Automatic Continuity of Derivations on Semidirect Products of Banach Algebras
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Automatic Continuity of Derivations on Semidirect Products of Banach Algebras Hamid Farhadi1
· Hoger Ghahramani1
Received: 26 June 2020 / Revised: 3 July 2020 / Accepted: 8 October 2020 © Iranian Mathematical Society 2020
Abstract Let A and U be Banach algebras such that U is also a Banach A-bimodule with compatible algebra operations, module actions and compatible norm. By defining an appropriate multiplication, we turn 1 -direct product A × U into a Banach algebra so that A is a closed subalgebra and U is a closed ideal of it. This algebra is, in fact, the semidirect product of A and U which we denote by A U. In this paper, we study automatic continuity of derivations on A U in a general setting. As an application of our results, we present various results about the automatic continuity of derivations of module extension Banach algebras and θ -Lau products of Banach algebras. Some examples are also given. Keywords Semidirect product · Banach algebra · Derivation · Automatic continuity Mathematics Subject Classification 46H40 · 46H25
1 Introduction Let A be a Banach algebra (over the complex field C), and U be a Banach A-bimodule. A linear map D : A → U is called a derivation if D(ab) = D(a)b + a D(b) holds for all a, b ∈ A. For any x ∈ U, the map ad x : A → U given by ad x (a) = ax − xa is a continuous derivation called inner. Derivations and their various properties are significant subjects in study of Banach algebras. Among the most important problems
Communicated by Mohammad B. Asadi.
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Hamid Farhadi [email protected] Hoger Ghahramani [email protected] ; [email protected]
1
Department of Mathematics, University of Kurdistan, P. O. Box 416, Sanandaj, Iran
123
Bulletin of the Iranian Mathematical Society
related to the theory of derivations is this question; under what conditions is a derivation D : A → U continuous? The problem of continuity of derivations lies in the theory of automatic continuity which has been developed extensively in the context of Banach algebras and many studies have been performed in this regard. The reader can find more information in [6] which is a detailed source in this context. Here, we mention the most important results obtained concerning the automatic continuity of derivations. Johnson and Sinclair in [11] have shown that every derivation on a semisimple Banach algebra is continuous. Ringrose [17] showed that every derivation from a C ∗ -algebra A into Banach Abimodule U is continuous. In [5], Christensen proved that every derivation from a nest algebra on Hilbert space H into B(H) is continuous. Additionally, some results on automatic continuity of derivations on prime Banach algebras have been established by Villena in [21] and [22] . In [9], the authors have investigated the automatic continuity of derivations on triangular Banach algebras. A generalization of triangular Banach algebras is the class of module extensions of Banach algebras whose derivations and some related notions are studied by several authors; see for instance [15,24]. A
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