# Biderivations and Commuting Linear Maps on Topologically Simpleℒ∗-Algebras

Authors: Abdelaziz Ben Yahya
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

Letℒ be a topologically simple ℒ∗-algebra of arbitrary dimension. In this paper, we introduce the notion of semi-inner biderivation in order to prove that every continuous commuting linear mapping on ℒ is a scalar multiple of the identity mapping.


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## Body

## 1. Introduction

Letℒ,.,. be a Lie algebra over a field F of a characteristic different from two. A linear map D:ℒ⟶ℒ is called derivation on ℒ,.,. if it satisfies the following identity:(1)Dx,y=Dx,y+x,Dy,for all x,y∈ℒ.A bilinear mapδ:ℒ×ℒ⟶ℒ is called biderivation on ℒ,.,. if it satisfies the following identities:(2)δx,y,z=x,δy,z+δx,z,y,δx,y,z=δx,y,z+y,δx,z.For allx,y,z∈ℒ, which means that it is a derivation with respect to both components. In addition, if δx,y=−δy,x, δ will be called skew-symmetric biderivation. Let λ∈F and f be the bilinear map f:ℒ×ℒ⟶ℒ sending x,y to λx,y; it is straightforward to prove that f is a biderivation of ℒ,.,.; and the biderivations of this type are called inner biderivations of ℒ,.,.. A linear map ϕ:ℒ⟶ℒ is called a commuting linear map on ℒ,.,. if it satisfies the following identity: ϕx,x=0, for all x∈ℒ. It is easy to show that(3)ϕx,y=x,ϕy,∀x,y∈ℒ,which implies that the bilinear map δ defined by(4)δx,y=ϕx,y=x,ϕy,is a skew-symmetric biderivation on ℒ,.,..Commuting maps and biderivations arose first in the associative ring theory [1, 2]. Since then, many authors have made considerable efforts to make their study very successful (see, for example, [3–10]). The way used in [8] requires the finiteness of the dimension of the simple Lie algebra. However, the purpose of this paper is to extend the results given in [8] concerning the commuting linear maps to topologically simple ℒ∗-algebras, which are of arbitrary dimension. To overcome the problem of the nonfiniteness of the dimension, we use some techniques related to these algebras. The ℒ∗-algebras are introduced by Schue in [11]. We recall that an ℒ∗-algebra over ℂ (the complex field) is a Lie algebra ℒ, which is also a complex Hilbert space with the inner product .,. endowed with a (conjugate-linear) algebra involution ∗ such that x,y,z=y,x∗,z, for all x,y,z in ℒ.Letℒ be an ℒ∗-algebra; for subsets M and N of ℒ, we recall that M,N denotes the closed subspace spanned by m,n:m∈M,n∈N. ℒ is said to be semisimple as an ℒ∗-algebra if and only if ℒ=ℒ,ℒ. From [11], a finite-dimensional Lie algebra ℒ is semisimple as an ℒ∗-algebra if and only if it is semisimple in the usual sense. The ℒ∗-algebra ℒ will be called topologically simple if and only if there are no nontrivial closed ideals. In [11], the author shows that the ℒ∗-algebras are reductive, the semisimple ones are the Hilbert space direct sum of its closed topologically simple ideals, and the author also gives the classification of the topologically simple ℒ∗-algebras in the separable case. The classification of the topologically simple ℒ∗-algebras in the arbitrary dimensional case can be found in [12]. In [13], Schue shows that every semisimple ℒ∗-algebra has a Cartan decomposition relative to a Cartan subalgebra. We recall that a Cartan subalgebra of a semisimple ℒ∗-algebra is defined as a maximal self-adjoint abelian subalgebra.The paper is organized as follows. In the second section, we give some definitions and basic results related toℒ∗-algebras. In Section 3, we introduce the notion of semi-inner biderivation in order to show that every semi-inner biderivation on an arbitrary dimensional topologically simple ℒ∗-algebra is inner; using this result, we determine all continuous commuting linear maps on an arbitrary dimensional topologically simple ℒ∗-algebras.


## 2. Preliminaries

In this section, we summarize some basic results related toℒ∗-algebras, collected from [14, 15]. First of all, we point out that the notation concerning Lie algebras follows principally from [8, 14]. Let ℂ be the complex numbers field, ℒ a semisimple ℒ∗-algebra, H a fixed Cartan subalgebra of ℒ, and − denotes the conjugation operator on ℂ, a root of ℒ relative to H is a linear form commuting with the involution:(5)α:H,∗⟶ℂ,−.That is,αh∗=αh¯ for any h∈H, such that there exists vα∈ℒ, vα≠0 satisfying h,vα=αhvα for any h∈H. The subspace(6)Vα=vα∈ℒ:h,vα=αhvαfor allh∈H,is called the root space associated to α; it follows from this that if α is a root, then −α is also one and Vα∗=V−α. The root space associated to the zero root is equal to the Cartan subalgebra H, using the Jacobi identity; one proves that if α+β is a root, then ℒα,ℒβ⊆ℒα+β, and if α+β is not a root, then ℒα,ℒβ=0. Let Φ denote the set of nonzero roots of ℒ relative to H, then we have the following Cartan decomposition ℒ=H⊕∑α∈Φ⊕Vα, where ⊕ is the usual Hilbert space direct sum.Letα be a root of ℒ relative to H, then α is a linear functional on H; this implies that there exists a unique vector hα∈H such that αh=h,hα where .,. denotes the inner product of ℒ. Consequently, hα is self-adjoint, which means that hα∗=hα and hα=vα,vα∗ for any vα∈Vα with vα=1. Then, we have the following result.Lemma 1 (see [15]).
The sethα:α∈Φ is total in H, i.e., for any h∈H, h,hα=0 for all hα, implies h=0.Letℋ be a Hilbert space, the orthogonal dimension of ℋ is denoted by dimℋ, i.e., the cardinality of an orthonormal basis for ℋ. We will denote the cardinality of an arbitrary set E by E.Now, we will define the root system relative to a Cartan subalgebra of the semisimpleℒ∗-algebra ℒ.Definition 1.
Letℒ be a semisimple ℒ∗-algebra, H a Cartan subalgebra of ℒ, and Φ the set of nonzero roots of ℒ relative to H. A subset Φ0 of Φ will be called a root system of Φ if the following conditions are satisfied:(i)
Ifα∈Φ0, then −α∈Φ0(ii)
Ifα,β∈Φ0, such that α+β∈Φ, then α+β∈Φ0
We need some further notations;ℝ and ℚ will refer to the real and the rational fields, respectively. For a subset S of Φ, the set of all ℂ-linear combinations of elements of S will be denoted by SpℂS=SpS and the set of all ℚ-linear combinations of elements of S by SpℚS. If we write Sp^S=SpS∩Φ, then Sp^S is obviously a root system. The following results will be useful in our main proofs.Lemma 2.
Letℒ be a topologically simple ℒ∗-algebra and Φ the set of its nonzero roots relative to some Cartan subalgebra H. For any subset S of Φ, there exists a topologically simple ℒ∗-subalgebra ℒS of ℒ, with Cartan subalgebra HS, such that(i)
S⊆ΦS, where ΦS is the set of roots of ℒS (relative to HS) and ΦS=Sp^S=SpℚS∩Φ(ii)
ℒS is finite-dimensional if S is finite(iii)
ℒS is infinite-dimensional and dimℒS=S if S is infiniteProof.
See Proposition 3 in [14].Definition 2.
Letℒ be a topologically simple ℒ∗-algebra and Φ the set of nonzero roots relative to a Cartan subalgebra H. Let α,β∈Φ; we say that α is connected to β if there are some γ1,γ2,…,γk∈Φ such that α+γ1,γ1+γ2,…,γk+β∈Φ∪0.
Obviously, the connected relation is an equivalence relation onΦ.Lemma 3.
Any two roots of a topologically simpleℒ∗-algebra are connected.Proof.
LetS=β1,βl⊂Φ, then by Lemma 2 there exists a finite-dimensional simple ℒ∗-algebra ℒS of ℒ, with Cartan subalgebra HS, such that β1andβl are roots of ℒS. Using Lemma 1.3 in [8], we obtain that β1 and βl are connected in ℒS. Since ΦS⊂Φ, then β1 and βl are connected in ℒ.


## 3. Semi-Inner Biderivations onℒ∗-Algebras and Commuting Linear Maps on Topologically Simple L∗-Algebras

In this section, we introduce the notion of semi-inner biderivation in order to show that every continuous commuting linear map on a topologically simpleℒ∗-algebra ℒ is a scalar multiple of the identity mapping. The aim of the first main theorem of this section is to prove that every semi-inner biderivation of a topologically simple ℒ∗-algebra is inner. To get this result, we have to show two lemmas.Definition 3.
Letℒ be an ℒ∗-algebra and f:ℒ×ℒ⟶ℒ a biderivation of ℒ; f is said to be a semi-inner biderivation of ℒ if there exists two continuous linear maps ϕ:ℒ⟶ℒ and ψ:ℒ⟶ℒ such that(7)fx,y=ϕx,y=x,ψy,∀x,y∈ℒ,where f will be denoted by fϕ,ψ.Remark 1.
By using Lemma 2.1 in [8], any biderivation of a finite-dimensional simple complex Lie algebra is semi-inner (a linear map between two finite-dimensional vector spaces is continuous).
From now on,ℒ will represent an infinite-dimensional topologically simple ℒ∗-algebra of arbitrary dimension and ℒ=H⊕∑α∈Φ⊕Vα where its Cartan decomposition is relative to a Cartan subalgebra H (Φ is the set of nonzero roots relative to H).Lemma 4.
Letfϕ,ψ:ℒ×ℒ⟶ℒ be a semi-inner biderivation of ℒ, then for any h∈H, we have ϕh,ψh∈H.Proof.
For anyα∈Φ+, we select xα∈Vα, such that xα=1, xα∗=x−α∈V−α, and h∈H. Then, we have(8)xα,x−α=hα,hα,xα=αhαxα,hα,x−α=−αhαx−α.
Letα∈Φ, we denote by Φ˜α the set Φ\α,−α. Let(9)ϕhα=a1h1+a2xα+a3x−α+∑β∈Φ˜αkβxβ,(10)ϕxα=b1h2+b2xα+b3x−α+∑β∈Φ˜αtβxβ,(11)ϕx−α=c1h3+c2xα+c3x−α+∑β∈Φ˜αtβxβ,(12)ψhα=s1h4+s2xα+s3x−α+∑β∈Φ˜αmβxβ,(13)ψxα=p1h5+p2xα+p3x−α+∑β∈Φ˜αnβxβ,(14)ψx−α=q1h6+q2xα+q3x−α+∑β∈Φ˜αrβxβ.
For someai,bi,ci,si,pi,qi,kβ,tβ,lβ,mβ,nβ,rβ∈ℂ, the sums are orthogonal, i=1,2,3,β∈Φ˜α, and hj∈H, j=1,2,…,6.
By equations (10) and (14), we have(15)fhα,xα=ϕhα,xα=a1αh1xα−a3hα+∑β∈Φ˜αkβxβ,xα,fhα,xα=hα,ψxα=αhαp2xα−αhαp3x−α+∑β∈Φ˜αmββhαxβ.
If we compare the two equations above andxβ,xα∈Lα+β≠H since β∈Φ˜α, we obtain a3=0. In the same way, by considering fhα,x−α with equations (10) and (14), a2=0. Similarly, considering the images fxα,hα and fx−α,hα, we obtain s2=s3=0, by equations (11)–(13).
If we puta1h1=h^α∈η and s1h4=h⌣α∈η, then equations (9) and (12) can be written as follows:(16)ϕhα=h^α+∑β∈Φ˜αkβxβ,(17)ψhα=h⌣α+∑β∈Φ˜αmβxβ.
Now, for any rootγ∈Φ˜α,the setΦ\α,−α,γ,−γ is denoted by Φ˜α,γ. By equations (16) and (17), we can write(18)ϕhα=h^α+kγxγ+k−γx−γ+∑β∈Φ˜α,γkβxβ,ψhγ=h⌣γ+μαxα+μ−αx−α+∑β∈Φ˜α,γμβxβ.
The two equations above imply that(19)fhα,hγ=ϕhα,hγ=−γhγkγxγ+γhγk−γx−γ−∑β∈Φ˜α,γkββhγxβ,fhα,hγ=hα,ψhγ=αhαμαxα−αhαμ−αx−α+∑β∈Φ˜α,γμββhαxβ.
By comparing, one haskγ=k−γ=μα=μ−α=0. Due to the arbitrariness of γ, we obtain φhα=h^α∈H and ψhα=h⌣α∈H. By Lemma 1, the set hα,α∈Φ is total in H and ϕ and ψ are continuous, and we have ϕH⊂H and ψH⊂H.Lemma 5.
Letfϕ,ψ:ℒ×ℒ⟶ℒ be a semi-inner biderivation of ℒ, then there is a complex number λ such that(20)ϕx=ψx=λx,∀x∈ℒα,α∈Φ.Proof.
Leth∈H and α,γ∈Φ such that α≠γ, then we have(21)fxα,h=ϕxα,h=−h,ϕxα,(22)fxα,h=xα,ψh=−ψh,xα=−αψhxα.
Let(23)ϕxα=h^α+∑β∈Φtβxβ,where tβ∈ℂ and h^α∈H. Using equations (21) and (23), we obtain(24)fxα,h=−∑β∈Φtββhxβ.
Combining equations (22) and (24), we obtain tααh=αψh and tββh=0 for every β∈Φ\β.
For anyβ∈Φ\α, if we replace h by hβ in equation (24), we get tβ=0; this implies that(25)ϕxα=tαxα+h^α,αhtα=αψh.
The image offh,xα is computed. Similarly, we get(26)ψxα=kαxα+h⌣α,αhkα=αϕh,where kβ∈ℂ and h⌣α∈H.
For anyα,β∈Φ, using equations (25) and (26), we have(27)fxα,xβ=ϕxα,xβ=tαxα,xβ+βh^αxβ,(28)fxα,xβ=xα,ψxβ=kβxα,xβ−αh⌣βxα.
By combining equations (27) with (28), we first see that βh^α=αh⌣β=0 if α≠β. However, −αh^α=αh⌣−α=0 by taking β=−α. This means that βh^α=0 for all β∈Φ, i.e., h^α∈∩β∈Φ ker β, which gives h^α=0. Similarly, by taking α=−β, we have h⌣β=0. Therefore, by equations (25) and (26), one can obtain(29)ϕxα=tαxα,ψxα=kαxα.
Using equation (29) and fxα,x−α=ϕxα,x−α=xα,ψx−α, it follows that(30)tα=k−α,∀α∈Φ.
Then, comparing equations (27) with (28), we obtain(31)tα=kβ,forα+β∈Φ.
LetS=α⊆Φ, then by Lemma 2 there exists a finite-dimensional simple ℒ∗-algebra ℒS of ℒ, with Cartan subalgebra HS, such that α is a root of ℒS. Now, let us prove that fϕ,ψℒS×ℒS⊆ℒS indeed; let x,y∈ℒS, then x=hx+∑β∈ΦSxβandy=hy+∑γ∈ΦSyγ.(32)fϕ,ψx,y=ϕx,y=∑γ∈ΦSϕhx,yγ+∑β∈ΦStβxβ,hy+∑β∈ΦS∑γ∈ΦStβxβ,yγ.
Then,fϕ,ψ╱ℒS×ℒS is a biderivation of ℒS; by Theorem 2.4 in [8], there exists μ∈ℂ such that fϕ,ψx,y=μx,y for any x,y∈ℒS. Then, μxβ,y=tβxβ,y=−kβy,xβ for any β∈ΦS and y∈ℒS; this implies that(33)tα=t−α=kα=k−α=μ,for arbitraryα∈Φ.
Equations (30)–(33) imply that if α,β∈Φ such that α+β∈Φ∪0,thentα=kβ=tβ=kα. Additionally, for arbitrary connected roots α,β∈Φ,tα=kβ=tβ=kα, from Lemma 3, we conclude that(34)tα=tα′=kβ=kβ′,∀α,α′,β,β′∈Φ.
If we posetα=λ in equation (34), we get our result.Remark 2.
In the above proof, there is another method to show thattγ=0 if γ≠α: Indeed, using equation (23), we have(35)fxα,hγ,xγ=−hγ,ϕxα,xγ=−ϕxα,hγ∗,xγ=h^α+∑β∈Φtβxβ,γhγ∗xγ=tγγhγ∗xγ,xγ.
We also havefxα,hγ,xγ=−αψhγxα,xγ=0. Then, tγ=0 if γ≠α.Remark 3.
In the case whereℒ is separable, Theorem 2 in [16] will facilitate some difficulties encountered in the above proof (the above proof will look like the proof of Lemma 2.2 in [8]).
Thanks to the above Lemmas, we can state our first main theorem.Theorem 1.
Letℒ be a topologically simple ℒ∗-algebra. Then, f is a semi-inner biderivation of ℒ if and only if it is inner, i.e., there is a complex number λ such that fx,y=λx,y.Proof.
Letℒ be a topologically simple ℒ∗-algebra, any inner biderivation f of ℒ such that fx,y=λx,y is a semi-inner biderivation fϕ,ψ with(36)ϕ=ψ=λidℒ.
Now, let us prove the “only if direction.”
The first case: ifℒ is finite-dimensional, the result follows from Theorem 2.4 in [8]. The second case: if ℒ is infinite-dimensional, then, for fϕ,ψ a semi-inner biderivation of ℒ, by using Lemma 5, there is λ∈ℂ such that ϕx=ψx=λx,∀x∈Lα,α∈Φ. For any h∈H and α∈Φ, we have fh,xα=ϕh,xα=h,ψxα. This implies that αϕhxα=λαhxα which means that(37)αλh−ϕh=0,∀α∈Φ,this implies ϕh=λh for all h∈H. For any x,y∈L, with x=h+∑α∈Φlαxα, where lα∈ℂ, with lα=0 except for a countable number. Therefore,(38)fx,y=ϕx,y=∑α∈Φlαϕxα+ϕh,y=∑α∈Φlαλxα+λh,y=λx,y.
In recent years, many authors have studied commuting linear maps of certain algebra structures; for some of these achievements, refer to [3, 4, 6, 8, 9, 17, 18]. It should be noted that this subject is not new since it was studied in 1957, exactly in Posner’s works [19]. As we said in the introduction, if ϕ is a commuting linear map on ℒ, then ϕx,y=x,ϕy for any x,y∈ℒ, and fx,y=ϕx,y=x,ϕy is a biderivation of ℒ. Using Theorem 1, the present theorem aims to determine all continuous commuting linear maps on topologically simple L∗-algebras.Theorem 2.
Letℒ be a topologically simple ℒ∗-algebra. Then, every continuous linear map ϕ on ℒ is a commuting linear map if and only if it is a scalar multiplication map on ℒ.Proof.
Letϕ be a continuous commuting linear map on a topologically simple ℒ∗-algebra ℒ. Then, f defined by fx,y=ϕx,y=x,ϕy is a semi-inner biderivation of ℒ. By Theorem 1, we have fx,y=ϕx,y=λx,y for some λ∈ℂ. Since ℒ is topologically simple ℒ∗-algebra and y is arbitrary, therefore, we have ϕx=λx.
The following Lemma is one of the interesting results given in (see p.7 in [3]).Lemma 6.
Letℒ be a simple Lie algebra over an algebraically closed field F of characteristic different from 2 such that cardF>dimℒ (cardF is the cardinality of F and dimℒ is the dimension of ℒ). For any skew-symmetric biderivation f of ℒ, there exists λ∈F such that(39)fx,y=λx,y,for allx,y∈ℒ.Remark 4.
Letℒ be a simple complex Lie algebra of countable dimension and ϕ a commuting linear map on ℒ. Then, ϕ is of the form ϕx=λxfor allx∈ℒ where λ∈ℂ. Indeed, let ϕ be a commuting linear map on ℒ, then fx,y=ϕx,y=x,ϕy for all x,y∈ℒ which is a skew-symmetric biderivation of ℒ. By Lemma 6, there exists λ∈ℂ, such that f is of the form fx,y=λx,y for all x,y∈ℒ. Then, ϕ is of the form ϕx=λxfor allx∈ℒ.
We mention here that this proof does not seem to work in our case when the underlying Hilbert space of theℒ∗-algebra is infinite-dimensional. However, Corollary 2.4 in [3] may simplify some of the proofs in our paper.


## 4. Conclusions

This paper aimed to show that every continuous commuting linear map on a topologically simpleℒ∗-algebra ℒ is a scalar multiplication map on ℒ.



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