Asymptotic aspect of derivations in Banach algebras
- Jaiok Roh^{1} and
- Ick-Soon Chang^{2}Email author
https://doi.org/10.1186/s13660-017-1308-0
© The Author(s) 2017
Received: 5 November 2016
Accepted: 25 January 2017
Published: 6 February 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.
Keywords
MSC
1 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 Singer-Wermer 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 Bourgin-type 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.
2 Approximate left derivations
We first demonstrate the following proposition quoted in this work.
Proposition 2.1
[15], Proposition 1.6
- (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.
From now on, we suppose that \(\mathbb{T}_{\varepsilon} :=\{ e^{i \theta}: 0 \leq\theta\leq \varepsilon\}\). The commutator \(xy-yx\) will be denoted by \([x,y]\). We start our investigations for approximate ring left derivations with some results.
Theorem 2.2
Proof
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
Proof
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. □
3 Inequalities related to a linear derivation
In this section, we write a unit element of algebra \(\mathcal {A}\) by e.
Theorem 3.1
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
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 \(\lambda \in\mathbb{C}\).
Similarly, one can prove that η is linear. □
Theorem 3.4
Proof
The remainder of the proof can be carried out similarly to the corresponding part of Theorem 3.1. □
Theorem 3.5
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 \(x \in \mathcal {A}\).
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
Proof
Therefore, since \(\mathcal {A}\) is semisimple, we conclude that \(\delta_{1}\) is continuous; see [19]. This completes the proof. □
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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