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Roh and Chang Journal of Inequalities and Applications (2017) 2017:36
DOI 10.1186/s13660-017-1308-0
R E S E A R C H Open Access
Asymptotic aspect of derivations in
Banach algebras
Jaiok Roh1and Ick-Soon Chang2*
*Correspondence:
ischang@cnu.ac.kr
2Department of Mathematics,
Chungnam National University, 99
Daehangno, Yuseong-gu, Daejeon,
34134, Korea
Full list of author information is
available at the end of the article
Abstract
We prove that every approximate linear left derivation on a semisimple Banach
algebra is continuous. Also, we consider linear derivations on Banach algebras and we
first study the conditions for a linear derivation on a Banach algebra. Then we examine
the functional inequalities related to a linear derivation and their stability. We finally
take central linear derivations with radical ranges on semiprime Banach algebras and
a continuous linear generalized left derivation on a semisimple Banach algebra.
MSC: 16N20; 16N60; 39B72; 39B82; 46H40; 47H99
Keywords: derivation; inequality; Banach algebra; stability; radical range
1 Introduction and preliminaries
Let Abe an algebra. A linear mapping δ:A→Ais called a left derivation (resp., deriva-
tion) if δ(xy)=xδ(y)+yδ(x)(resp.,δ(xy)=xδ(y)+δ(x)y) is fulfilled for all x,y∈A.Alinear
mapping δ:A→Ais said to be a left Jordan derivation if δ(x)=xδ(x)holdsforallx∈A.
A linear mapping δ:A→Ais called a generalized left derivation if there exists a linear
left derivation δ:A→Asuch that δ(xy)=xδ(y)+yδ(x) for all x,y∈A.Alinearmap-
ping δ:A→Ais said to be a generalized left Jordan derivation if there exists a linear left
Jordan derivation δ:A→Asuch that δ(x)=xδ(x)+xδ(x) for all x∈A.
Singer and Wermer [] obtained a fundamental result which started the investigation
of the ranges of linear derivations on Banach algebras. The result, which is called the
Singer-Wermer theorem, states that every continuous linear derivation on a commuta-
tive Banach algebra maps into the radical. In the same paper, they made a very insightful
conjecture: that the assumption of continuity is unnecessary. Thomas []provedthiscon-
jecture. Hence linear derivations on Banach algebras (if everywhere defined) genuinely
belong to the noncommutative setting.
On the other hand, the study of stability problems had been formulated by Ulam [].
Hyers [] had answered affirmatively the question of Ulam for Banach spaces. Hyers’ the-
orem was generalized by Aoki [] for additive mappings and by Rassias [] for linear map-
pings by considering an unbounded difference. In particular, the stability result concern-
ing derivations between operator algebras was first obtained by Šemrl []. Badora gave a
generalization of the Bourgin result and he also dealt with the stability and the supersta-
bility of Bourgin-type for derivations; see [–] and the references therein. Recently, the
stability problems for derivations are considered by some authors in [–].
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Inthiswork, wefirst take intoaccountthefunctional inequalitywhich expandsthefunc-
tional inequality in []. It is well known that every ring left derivation (resp., ring left Jor-
dan derivation)on asemiprimering maps intoits center;see[, ]. Consideringthebase
of the previous result, we show that every approximate ring left derivation on a semiprime
normed algebra maps into its center and then, by using this fact, we prove that every ap-
proximate linear left derivation on a semisimple Banach algebra is continuous. We also
establish the functional inequalities related to a linear derivation and their stability. In
particular, mappings satisfying such functional inequalities on a semiprime Banach alge-
bra are linear derivations which map into the intersection of the center and the radical.
We finally investigate a linear generalized left Jordan derivation on a semisimple Banach
algebra with application.
2 Approximate left derivations
We first demonstrate the following proposition quoted in this work.
Proposition . ([], Proposition .) Let Rbe a ring,Xbe a left R-module,and δ:
R→Xbe a left derivation.
(i) Suppose that aRx=with a∈R,x∈Ximplies a=or x=.If δ=,then Ris
commutative.
(ii) Suppose that X=Ris a semiprime ring.Then δis a derivation which maps Rinto
its center.
Let Abe a normed algebra. An additive mapping δ:A→Ais said to be an approximate
ring derivation (resp., approximate ring left derivation) if for some ε≥,
δ(xy)–xδ(y)–δ(x)y
≤εresp.,
δ(xy)–xδ(y)–yδ(x)
≤ε
for all x,y∈A. In addition, if δ(λx)=λδ(x) for all x∈Aand λ∈C,thenδis called an
approximate linear derivation (resp., approximate linear left derivation).
From now on, we suppose that Tε:= {eiθ:≤θ≤ε}.Thecommutatorxy –yx will be
denoted by [x,y]. We start our investigations for approximate ring left derivations with
some results.
Theorem . Let Abe a semiprime normed algebra.Assume that l ≥is a fixed integer
and s,s,...,slare fixed positive real numbers,where sj>(j=,)and s=.Suppose
that δ:A→Ais a mapping such that
l
j= sjδ(xj)
≤
δl
j= sjxj
(.)
for all x,x,...,xl∈Aand for some ε≥,
δ(xy)–xδ(y)–yδ(x)
≤ε(.)
for all x,y∈A.Then δis an approximate ring derivation which maps Ainto its center
Z(A).
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Proof By letting x=x=···=xl=in(.), we get δ() = . And we put x=x=···=
xl=in(.)andthensetx=x,x=y,x=z,s=s,s=tto obtain
sδ(x)+tδ(y)+δ(z)
≤
δ(sx+ty +z)
for all x,y,z∈A.(.)
It follows by the result of []thatδis additive. In particular, in view of (.), we find that
δis an approximate ring left derivation.
By virtue of (.), we see that
δ(yx)–yδ(x)–xδ(y)
≤ε(.)
for all x,y∈A. Combining (.)and(.), we get
δ(xy)–δ(yx)
≤
δ(xy)–xδ(y)–yδ(x)
+
δ(yx)–yδ(x)–xδ(y)
≤ε(.)
for all x,y∈A. It follows from (.)and(.)that
[x,y]δ(x)
≤
δ(x·yx)–xδ(yx)–yxδ(x)
+
δ(xy ·x)–xyδ(x)–xδ(xy)
+x
δ(xy)–δ(yx)
≤εx+
(.)
for all x,y∈A.Replacingxby nx in (.) and then dividing on both sides by n,wehave
[x,y]δ(x)
≤x
n+
nε
for all x,y∈Aand all positive integer n.Takingthelimitasn→∞in the above relation,
we see that
[x,y]δ(x) = for all x,y∈A.(.)
Just proceeding as in the proof of Proposition .,weget[δ(w),x] = for all x,w∈A.That
is, δ(w)belongtoitscenterZ(A). So δis an approximate ring derivation. Therefore we
arrive at the desired conclusion.
Theorem . Let Abe a noncommutative prime normed algebra.Assume that l ≥is
a fixed integer and s,s,...,slare fixed positive real numbers,where sj>(j=,)and
s=.Suppose that δ:A→Ais a mapping subject to the conditions (.)and (.). Then
δis identically zero.
Proof Employing the same argument as the proof Theorem .,wefeelthatδsatisfies
equation (.). Since Ais noncommutative, choose a zthat does not belong to the center
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of A. Using the same method in the proof of Proposition .,weseethatδ=,which
completes the proof.
Theorem . Let Abe a semisimple Banach algebra.Assume that l ≥is a fixed inte-
ger and s,s,...,slare fixed positive real numbers,where s=λs(s>),s>and s=.
Suppose that δ:A→Ais a mapping subject to
l
j= sjδ(xj)
≤
δl
j= sjxj
(.)
for all x,x,...,xl∈Aand all λ∈Tε,where x=λz(z∈A)and the inequality (.). Then
δis a continuous.
Proof As we did in the proof of Theorem .,wegetδ() = . We take x=x=···=xl=
in (.)andthenputx=x,x=y,s=tto have
λsδ(x)+tδ(y)+δ(λz)
≤
δ(λsx+ty +λz)
(.)
for all x,y,z∈Aand all λ∈Tε. Now we consider λ=in(.)andsoδsatisfies the in-
equality (.). Hence we find that δis additive [].
Next, setting x=x
s,y=andz=–xin (.), we obtain sδ(x
s)=δ(x). Letting x=x
s,y=,
and z=–xin (.), we get δ(λx)=λδ(x) for all x∈Aand all λ∈Tεand so we see that δis
linear [].
Since semisimple algebras are semiprime [], Theorem . guarantees that δis an ap-
proximate linear derivation. Therefore δis continuous []. The proof is complete.
3 Inequalities related to a linear derivation
In this section, we write a unit element of algebra Aby e.
Theorem . Let Abe a semiprime unital Banach algebra.Suppose that δ:A→Ais a
mapping subject to the inequality (.)and for some ε≥,
δx–xδ(x)
≤ε(.)
for all x ∈A.Then δis a linear derivation which maps Ainto the intersection of its center
Z(A)and its radical rad(A).
Proof Employing the same way in the proof Theorem .,wefindthatδis linear. By lin-
earization of (.) and additivity of δ,weget
δx+δ(xy)+δ(yx)+δy–xδ(x)–xδ(y)–yδ(x)–yδ(y)
≤ε(.)
for all x,y∈A. Substituting –xfor xin (.), we have
δx–δ(xy)–δ(yx)+δy–xδ(x)+xδ(y)+yδ(x)–yδ(y)
≤ε(.)
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for all x,y∈A.Equations(.)and(.)yield
δ(xy)+δ(yx)–xδ(y)–yδ(x)
≤
δx+δ(xy)+δ(yx)+δy–xδ(x)–xδ(y)–yδ(x)–yδ(y)
+
δx–δ(xy)–δ(yx)+δy–xδ(x)+xδ(y)+yδ(x)–yδ(y)
≤ε
for all x,y∈A. We have therefore
δ(xy +yx)–xδ(y)–yδ(x)
≤εfor all x,y∈A.(.)
Putting xy +yx for yin (.), we obtain
δx(xy +yx)+(xy +yx)x–xδ(xy+yx)–(xy +yx)δ(x)
≤ε(.)
for all x,y∈A. On the other hand, we have from (.)andtheequation
x(xy +yx)+(xy +yx)x=xy+yx+xyx
the result
δx(xy +yx)+(xy +yx)x–xδ(y)–yδx–δ(xyx)
=
δxy+yx–xδ(y)–yδx
≤ε(.)
for all x,y∈A.Bycomparing(.)and(.), we arrive at
xδ(xy +yx)+(xy +yx)δ(x)–xδ(y)–yδx–δ(xyx)
≤
δx(xy +yx)+(xy +yx)x–xδ(y)–yδx–δ(xyx)
+
δx(xy +yx)+(xy +yx)x–xδ(xy +yx)–(xy +yx)δ(x)
≤ε(.)
for all x,y∈A. Applying equation (.)with(.)and(.), we have
xδ(y)+xyδ(x)–yxδ(x)–δ(xyx)
≤x
δ(xy +yx)–xδ(y)–yδ(x)
+y
δx–xδ(x)
+
xδ(xy +yx)+(xy +yx)δ(x)–xδ(y)–yδx–δ(xyz)
≤x+y+
ε(.)
for all x,y∈A. Letting x=nx,y=ny in (.) and then dividing the resulting inequality
by n,weget
xδ(y)+xyδ(x)–yxδ(x)–δ(xyx)
≤x
n+y
n+
nε(.)
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for all x,y∈Aand all positive integers n.Takingthelimitn→∞of (.), it is reduced to
the equation
δ(xyx)=xδ(y)+xyδ(x)–yxδ(x) for all x,y∈A.(.)
Putting x=y=z=ein (.), we get δ(e) = . Again, considering y=ein (.), we easily
prove that
δx=xδ(x) for all x∈A.
This means that δis a linear left Jordan derivation.
On the other hand, from Vukman’s result [], we see that δis a linear derivation with
δ(A)⊆Z(A). Since Z(A) is a commutative Banach algebra, the Singer-Wermer theorem
tellsusthatδ|Z(A)maps Z(A)intorad(Z(A)) = Z(A)∩rad(A)andthusδ(A)⊆rad(A).
Using the semiprimeness of rad(A) as well as the identity
δ(x)yδ(x)=δ(xyx)–xδ(yx)–δ(xy)x+xδ(y)x(x,y∈A),
we have δ(A)⊆rad(A). Therefore δ(A)⊆Z(A)∩rad(A), which concludes the proof.
As consequences of Theorem .,wegetthefollowing.
Corollary . Let Abe a unital semisimple Banach algebra.Assume that a mapping δ:
A→Asatisfies the assumptions of Theorem ..Then δis identically zero.
Now we consider the result which is needed in the following theorems.
Lemma. LetAbe aBanach algebra.Suppose that L:A×A→Ais abilinear mapping
and that ξand ηare mappings satisfying L(x,y)=xξ(y)+yη(x)for all x,y∈A.If Ais
semiprime or unital,then ξand ηare linear mappings.
Proof Note that, for all x,y∈Aand all λ∈C,
xξ(λy)+λyη(x)=L(x,λy)=λL(x,y)=λxξ(y)+yη(x).
Hence we see that, for all x,y∈A,
xξ(λy)–λξ(y)=. (.)
If Ais unital, then we see that ξ(λy)=λη(y) by letting x=ein (.).
If Ais nonunital, then ξ(λy)–λξ(y) lies in the right annihilator ran(A)ofA.IfAis
semiprime, then ran(A)=,sothatξ(λy)=λξ (y) for all y∈Aand all λ∈C.
Observe that, for all x,y,z∈A,
xξ(y+z)+(y+z)η(x)=L(x,y+z)=L(x,y)+L(x,z)
=xξ(y)+yη(x)+xξ(z)+zη(x).
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Hence x(ξ(y+z)–ξ(y)–ξ(z)) = for all y,z∈A.Asabove,wegetξ(x+z)=ξ(x)+ξ(z)for
all x,z∈A,sothatξis linear.
Similarly, one can prove that ηis linear.
Theorem . Let Abe a semiprime Banach algebra.Assume that l ≥is a fixed inte-
ger and s,s,...,slare fixed positive real numbers,where s=λs(s>),s>and s=.
Suppose that δ:A→Ais a mapping with δ() = such that,for some ε≥,
l
j= sjδ(xj)
≤
δl
j= sjxj
+ε(.)
for all x,x,...,xl∈Aand all λ∈Tε,where x=λz(z∈A)and
δ(xy +yx)–xδ(y)–yδ(x)
≤θ(.)
for some θ≥and all x,y∈A.Then δis a linear derivation which maps Ainto the inter-
section of its center Z(A)and its radical rad(A).
Proof We let x=x=···=xl=in(.)andthenputx=x,x=y,s=tto have
λsδ(x)+tδ(y)+δ(λz)
≤
δ(λsx+ty +λz)
+ε(.)
for all x,y,z∈Aand all λ∈Tε. Now we consider λ=in(.). It follows from the result
in [] that there exists a unique additive mapping D:A→Adefined by
D(x):= lim
n→∞
δ(snx)
snfor all x∈A. (.)
Moreover, sD(x
s)=D(x)holdsforallx∈A.
Letting x=x
s,y=,andz=–xin (.), we find that
λsδx
s–δ(λx)
≤ε
for all x∈Aand all λ∈Tε.Thisimpliesthat
lim
n→∞
sn
λsδsnx
s–δλsnx
≤lim
n→∞
ε
sn=.
Thus λsD(x
s)=D(λx), so that D(λx)=λD(x) for all x∈Aand all λ∈Tε.Thusweseethat
Dis linear [].
By (.), we see that
lim
n→∞
δ(sn(xy +yx))
sn–xδ(y)–yδ(snx)
sn
≤lim
n→∞
θ
sn=.
Hence we arrive at
D(xy +yx)=xδ(y)+yD(x) for all x,y∈A.
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It follows from Lemma . that δis linear. Then we have by (.)thatD=δ. Therefore
δx=xδ(x) for all x∈A.
That is, δis a linear left Jordan derivation.
The remainder of the proof can be carried out similarly to the corresponding part of
Theorem ..
Theorem . Let Abe a unital Banach algebra.Assume that l ≥is a fixed integer and
s,s,...,slare fixed positive real numbers,where s=λs(s>),s>and s=.Suppose
that δ:A→Ais a mapping with δ() = such that,for some ε≥,
l
j= sjδ(xj)
≤
δl
j= sjxj
+ε(.)
for all x,x,...,x∈Aand all λ∈S:= {,i},where x=λz(z∈A)and (.). If δ(pe)=
for all irrational numbers p,then δis a linear left Jordan derivation.In this case Ais a
semiprime unital Banach algebra,δis a linear derivation which maps Ainto the intersec-
tion of its center Z(A)and its radical rad(A).
Proof We first consider λ=in(.). We see by the result in [] that there is a unique
additive mapping D:A→Adefined by (.). In addition, sD(x
s)=D(x) for all x∈A.
Also we set λ=iin (.). And we take x=x=···=xl=in(.)andthenletx=
x,x=y,s=tto have
isδ(x)+tδ(y)+δ(iz)
≤
δ(isx+ty+iz)
+ε(.)
for all x,y,z∈A. Putting x=x
s,y=andz=–xin (.), we obtain
isδx
s–δ(ix)
≤ε
for all x∈A, which shows that
lim
n→∞
sn
isδsnx
s–δisnx
≤lim
n→∞
ε
sn=.
Hence isD(x
s)=D(ix). So we have D(ix)=iD(x) for all x∈A.
We have by (.)
lim
n→∞
δ(sn(xy +yx))
sn–xδ(sny)
sn–yδ(snx)
sn
≤lim
n→∞
θ
sn=.
This implies that
D(xy +yx)=xD(y)+yD(x) for all x,y∈A.(.)
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Again, by virtue of (.), we see that
lim
n→∞
δ(sn(xy +yx))
sn–xδ(y)–yδ(snx)
sn
≤lim
n→∞
θ
sn=.
This implies that
D(xy +yx)=xδ(y)+yD(x) for all x,y∈A.(.)
Comparing (.)and(.), we arrive at xδ(y)=xD(y) for all x,y∈A.SinceAcontains
the unit element, we find that D=δ.Equation(.)canbewritten
δ(xy +yx)=xδ(y)+yδ(x) for all x,y∈A.(.)
Lettingx=y=ein (.),wehave δ(e) = . Nowwe obtain byadditivity of δδ(qx)=qδ(x)
for all q∈Qand all x∈A.Soδ(qe)=qδ(e) = for all q∈Q.Thisfactandtheassumption
of δimply that δ(te) = for all t∈R. Considering y=te in (.), we have δ(tx)=tδ(x)for
all t∈Rand all x∈A.Thusδis R-linear.Henceweseethat
δ(μx)=δ(t+ti)x=δ(tx)+tδ(ix)=tδ(x)+tiδ(x)=(t+ti)δ(x)=μδ(x)
for all μ∈Cand all x∈A.Soweseethatδis C-linear. In view of (.), we get
δx=xδ(x) for all x∈A.
Thereby δis a linear left Jordan derivation.
On the other hand, if Ais semiprime unital Banach algebra, then the rest of the proof is
similartothecorrespondingpartofTheorem..
Theorem . Let Abe a semisimple Banach algebra.Assume that l ≥is a fixed inte-
ger and s,s,...,slare fixed positive real numbers,where s=λs(s>),s>and s=.
Suppose that,for each k =,,δk:A→Ais a mapping with δk() = such that,for some
ε≥,
l
j= sjδk(xj)
≤
δkl
j= sjxj
+ε(.)
for all x,x,...,xl∈Aand all λ∈Tε,where x=λz(z∈A)and
δ(xy +yx)–xδ(y)–yδ(x)
≤θ,(.)
δ(xy +yx)–xδ(y)–yδ(x)–xδ(y)–yδ(x)
≤θ,(.)
for some θ,θ≥and all x,y∈A.Then δis a linear generalized left Jordan derivation
associated with a linear left Jordan derivation δ.In this case,δis continuous.
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Proof It iswell knownthatsemisimplealgebras aresemiprime [].Aswe sawin the proof
of Theorem .,δis a linear left Jordan derivation. In addition, we see that there exists a
unique linear mapping D:A→Adefined by
D(x):= lim
n→∞
δ(snx)
snfor all x∈A.(.)
According to (.)and(.), we see that
lim
n→∞
δ(sn(xy +yx))
sn–xδ(y)–yδ(snx)
sn–xδ(y)–yδ(x)
≤lim
n→∞
θ
sn=,
which implies that
D(xy +yx)=xδ(y)+yD(x)+xδ(y)+yδ(x) for all x,y∈A.(.)
So we obtain from (.)
D(xy +yx)–xδ(y)–yδ(x)=xδ(y)+yD(x) for all x,y∈A.(.)
In particular, the left-side of equation (.) is a bilinear mapping. Lemma . guarantees
that δis linear. By (.), we have D=δ.Equation(.)gives
δx=xδ(x)+xδ(x) for all x∈A.
Thus δis a linear generalized left Jordan derivation.
Therefore, since Ais semisimple, we conclude that δis continuous; see []. This com-
pletes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematical Finance and Information Statistics, Hallym University, 1 Hallymdaehakggil, Chuncheon,
24252, Korea. 2Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon,
34134, Korea.
Acknowledgements
The authors would like to thank the referees for giving useful suggestions and for the improvement of this manuscript.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (NRF-2015R1D1A1A01059467).
Received: 5 November 2016 Accepted: 25 January 2017
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