#### Abstract

Let A be a unital -algebra and X be a unitary Banach A-bimodule. In this paper, we characterize continuous generalized derivations and generalized Jordan derivations as form through the action on zero product. In other words, we show that under some conditions on elements of A, a linear map on A can be a generalized Jordan derivation.

#### 1. Introduction and Preliminaries

Let be a unital Banach algebra with the unit and be a unitary (Banach) -bimodule. A linear map is said to be a derivation (resp., generalized derivation) if for each ,where “” denotes the Jordan product on , and “” denotes the Jordan product on defined through

The mentioned map is called a Jordan derivation (resp., generalized Jordan derivation) iffor all . By the usual polarization, the Jordan derivation (resp., generalized Jordan derivation) identity is equivalent to assuming that

Clearly, each (generalized) derivation is a (generalized) Jordan derivation, but the converse is not true in general. There are plenty of known examples of Jordan derivations that are not a derivation and can be found in works of literature. For the correctness of the converse, Johnson in  (Theorem 6.3) proved that every continuous Jordan derivation from -algebra in any -bimodule is a derivation.

Recall from  that a -algebra is called a -algebra if it is a dual space as a Banach space. Note that every -algebra is unital.

The set of idempotents of given algebra is denoted by . Let be the subalgebra of generated by idempotents. We say that a Banach algebra is generated by idempotents, if , where denotes the closure of . Examples of Banach algebras with the last property include all -algebras, the group algebra for a compact group , and also topologically simple Banach algebras containing a nontrivial idempotent . In other words, such Banach algebras are generated by idempotents. Another classes of Banach algebras with the property that are available in .

Let be a Banach algebra and be an arbitrary Banach space. We say a continuous bilinear mapping preserves zero products if

The study of zero products preserving bilinear maps has been initiated by Alaminos et al. in  for a very special setting, and then, it was studied in  for the general case. Motivated by (5), the following concept was presented in .

Definition 1. A Banach algebra has the property if for every continuous bilinear mapping , where is an arbitrary Banach space, and condition (5) implies that , for all .
It follows from  (Theorem 2.11) that -algebras, group algebras, and Banach algebras that are generated by idempotents have the property , see also  (Lemma 2.1) for group algebras in a different view.
It should be pointed out that the property is a powerful tool for characterizing homomorphisms, derivations, and Jordan derivations on that class of algebras through the action on zero products. We refer the reader to [3, 512] for a full account of the topic and references therein.
The next result which is closely related to the property plays a key role in the sequel.

Theorem 1. (, Theorem 2.2). Let be a -algebra, be a Banach space, and be a continuous bilinear mapping such that

Then,for all .

In this paper, we consider the subsequent conditions on a linear map from a Banach algebra into a Banach -bimodule for each .(1) (2) (3) (4) (5)

Our purpose is to investigate whether the conditions above can characterize generalized derivations and generalized Jordan derivations. Indeed, which of the above are sufficient conditions to be the generalized derivation of a continuous linear map.

#### 2. Generalized Derivations on -Algebras

In this section, we characterize generalized derivations from unital -algebra into unitary Banach -bimodule that satisfy condition or , through their action on zero products.

Theorem 2. Let be a unital -algebra. If is a continuous linear map satisfying , then is a generalized derivation.

Proof. Define a continuous bilinear mapping by . Then, whenever , and so the property givesTaking in (8), we getfor all . Interchanging into in (9), we get for all . Thus, it follows from (9) thatfor all . Therefore, is a generalized derivation.
Note that if is a generalized derivation, then the linear map defined by is a derivation, and thus, we get the following result.

Corollary 1. (, Corollary 4.2). Let be a unital -algebra. If is a continuous linear map satisfying , then for all , and there is a derivation such that .

Assume that is a continuous linear map from unital -algebra into unitary -bimodule such that for all with . Then, is a generalized Jordan derivation. Indeed, by a similar proof of the preceding theorem, we can obtain

Taking , we find for all . Moreover, if , then is a Jordan derivation, and hence, it is a derivation by  (Theorem 6.3).

The following fundamental example has been demonstrated by Johnson in .

Example 1. LetWe make an -bimodule by definingConsider the linear map defined via . Then, for all , and . In particular, is a (generalized) Jordan derivation, but it is not a (generalized) derivation. Therefore, Johnson’s result is not valid for unital Banach algebras instead of -algebras in general.
In the next result, we characterize generalized derivations by using condition .

Theorem 3. Suppose that is a unital -algebra and is a continuous linear map such that condition holds. Then, is a generalized derivation.

Proof. Pick such that and define a continuous bilinear mapping by , for all . Then, whenever . Using the property with , we obtainWe have derived this identity under the assumption that such that . Let now be fixed. We may apply the property for given throughfor all , and hence, we conclude thatTaking in the above equality, we reachfor all . This completes the proof.

#### 3. Generalized Jordan Derivations on -Algebras

In this section, we prove that each linear mapping from unital -algebra into unitary Banach -bimodule that satisfies one of conditions , , and is a generalized Jordan derivation.

Our first main theorem is indicated as follows.

Theorem 4. Let be a -algebra and be a continuous linear map satisfying . Then, is a generalized Jordan derivation.

Proof. Define a bilinear mapping byThen, implies that . Applying Theorem 1 by putting , we arrivefor all . This means thatReplacing by in (20) givesfor all . Here, we claim that for all . Let be an idempotent in . Substituting by in (21), we haveWe multiply (22) by on the left, and we obtainand soSimilarly, by multiplying both sides of (22) by , we arrive atFrom (24) and (25), it follows that for all idempotent . By Lemma 1.7.5 and Proposition 1.3.1 of , every self-adjoint element is the limit of a sequence of linear combinations of projections in . Therefore, for each self-adjoint element in . Now, each arbitrary element can be written as where are self-adjoint elements of . Hence,for all . Thus, equality (21) implies thatfor all . Consequently, is a generalized Jordan derivation.
It is well-known that there are two products on , the second dual space of a Banach algebra , called the first and second Arens products which make into a Banach algebra (, Definition 2.6.16). If these products coincide on , then is said to be Arens regular. It is shown in Chapter 2 of  that every -algebra is Arens regular. Moreover, the second dual of each -algebra is a -algebra.
According to , for each Banach -bimodule , turns into a Banach -bimodule where equipped with the first Arens product. The module actions are defined bywhere and are nested in and that converge, in -topologies, to and , respectively. One may refer to the monograph of Dales  for a full account of Arens product and -continuity of the above structures.
There exists another related concept of generalized derivation, which appeared in  for the first time. Let be a Banach algebra and be an -bimodule. A linear operator is said to be a generalized derivation if there exists such thatIt should be noted that if is unital and is a generalized derivation, then by  (Proposition 4.2) equality (29) converts tofor all , and hence, is a generalized derivation in the usual sense. Motivated by (29), we introduce the concept of generalized Jordan derivation as follows. A linear operator is said to be a generalized Jordan derivation if there exists such thatfor all . Similarly, if is unital and satisfies in (31) for some , thenIn what follows, we prove Theorem 4 to the -algebra case. First, note that the linear span of projections is dense in a unital -algebra of real rank zero ; hence, the conclusion of Theorem 4 is also valid for such -algebras.
Applying the techniques in the proof of Theorem 4.1 from , we have the upcoming result.

Theorem 5. Let be a unital -algebra and be a continuous linear map satisfying . Then, is a generalized Jordan derivation.

Proof. It follows from Theorem 1 that the bilinear map defined throughfulfillsfor all . Furthermore, the Arens regularity of , the --continuity of , and the separate weak continuity of the module operations on necessitate thatfor all . Take . Then, it follows from equality (35) by putting thatfor all . In particular, we haveSimilar to the proof of Theorem 4, we have for all . Hence, by (37),It remains to show that for all . In other words, it suffices to prove it for each positive element . Suppose that be a positive element and with . According to (38),Consequently, is a generalized Jordan derivation.

Corollary 2. Let be a unital -algebra. If is a continuous linear map satisfying , then is a derivation if and only if .

Proof. It is obvious that if is a derivation, then . On the other hand, it follows from Theorem 5 that is a Jordan derivation when , and therefore, is a derivation by Johnson’s result.
Here, we mention that each condition and the implicationimply , and hence, Theorems 4, 5 and Corollary 2 still work with condition replaced by or .

Theorem 6. Let be a -algebra and be a continuous linear map satisfying . Then, is a generalized Jordan derivation.

Proof. Since for all idempotent , condition implies thatMultiplying both sides of (41) by on the left, we obtainand soSimilarly, by multiplying (41) on the right by , we arrive atFrom (43) and (44), we get for all idempotent . Now, similar to the proof of Theorem 4, we conclude thatApplying Theorem 1 to the bilinear map defined bywe findfor all . Switching by and using (45), we reachReplacing by in (48), it concludes thatfor all . Thus, is a generalized Jordan derivation.
Similar to the proof of Theorem 5, we can obtain the following corollary.

Corollary 3. Let be a unital -algebra and be a continuous linear map satisfying . Then, is a generalized Jordan derivation.

It should be pointed out that each of the conditionsimplies , and therefore, Theorem 6 and Corollary 3 are true if the condition replaced by either or .

Theorem 7. Let be a -algebra and be a continuous linear map such that the condition holds. Then, for all . In particular, is a generalized derivation.

Proof. It is obvious that for every idempotent , . Hence,A simple calculation shows that . Let denote the set of self-adjoint elements of and . Then, , where are real numbers and is an orthogonal family of projections in . Since for , condition implies that for all with . Thus, for each ,It follows from the linearity of that for each . Now, each arbitrary element can be written as for . Therefore,Thus, for all . Replacing by , we obtain for all .
The next result is a direct consequence of Theorem 7. We include it without proof.

Corollary 4. Let be a unital -algebra. If is a continuous linear map satisfying , then for all . Moreover, if , then .

One should remember that Theorem 1 has an important role in characterizing generalized Jordan derivations on -algebras. The following result is another analogous criterion for Banach algebra with the property that .

Theorem 8. (, Corollary 3.6). Let be a unital Banach algebra, be a Banach space, and be a continuous bilinear mapping such that

Then,for all and . In particular, if is generated by idempotents, then

Since all Banach algebras which are generated by idempotents have the property , Theorems 2 and 3 remain valid for such algebras. On the other hand, by using the preceding theorem, some results of the current paper can be proved for unital (Arens regular) Banach algebras with the property that .

#### Data Availability

Not applicable. In fact, all results are obtained without any software and found by manual computations. In other words, the manuscript is in the pure mathematics (mathematical analysis) category.

#### Conflicts of Interest

The authors do not have any conflicts of interest regarding this article.

#### Authors’ Contributions

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.