a subalgebra which is a matrix algebra , .

Corollary 9. Let , be automorphisms of . If, for any , , there exists a unital subalgebra of which contains , and is isomorphic to a matrix algebra, then every local generalized -derivation (local -derivation, resp.) from an algebra into any -bimodule is a generalized -derivation (-derivation, resp.).

Let and be complex Hausdorff topological linear spaces and let be the algebra of all continuous linear mappings from into . We say that a subset of is reflexive if whenever and for any , where denotes the topological closure of . By a subspace lattice on we mean a collection of closed subspaces of containing and such that, for each family of elements of , both and belong to , where denotes the closed linear span of . If is a subspace lattice, then we denote the algebra of all operators on , that leave invariant each element of by . A totally ordered subspace lattice is called a nest and the associated reflexive algebra is called a nest algebra.

Now we consider local generalized -derivations on a reflexive subalgebra in a factor von Neumann algebra. The proof of the following corollaries uses Theorem 4 and arguments similar to those in the proof of [8, Theorem 2.17, Theorem 2.18].

Corollary 10. Let be a subspace lattice in a factor von Neumann algebra on with and and let , be automorphisms of . Then every local generalized -derivation (local -derivation, resp.) from into is a generalized -derivation (-derivation, resp.).

Corollary 11. Let be a nest in a factor von Neumann algebra on and let , be automorphisms of . Then every local generalized -derivation (local -derivation, resp.) from into is a generalized -derivation (-derivation, resp.).

Suppose that is topologically generated by its idempotents (i.e., the subalgebra of generated by its idempotents is dense in ) and suppose that is a continuous local generalized -derivation, where are automorphisms of . Let , for some idempotents , and some scalars . Then, by Lemma 3, we have and since is continuous and is generated by its idempotents, we have the following proposition.

Proposition 12. Let , be automorphisms of . If is topologically generated by its idempotents and is a topological -bimodule, then every continuous local generalized -derivation (local -derivation, resp.) from into is a generalized -derivation (-derivation, resp.).

Hadwin and Li proved that if is a nest in a von Neumann algebra and , then the linear span of all idempotents in is -dense in (see [8, Proposition 2.3]). Thus, by Proposition 12, we have the next corollary.

Corollary 13. Let be a nest in a von Neumann algebra , , and let , be automorphisms of . Then every -continuous local generalized -derivation (resp. local -derivation) from into is a generalized -derivation (resp. -derivation).

At the end, let us prove that the set of all continuous generalized -derivations from into , denoted by , is reflexive.

Corollary 14. Let , , , be as in Theorem 4. Then is reflexive.

Proof. Let be a continuous linear mapping such that for any , Now, let , be arbitrary idempotents, any element from , and . Then, according to above observations, there exists a sequence such that Therefore, According to Lemma 2 and Theorem 4, this yields that is a generalized -derivation; that is, . The proof is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.