# Generalized Derivations and Generalized Jordan Derivations onC∗-Algebras through Zero Products

Authors: Abbas Zivari-Kazempour; Abasalt Bodaghi
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetA be a unital C∗-algebra and X be a unitary Banach A-bimodule. In this paper, we characterize continuous generalized derivations and generalized Jordan derivations as form D:A⟶X 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.


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## Body

## 1. Introduction and Preliminaries

LetA be a unital Banach algebra with the unit eA and X be a unitary (Banach) A-bimodule. A linear map D:A⟶X is said to be a derivation (resp., generalized derivation) if for each a,b∈A,(1)Dab=Da⋅b+a⋅Db,resp.,Dab=Da⋅b+a⋅Db−a⋅DeA⋅b,where “∘” denotes the Jordan product a∘b=ab+ba on A, and “·” denotes the Jordan product on X defined through(2)a·x=x·a=a⋅x+x⋅a,a∈A,x∈X.The mentioned map is called a Jordan derivation (resp., generalized Jordan derivation) if(3)Da∘b=Da·b+a·Db,resp.,Da∘b=Da·b+a·Db−a⋅DeA⋅b−b⋅DeA⋅a,for all a,b∈A. By the usual polarization, the Jordan derivation (resp., generalized Jordan derivation) identity is equivalent to assuming that(4)Da2=Da·a,resp.,Da2=Da·a−aDeAa,a∈A.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 [1] (Theorem 6.3) proved that every continuous Jordan derivation from C∗-algebra A in any A-bimodule X is a derivation.Recall from [2] that a C∗-algebra A is called a W∗-algebra if it is a dual space as a Banach space. Note that every W∗-algebra is unital.The set of idempotents of given algebraA is denoted by ℐA. Let algℐA be the subalgebra of A generated by idempotents. We say that a Banach algebra A is generated by idempotents, if A=algℐA¯, where algℐA¯ denotes the closure of algℐA. Examples of Banach algebras with the last property include all W∗-algebras, the group algebra L1G for a compact group G, and also topologically simple Banach algebras containing a nontrivial idempotent [3]. In other words, such Banach algebras are generated by idempotents. Another classes of Banach algebras with the property that A=algℐA¯ are available in [3].LetA be a Banach algebra and X be an arbitrary Banach space. We say a continuous bilinear mapping ϕ:A×A⟶X preserves zero products if(5)ab=0⟹ϕa,b=0,a,b∈A.The study of zero products preserving bilinear maps has been initiated by Alaminos et al. in [4] for a very special setting, and then, it was studied in [3] for the general case. Motivated by (5), the following concept was presented in [3].Definition 1.
A Banach algebraA has the property B if for every continuous bilinear mapping ϕ:A×A⟶X, where X is an arbitrary Banach space, and condition (5) implies that ϕab,c=ϕa,bc, for all a,b,c∈A.
It follows from [3] (Theorem 2.11) that C∗-algebras, group algebras, and Banach algebras that are generated by idempotents have the property B, see also [5] (Lemma 2.1) for group algebras in a different view.
It should be pointed out that the propertyB 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, 5–12] for a full account of the topic and references therein.
The next result which is closely related to the propertyB plays a key role in the sequel.Theorem 1.
([6], Theorem 2.2). Let A be a C∗-algebra, X be a Banach space, and ϕ:A×A⟶X be a continuous bilinear mapping such that(6)ab=ba=0⇒ϕa,b=0,a,b∈A.Then,(7)ϕxa,yb+ϕbx,ay=ϕx,ayb+ϕbxa,y,for all a,b,x,y∈A.In this paper, we consider the subsequent conditions on a linear mapD from a Banach algebra A into a Banach A-bimodule X for each a,b,c∈A.(1)
G1ab=0⇒Da⋅b+a⋅Db=0(2)
G2ab=bc=0⇒a⋅Db⋅c=0(3)
J1ab=ba=0⇒Da⋅b+a⋅Db=0(4)
J2ab=ba=0⇒Da·b+a·Db=0(5)
J3ab=ba=0⇒a⋅Db+b⋅Da=0.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 onC∗-Algebras

In this section, we characterize generalized derivations from unitalC∗-algebra A into unitary Banach A-bimodule X that satisfy condition G1 or G2, through their action on zero products.Theorem 2.
LetA be a unital C∗-algebra. If D:A⟶X is a continuous linear map satisfying G1, then D is a generalized derivation.Proof.
Define a continuous bilinear mappingϕ:A×A⟶X by ϕa,b=Da⋅b+a⋅Db. Then, ϕa,b=0 whenever ab=0, and so the property B gives(8)Da⋅bc+a⋅Dbc=ϕa,bc=ϕab,c=Dab⋅c+ab⋅Dc,a,b,c∈A.
Takingc=eA in (8), we get(9)Da⋅b+a⋅Db=Dab+ab⋅DeA,for all a,b∈A. Interchanging a into eA in (9), we get DeA⋅b=b⋅DeA for all b∈A. Thus, it follows from (9) that(10)Dab=Da⋅b+a⋅Db−a⋅DeA⋅b,for all a,b∈A. Therefore, D is a generalized derivation.
Note that ifD is a generalized derivation, then the linear map δ:A⟶X defined by δa=Da−DeA⋅a is a derivation, and thus, we get the following result.Corollary 1.
([4], Corollary 4.2). Let A be a unital C∗-algebra. If D:A⟶X is a continuous linear map satisfying G1, then a⋅DeA=DeA⋅a for all a∈A, and there is a derivation δ:A⟶X such that Da=DeA⋅a+δa.Assume thatD is a continuous linear map from unital C∗-algebra A into unitary A-bimodule X such that Dba+bDa=0 for all a,b∈A with ab=0. Then, D is a generalized Jordan derivation. Indeed, by a similar proof of the preceding theorem, we can obtain(11)Dab=Db⋅a+b⋅Da−DeA⋅ab,a,b∈A.Takinga=b, we find Da2=Da·a−aDeAa for all a∈A. Moreover, if DeA=0, then D is a Jordan derivation, and hence, it is a derivation by [1] (Theorem 6.3).The following fundamental example has been demonstrated by Johnson in [1].Example 1.
Let(12)A=a11a120a22:a11,a12,a22∈ℂ.
We makeX=ℂ an A-bimodule by defining(13)aλ=a22λ,λa=λa11,λ∈ℂ,a∈A.
Consider the linear mapD:A⟶X defined via Da=a12. Then, Dab=Db⋅a+b⋅Da for all a,b∈A, and DeA=0. In particular, D 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 C∗-algebras in general.
In the next result, we characterize generalized derivations by using conditionG2.Theorem 3.
Suppose thatA is a unital C∗-algebra and D:A⟶X is a continuous linear map such that condition G2 holds. Then, D is a generalized derivation.Proof.
Picka0,b0∈A such that a0b0=0 and define a continuous bilinear mapping ϕ:A×A⟶X by ϕa,b=aDba0b0, for all a,b∈A. Then, ϕa,b=0 whenever ab=0. Using the property B with c=eA, we obtain(14)abDa0b0=ϕab,eA=ϕa,b=a⋅Dba0⋅b0,a,b∈A.
We have derived this identity under the assumption thata0,b0∈A such that a0b0=0. Let now a1,b1∈A be fixed. We may apply the property B for ψ:A×A⟶X given through(15)ψa,b=a1b1⋅Da⋅b−a1⋅Db1a⋅b,for all a,b∈A, and hence, we conclude that(16)a1b1⋅Dab−a1⋅Db1ab⋅c=a1b1⋅Dabc−a1Db1a⋅bc.
Takinga1=c=a=eA in the above equality, we reach(17)Db1b=Db1⋅b+b1⋅Db−b1⋅DeA⋅b,for all b1,b∈A. This completes the proof.


## 3. Generalized Jordan Derivations onC∗-Algebras

In this section, we prove that each linear mappingD from unital C∗-algebra A into unitary Banach A-bimodule X that satisfies one of conditions J1, J2, and J3 is a generalized Jordan derivation.Our first main theorem is indicated as follows.Theorem 4.
LetA be a W∗-algebra and D:A⟶X be a continuous linear map satisfying J1. Then, D is a generalized Jordan derivation.Proof.
Define a bilinear mappingϕ:A×A⟶X by(18)ϕa,b=Da⋅b+a⋅Db,a,b∈A.
Then,ab=ba=0 implies that ϕa,b=0. Applying Theorem 1 by putting x=y=eA, we arrive(19)ϕa,b+ϕb,a=ϕeA,ab+ϕba,eA,for all a,b∈A. This means that(20)a·Db+Da·b=Da∘b+DeA⋅ab+ba⋅DeA.
Replacingb by a in (20) gives(21)2Da2=2Da·a−DeA⋅a2−a2⋅DeA,for all a∈A. Here, we claim that a⋅DeA=DeA⋅a for all a∈A. Let p be an idempotent in A. Substituting a by p in (21), we have(22)2Dp=2p⋅Dp+2Dp⋅p−DeA⋅p−p⋅DeA.
We multiply (22) by p on the left, and we obtain(23)2p⋅Dp=2p⋅Dp+2p⋅Dp⋅p−p⋅DeA⋅p−p⋅DeA,and so(24)2p⋅Dp⋅p−p⋅DeA⋅p−p⋅DeA=0.
Similarly, by multiplying both sides of (22) by p, we arrive at(25)2p⋅Dp⋅p−DeA⋅p−p⋅DeA⋅p=0.
From (24) and (25), it follows that p⋅DeA=DeA⋅p for all idempotent p∈A. By Lemma 1.7.5 and Proposition 1.3.1 of [2], every self-adjoint element x∈A is the limit of a sequence of linear combinations of projections in A. Therefore, x⋅DeA=DeA⋅x for each self-adjoint element x in A. Now, each arbitrary element a∈A can be written as a=x+iy where x,y are self-adjoint elements of A. Hence,(26)a⋅DeA=x+iy⋅DeA=DeA⋅x+iy=DeA⋅a,for all a∈A. Thus, equality (21) implies that(27)Da2=Da·a−a⋅DeA⋅a,for all a∈A. Consequently, D is a generalized Jordan derivation.
It is well-known that there are two products onA∗∗, the second dual space of a Banach algebra A, called the first and second Arens products which make A∗∗ into a Banach algebra ([13], Definition 2.6.16). If these products coincide on A∗∗, then A is said to be Arens regular. It is shown in Chapter 2 of [13] that every C∗-algebra A is Arens regular. Moreover, the second dual of each C∗-algebra is a W∗-algebra.
According to [14], for each Banach A-bimodule X, X∗∗ turns into a Banach A∗∗-bimodule where A∗∗ equipped with the first Arens product. The module actions are defined by(28)Φ⋅u=w∗−limilimjai⋅xj,u⋅Φ=w∗−limjlimixj⋅ai,Φ∈A∗∗,u∈X∗∗,where aii∈I and xij∈I are nested in A and X that converge, in w∗-topologies, to Φ and u, respectively. One may refer to the monograph of Dales [13] for a full account of Arens product and w∗-continuity of the above structures.
There exists another related concept of generalized derivation, which appeared in [15] for the first time. Let A be a Banach algebra and X be an A-bimodule. A linear operator D:A⟶X is said to be a generalized derivation if there exists ξ∈X∗∗ such that(29)Dab=Da⋅b+a⋅Db−a⋅ξ⋅b,a,b∈A.
It should be noted that ifA is unital and D is a generalized derivation, then by [3] (Proposition 4.2) equality (29) converts to(30)Dab=Da⋅b+a⋅Db−a⋅DeA⋅b,for all a,b∈A, and hence, D is a generalized derivation in the usual sense. Motivated by (29), we introduce the concept of generalized Jordan derivation as follows. A linear operator D:A⟶X is said to be a generalized Jordan derivation if there exists ξ∈X∗∗ such that(31)Da∘b=Da·b+a·Db−a⋅ξ⋅b−b⋅ξ⋅a,for all a,b∈A. Similarly, if A is unital and D satisfies in (31) for some ξ∈X∗∗, then(32)Dab=Da·b+a·Db−a⋅DeA⋅b−b⋅DeA⋅a,a,b∈A.
In what follows, we prove Theorem4 to the C∗-algebra case. First, note that the linear span of projections is dense in a unital C∗-algebra of real rank zero [16]; hence, the conclusion of Theorem 4 is also valid for such C∗-algebras.
Applying the techniques in the proof of Theorem 4.1 from [6], we have the upcoming result.Theorem 5.
LetA be a unital C∗-algebra and D:A⟶X be a continuous linear map satisfying J1. Then, D is a generalized Jordan derivation.Proof.
It follows from Theorem1 that the bilinear map ϕ:A×A⟶X defined through(33)ϕa,b=Da⋅b+a⋅Dbfulfills(34)xa⋅Dyb+Dxa⋅yb+bx⋅Day+Dbx⋅ay−x⋅Dayb−Dx⋅ayb−bax⋅Dy−Dbxa⋅y=0,for all a,b,x,y∈A. Furthermore, the Arens regularity of A, the w∗-w∗-continuity of D∗∗, and the separate weak continuity of the module operations on X∗∗ necessitate that(35)xa⋅D∗∗yb+D∗∗xa⋅yb+bx⋅D∗∗ay+D∗∗bx⋅ay−x⋅D∗∗ayb−D∗∗x⋅ayb−bax⋅D∗∗y−D∗∗bxa⋅y=0,for all a,b,x,y∈A∗∗. Take ξ=D∗∗eA∗∗∈X∗∗. Then, it follows from equality (35) by putting x=y=eA∗∗ that(36)a·D∗∗b+D∗∗a·b=D∗∗a∘b+ξ⋅ab+ba⋅ξ,for all a,b∈A∗∗. In particular, we have(37)Da∘b=a·Db+Da·b−ξ⋅ab−ba⋅ξ,a,b∈A.
Similar to the proof of Theorem4, we have a⋅ξ=ξ⋅a for all a∈A∗∗. Hence, by (37),(38)Da∘b=a·Db+Da·b−a⋅ξ⋅b−b⋅ξ,a,b∈A.
It remains to show thatξ⋅a∈X for all a∈A. In other words, it suffices to prove it for each positive element a∈A. Suppose that a∈A be a positive element and b∈A with a=b2. According to (38),(39)ξ⋅a=ξ⋅b2=Db·b−Da∈X.
Consequently,D is a generalized Jordan derivation.Corollary 2.
LetA be a unital C∗-algebra. If D:A⟶X is a continuous linear map satisfying J1, then D is a derivation if and only if DeA=0.Proof.
It is obvious that ifD is a derivation, then DeA=0. On the other hand, it follows from Theorem 5 that D is a Jordan derivation when DeA=0, and therefore, D is a derivation by Johnson’s result.
Here, we mention that each conditionG1 and the implication(40)a∘b=0⇒Dab+aDb=0,a,b∈AD1,imply J1, and hence, Theorems 4, 5 and Corollary 2 still work with condition J1 replaced by G1 or D1.Theorem 6.
LetA be a W∗-algebra and D:A⟶X be a continuous linear map satisfying J2. Then, D is a generalized Jordan derivation.Proof.
SincepeA−p=eA−pp=0 for all idempotent p∈A, condition J2 implies that(41)2Dp−2p⋅Dp−2Dp⋅p+p⋅DeA+DeA⋅p=0.
Multiplying both sides of (41) by p on the left, we obtain(42)2p⋅Dp−2p⋅Dp−2p⋅Dp⋅p+p⋅DeA⋅p+p⋅DeA⋅p=0,and so(43)2p⋅Dp⋅p−p⋅DeA⋅p−p⋅DeA=0.
Similarly, by multiplying (41) on the right by p, we arrive at(44)2p⋅Dp⋅p−DeA⋅p−p⋅DeA⋅p=0.
From (43) and (44), we get p⋅DeA=DeA⋅p for all idempotent p∈A. Now, similar to the proof of Theorem 4, we conclude that(45)a⋅DeA=DeA⋅a,a∈A.
Applying Theorem1 to the bilinear map ϕ:A×A⟶X defined by(46)ϕa,b=Da·b+a·Db,a,b∈A,we find(47)xa·Dyb+Dxa·yb+bx·Day+Dbx·ay=x·Dayb+Dx·ayb+bax·Dy+Dbxa·y,for all a,b,x,y∈A. Switching x,y by eA and using (45), we reach(48)Da·b+a·Db=Dab+ab⋅DeA+Dba+ba⋅DeA.
Replacingb by a in (48), it concludes that(49)Da2=Da·a−a⋅DeA⋅a,for all a∈A. Thus, D is a generalized Jordan derivation.
Similar to the proof of Theorem5, we can obtain the following corollary.Corollary 3.
LetA be a unital C∗-algebra and D:A⟶X be a continuous linear map satisfying J2. Then, D is a generalized Jordan derivation.It should be pointed out that each of the conditions(50)ab=0⇒Da·b+a·Db=0,a,b∈AD2,a∘b=0⇒Da·b+a·Db=0,a,b∈AD3,implies J2, and therefore, Theorem 6 and Corollary 3 are true if the condition J2 replaced by either D2 or D3.Theorem 7.
LetA be a W∗-algebra and D:A⟶X be a continuous linear map such that the condition J3 holds. Then, Da=a⋅DeA for all a∈A. In particular, D is a generalized derivation.Proof.
It is obvious that for every idempotentp∈A, peA−p=eA−pp=0. Hence,(51)p⋅DeA−p+eA−p⋅Dp=0.
A simple calculation shows thatDp=p⋅Dp. Let Asa denote the set of self-adjoint elements of A and x∈Asa. Then, x=∑k=1nλkpk, where λk are real numbers and pk is an orthogonal family of projections in A. Since pipj=pjpi=0 for i≠j, condition J3 implies that pi⋅Dpj+pj⋅Dpi=0 for all i,j with i≠j. Thus, for each x∈Asa,(52)Dx2=D∑k=1nλk2pk2=∑k=1nλk2Dpk2=∑k=1nλkpk⋅∑k=1nλkDpk=x⋅Dx.
It follows from the linearity ofD that Dxy+yx=x⋅Dy+y⋅Dx for each x,y∈Asa. Now, each arbitrary element a∈A can be written as a=x+iy for x,y∈Asa. Therefore,(53)Da2=Dx2−y2+ixy+yx=Dx2−Dy2+iDxy+yx=x⋅Dx−y⋅Dy+ix⋅Dy+y⋅Dx=a⋅Da.
Thus,Da2=a⋅Da for all a∈A. Replacing a by a+eA, we obtain Da=aDeA for all a∈A.
The next result is a direct consequence of Theorem7. We include it without proof.Corollary 4.
LetA be a unital C∗-algebra. If D:A⟶X is a continuous linear map satisfying J3, then Da=a⋅DeA for all a∈A. Moreover, if DeA=0, then D≡0.One should remember that Theorem1 has an important role in characterizing generalized Jordan derivations on C∗-algebras. The following result is another analogous criterion for Banach algebra with the property that A=algℐA¯.Theorem 8.
([7], Corollary 3.6). Let A be a unital Banach algebra, X be a Banach space, and ϕ:A×A⟶X be a continuous bilinear mapping such that(54)ab=ba=0⇒ϕa,b=0,a,b∈A.Then,(55)ϕa,x+ϕx,a=ϕax,eA+ϕeA,xa,for all a∈A and x∈ℐA. In particular, if A is generated by idempotents, then(56)ϕa,b+ϕb,a=ϕab,eA+ϕeA,ba,a,b∈A.Since all Banach algebras which are generated by idempotents have the propertyB, 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 A=algℐA¯.



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