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Asymptotic aspect of derivations in Banach algebras
Journal of Inequalities and Applications volume 2017, Article number: 36 (2017)
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.
Introduction and preliminaries
Let \(\mathcal {A}\) be an algebra. A linear mapping \(\delta: \mathcal {A}\to \mathcal {A}\) is called a left derivation (resp., derivation) if \(\delta(xy)=x\delta(y)+y\delta(x)\) (resp., \(\delta(xy)=x\delta(y)+\delta(x)y\)) is fulfilled for all \(x,y \in \mathcal {A}\). A linear mapping \(\delta: \mathcal {A}\to \mathcal {A}\) is said to be a left Jordan derivation if \(\delta(x^{2})=2x\delta(x)\) holds for all \(x \in \mathcal {A}\). A linear mapping \(\delta_{1}:\mathcal {A}\to \mathcal {A}\) is called a generalized left derivation if there exists a linear left derivation \(\delta_{0}:\mathcal {A}\to \mathcal {A}\) such that \(\delta_{1}(xy)=x\delta_{1}(y)+y\delta_{0}(x)\) for all \(x,y \in \mathcal {A}\). A linear mapping \(\delta_{1}:\mathcal {A}\to \mathcal {A}\) is said to be a generalized left Jordan derivation if there exists a linear left Jordan derivation \(\delta_{0}:\mathcal {A}\to \mathcal {A}\) such that \(\delta_{1}(x^{2})=x\delta _{1}(x)+x\delta_{0}(x)\) for all \(x \in \mathcal {A}\).
Singer and Wermer [1] obtained a fundamental result which started the investigation of the ranges of linear derivations on Banach algebras. The result, which is called the SingerWermer theorem, states that every continuous linear derivation on a commutative Banach algebra maps into the radical. In the same paper, they made a very insightful conjecture: that the assumption of continuity is unnecessary. Thomas [2] proved this conjecture. 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 [3]. Hyers [4] had answered affirmatively the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [5] for additive mappings and by Rassias [6] for linear mappings by considering an unbounded difference. In particular, the stability result concerning derivations between operator algebras was first obtained by Šemrl [7]. Badora gave a generalization of the Bourgin result and he also dealt with the stability and the superstability of Bourgintype for derivations; see [8–10] and the references therein. Recently, the stability problems for derivations are considered by some authors in [11–13].
In this work, we first take into account the functional inequality which expands the functional inequality in [14]. It is well known that every ring left derivation (resp., ring left Jordan derivation) on a semiprime ring maps into its center; see [15, 16]. Considering the base 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 approximate 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 algebra 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.
Approximate left derivations
We first demonstrate the following proposition quoted in this work.
Proposition 2.1
[15], Proposition 1.6
Let \(\mathcal {R}\) be a ring, \(\mathcal {X}\) be a left \(\mathcal {R}\)module, and \(\delta : \mathcal {R}\to \mathcal {X}\) be a left derivation.

(i)
Suppose that \(a \mathcal {R}x =0\) with \(a \in \mathcal {R}, x \in \mathcal {X}\) implies \(a=0\) or \(x=0\). If \(\delta \ne0\), then \(\mathcal {R}\) is commutative.

(ii)
Suppose that \(\mathcal {X}=\mathcal {R}\) is a semiprime ring. Then δ is a derivation which maps \(\mathcal {R}\) into its center.
Let \(\mathcal {A}\) be a normed algebra. An additive mapping \(\delta :\mathcal {A}\to \mathcal {A}\) is said to be an approximate ring derivation (resp., approximate ring left derivation) if for some \(\varepsilon\ge0\),
for all \(x,y \in \mathcal {A}\). In addition, if \(\delta (\lambda x)=\lambda \delta (x)\) for all \(x \in \mathcal {A}\) and \(\lambda \in\mathbb{C}\), then δ is called an approximate linear derivation (resp., approximate linear left derivation).
From now on, we suppose that \(\mathbb{T}_{\varepsilon} :=\{ e^{i \theta}: 0 \leq\theta\leq \varepsilon\}\). The commutator \(xyyx\) will be denoted by \([x,y]\). We start our investigations for approximate ring left derivations with some results.
Theorem 2.2
Let \(\mathcal {A}\) be a semiprime normed algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{j}> 1\) (\(j=1,2\)) and \(s_{3}=1\). Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping such that
for all \(x_{1}, x_{2}, \ldots, x_{l}\in \mathcal {A}\) and for some \(\varepsilon\ge0\),
for all \(x,y \in \mathcal {A}\). Then δ is an approximate ring derivation which maps \(\mathcal {A}\) into its center \(Z(\mathcal {A})\).
Proof
By letting \(x_{1}=x_{2}=\cdots=x_{l}=0\) in (2.1), we get \(\delta(0)=0\). And we put \(x_{4}=x_{5}=\cdots=x_{l}=0\) in (2.1) and then set \(x_{1}=x, x_{2}=y,x_{3}=z,s_{1}=s,s_{2}=t\) to obtain
It follows by the result of [14] that δ is additive. In particular, in view of (2.2), we find that δ is an approximate ring left derivation.
By virtue of (2.2), we see that
for all \(x,y \in \mathcal {A}\). Combining (2.2) and (2.4), we get
for all \(x,y \in \mathcal {A}\). It follows from (2.2) and (2.5) that
for all \(x,y \in \mathcal {A}\). Replacing x by nx in (2.6) and then dividing on both sides by \(n^{2}\), we have
for all \(x,y \in \mathcal {A}\) and all positive integer n. Taking the limit as \(n \to\infty\) in the above relation, we see that
Just proceeding as in the proof of Proposition 2.1, we get \([\delta(w),x]=0\) for all \(x,w \in \mathcal {A}\). That is, \(\delta(w)\) belong to its center \(Z(\mathcal {A})\). So δ is an approximate ring derivation. Therefore we arrive at the desired conclusion. □
Theorem 2.3
Let \(\mathcal {A}\) be a noncommutative prime normed algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{j}> 1\) (\(j=1,2\)) and \(s_{3}=1\). Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping subject to the conditions (2.1) and (2.2). Then δ is identically zero.
Proof
Employing the same argument as the proof Theorem 2.2, we feel that δ satisfies equation (2.7). Since \(\mathcal {A}\) is noncommutative, choose a z that does not belong to the center of \(\mathcal {A}\). Using the same method in the proof of Proposition 2.1, we see that \(\delta=0\), which completes the proof. □
Theorem 2.4
Let \(\mathcal {A}\) be a semisimple Banach algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{1}=\lambda s\) (\(s>1\)), \(s_{2}> 1\) and \(s_{3}=1\). Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping subject to
for all \(x_{1}, x_{2}, \ldots, x_{l}\in \mathcal {A}\) and all \(\lambda\in \mathbb {T}_{\varepsilon}\), where \(x_{3}= \lambda z\) (\(z \in \mathcal {A}\)) and the inequality (2.2). Then δ is a continuous.
Proof
As we did in the proof of Theorem 2.2, we get \(\delta(0)=0\). We take \(x_{4}=x_{5}=\cdots=x_{l}=0\) in (2.8) and then put \(x_{1}=x, x_{2}=y,s_{2}=t\) to have
for all \(x,y,z \in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\). Now we consider \(\lambda=1\) in (2.9) and so δ satisfies the inequality (2.3). Hence we find that δ is additive [14].
Next, setting \(x=\frac{x}{s}, y=0\) and \(z=x\) in (2.3), we obtain \(s\delta(\frac{x}{s})=\delta(x)\). Letting \(x=\frac{x}{s}, y=0\), and \(z=x\) in (2.9), we get \(\delta(\lambda x)=\lambda\delta(x)\) for all \(x\in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\) and so we see that δ is linear [17].
Since semisimple algebras are semiprime [18], Theorem 2.2 guarantees that δ is an approximate linear derivation. Therefore δ is continuous [14]. The proof is complete. □
Inequalities related to a linear derivation
In this section, we write a unit element of algebra \(\mathcal {A}\) by e.
Theorem 3.1
Let \(\mathcal {A}\) be a semiprime unital Banach algebra. Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping subject to the inequality (2.8) and for some \(\varepsilon\ge0\),
for all \(x \in \mathcal {A}\). Then δ is a linear derivation which maps \(\mathcal {A}\) into the intersection of its center \(Z(\mathcal {A})\) and its radical \(\operatorname{rad}(\mathcal {A})\).
Proof
Employing the same way in the proof Theorem 2.4, we find that δ is linear. By linearization of (3.1) and additivity of δ, we get
for all \(x,y \in \mathcal {A}\). Substituting −x for x in (3.2), we have
for all \(x,y \in \mathcal {A}\). Equations (3.2) and (3.3) yield
for all \(x,y \in \mathcal {A}\). We have therefore
Putting \(xy+yx\) for y in (3.4), we obtain
for all \(x,y \in \mathcal {A}\). On the other hand, we have from (3.4) and the equation
the result
for all \(x,y \in \mathcal {A}\). By comparing (3.5) and (3.6), we arrive at
for all \(x,y \in \mathcal {A}\). Applying equation (3.7) with (3.1) and (3.4), we have
for all \(x,y \in \mathcal {A}\). Letting \(x=nx, y=ny\) in (3.8) and then dividing the resulting inequality by \(n^{3}\), we get
for all \(x,y \in \mathcal {A}\) and all positive integers n. Taking the limit \(n \to\infty\) of (3.9), it is reduced to the equation
Putting \(x=y=z=e\) in (3.10), we get \(\delta(e)=0\). Again, considering \(y=e\) in (3.10), we easily prove that
This means that δ is a linear left Jordan derivation.
On the other hand, from Vukman’s result [16], we see that δ is a linear derivation with \(\delta(\mathcal {A}) \subseteq Z(\mathcal {A})\). Since \(Z(\mathcal {A})\) is a commutative Banach algebra, the SingerWermer theorem tells us that \(\delta_{Z(\mathcal {A})}\) maps \(Z(\mathcal {A})\) into \(\operatorname{rad}(Z(\mathcal {A}))=Z(\mathcal {A}) \cap \operatorname{rad}(\mathcal {A})\) and thus \(\delta^{2}(\mathcal {A}) \subseteq \operatorname{rad}(\mathcal {A})\). Using the semiprimeness of \(\operatorname{rad}(\mathcal {A})\) as well as the identity
we have \(\delta(\mathcal {A}) \subseteq \operatorname{rad}(\mathcal {A})\). Therefore \(\delta(\mathcal {A}) \subseteq Z(\mathcal {A}) \cap \operatorname{rad}(\mathcal {A})\), which concludes the proof. □
As consequences of Theorem 3.1, we get the following.
Corollary 3.2
Let \(\mathcal {A}\) be a unital semisimple Banach algebra. Assume that a mapping \(\delta : \mathcal {A}\to \mathcal {A}\) satisfies the assumptions of Theorem 3.1. Then δ is identically zero.
Now we consider the result which is needed in the following theorems.
Lemma 3.3
Let \(\mathcal {A}\) be a Banach algebra. Suppose that \(\mathcal{L}:\mathcal {A}\times \mathcal {A}\to \mathcal {A}\) is a bilinear mapping and that ξ and η are mappings satisfying \(\mathcal{L}(x,y)= x\xi(y)+ y\eta(x)\) for all \(x,y \in \mathcal {A}\). If \(\mathcal {A}\) is semiprime or unital, then ξ and η are linear mappings.
Proof
Note that, for all \(x,y \in \mathcal {A}\) and all \(\lambda \in \mathbb{C}\),
Hence we see that, for all \(x,y \in \mathcal {A}\),
If \(\mathcal {A}\) is unital, then we see that \(\xi(\lambda y)=\lambda \eta(y)\) by letting \(x=e\) in (3.11).
If \(\mathcal {A}\) is nonunital, then \(\xi(\lambda y)\lambda \xi(y)\) lies in the right annihilator \(\operatorname{ran}(\mathcal {A})\) of \(\mathcal {A}\). If \(\mathcal {A}\) is semiprime, then \(\operatorname{ran}(\mathcal {A})=0\), so that \(\xi(\lambda y)=\lambda \xi(y)\) for all \(y \in \mathcal {A}\) and all \(\lambda \in\mathbb{C}\).
Observe that, for all \(x,y,z \in \mathcal {A}\),
Hence \(x(\xi(y+z)\xi(y)\xi(z))=0\) for all \(y,z \in \mathcal {A}\). As above, we get \(\xi(x+z)=\xi(x)+\xi(z)\) for all \(x,z \in \mathcal {A}\), so that ξ is linear.
Similarly, one can prove that η is linear. □
Theorem 3.4
Let \(\mathcal {A}\) be a semiprime Banach algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{1}=\lambda s\) (\(s>1\)), \(s_{2}> 1\) and \(s_{3}=1\). Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping with \(\delta(0)=0\) such that, for some \(\varepsilon\ge0\),
for all \(x_{1}, x_{2}, \ldots, x_{l} \in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\), where \(x_{3}= \lambda z\) (\(z \in \mathcal {A}\)) and
for some \(\theta\ge0\) and all \(x,y \in \mathcal {A}\). Then δ is a linear derivation which maps \(\mathcal {A}\) into the intersection of its center \(Z(\mathcal {A})\) and its radical \(\operatorname{rad}(\mathcal {A})\).
Proof
We let \(x_{4}=x_{5}=\cdots=x_{l}=0\) in (3.12) and then put \(x_{1}=x, x_{2}=y, s_{2}=t\) to have
for all \(x,y,z \in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\). Now we consider \(\lambda=1\) in (3.14). It follows from the result in [14] that there exists a unique additive mapping \(\mathcal{D}: \mathcal {A}\to \mathcal {A}\) defined by
Moreover, \(s\mathcal{D}(\frac{x}{s})=\mathcal{D}(x)\) holds for all \(x \in \mathcal {A}\).
Letting \(x=\frac{x}{s}, y=0\), and \(z=x\) in (3.14), we find that
for all \(x\in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\). This implies that
Thus \(\lambda s\mathcal{D}(\frac{x}{s})=\mathcal{D}(\lambda x)\), so that \(\mathcal{D}(\lambda x)=\lambda\mathcal{D}(x)\) for all \(x\in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\). Thus we see that \(\mathcal{D}\) is linear [17].
By (3.13), we see that
Hence we arrive at
It follows from Lemma 3.3 that δ is linear. Then we have by (3.15) that \(\mathcal {D}=\delta\). Therefore
That is, δ is a linear left Jordan derivation.
The remainder of the proof can be carried out similarly to the corresponding part of Theorem 3.1. □
Theorem 3.5
Let \(\mathcal {A}\) be a unital Banach algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{1}=\lambda s\) (\(s>1\)), \(s_{2}> 1\) and \(s_{3}=1\). Suppose that \(\delta: \mathcal {A}\to \mathcal {A}\) is a mapping with \(\delta(0)=0\) such that, for some \(\varepsilon\ge0\),
for all \(x_{1}, x_{2}, \ldots, x_{3} \in \mathcal {A}\) and all \(\lambda\in S:=\{1,i\}\), where \(x_{3}= \lambda z\) (\(z \in \mathcal {A}\)) and (3.13). If \(\delta(p e)=0\) for all irrational numbers p, then δ is a linear left Jordan derivation. In this case \(\mathcal {A}\) is a semiprime unital Banach algebra, δ is a linear derivation which maps \(\mathcal {A}\) into the intersection of its center \(Z(\mathcal {A})\) and its radical \(\operatorname{rad}(\mathcal {A})\).
Proof
We first consider \(\lambda=1\) in (3.16). We see by the result in [14] that there is a unique additive mapping \(\mathcal{D}: \mathcal {A}\to \mathcal {A}\) defined by (3.15). In addition, \(s\mathcal{D}(\frac{x}{s})=\mathcal{D}(x)\) for all \(x \in \mathcal {A}\).
Also we set \(\lambda=i\) in (3.16). And we take \(x_{4}=x_{5}=\cdots=x_{l}=0\) in (3.16) and then let \(x_{1}=x, x_{2}=y, s_{2}=t\) to have
for all \(x,y,z \in \mathcal {A}\). Putting \(x=\frac{x}{s}, y=0\) and \(z=x\) in (3.17), we obtain
for all \(x\in \mathcal {A}\), which shows that
Hence \(i s\mathcal{D}(\frac{x}{s})=\mathcal{D}(i x)\). So we have \(\mathcal{D}(i x)=i \mathcal{D}(x)\) for all \(x\in \mathcal {A}\).
We have by (3.13)
This implies that
Again, by virtue of (3.13), we see that
This implies that
Comparing (3.18) and (3.19), we arrive at \(x\delta (y)=x\mathcal{D}(y)\) for all \(x,y \in \mathcal {A}\). Since \(\mathcal {A}\) contains the unit element, we find that \(\mathcal {D}=\delta\). Equation (3.19) can be written
Letting \(x=y=e\) in (3.20), we have \(\delta(e)=0\). Now we obtain by additivity of δ \(\delta(q x)=q \delta(x)\) for all \(q \in\mathbb{Q}\) and all \(x \in \mathcal {A}\). So \(\delta(q e)=q \delta (e)=0\) for all \(q \in\mathbb{Q}\). This fact and the assumption of δ imply that \(\delta(t e)=0\) for all \(t \in\mathbb{R}\). Considering \(y=t e\) in (3.20), we have \(\delta(t x)=t \delta (x)\) for all \(t \in\mathbb{R}\) and all \(x \in \mathcal {A}\). Thus δ is \(\mathbb{R}\)linear. Hence we see that
for all \(\mu\in\mathbb{C}\) and all \(x \in \mathcal {A}\). So we see that δ is \(\mathbb{C}\)linear. In view of (3.20), we get
Thereby δ is a linear left Jordan derivation.
On the other hand, if \(\mathcal {A}\) is semiprime unital Banach algebra, then the rest of the proof is similar to the corresponding part of Theorem 3.1. □
Theorem 3.6
Let \(\mathcal {A}\) be a semisimple Banach algebra. Assume that \(l \ge3\) is a fixed integer and \(s_{1},s_{2}, \ldots, s_{l}\) are fixed positive real numbers, where \(s_{1}=\lambda s\) (\(s>1\)), \(s_{2}> 1\) and \(s_{3}=1\). Suppose that, for each \(k=0,1\), \(\delta_{k} : \mathcal {A}\to \mathcal {A}\) is a mapping with \(\delta_{k}(0)=0\) such that, for some \(\varepsilon\ge0\),
for all \(x_{1}, x_{2}, \ldots, x_{l} \in \mathcal {A}\) and all \(\lambda\in\mathbb{T}_{\varepsilon}\), where \(x_{3}= \lambda z\) (\(z \in \mathcal {A}\)) and
for some \(\theta_{0}, \theta_{1} \ge0\) and all \(x,y \in \mathcal {A}\). Then \(\delta_{1}\) is a linear generalized left Jordan derivation associated with a linear left Jordan derivation \(\delta_{0}\). In this case, \(\delta_{1}\) is continuous.
Proof
It is well known that semisimple algebras are semiprime [18]. As we saw in the proof of Theorem 3.4, \(\delta_{0}\) is a linear left Jordan derivation. In addition, we see that there exists a unique linear mapping \(\mathcal {D}_{1}: \mathcal {A}\to \mathcal {A}\) defined by
According to (3.23) and (3.24), we see that
which implies that
So we obtain from (3.25)
In particular, the leftside of equation (3.26) is a bilinear mapping. Lemma 3.3 guarantees that \(\delta_{1}\) is linear. By (3.24), we have \(\mathcal {D}_{1}=\delta_{1}\). Equation (3.25) gives
Thus \(\delta_{1}\) is a linear generalized left Jordan derivation.
Therefore, since \(\mathcal {A}\) is semisimple, we conclude that \(\delta_{1}\) is continuous; see [19]. This completes the proof. □
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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 (NRF2015R1D1A1A01059467).
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Roh, J., Chang, IS. Asymptotic aspect of derivations in Banach algebras. J Inequal Appl 2017, 36 (2017). https://doi.org/10.1186/s1366001713080
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DOI: https://doi.org/10.1186/s1366001713080
MSC
 16N20
 16N60
 39B72
 39B82
 46H40
 47H99
Keywords
 derivation
 inequality
 Banach algebra
 stability
 radical range