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2024 | OriginalPaper | Chapter

Central Power Values of Generalized Derivation and Structure of Unital Banach Algebra

Authors : Asma Ali, Kapil Kumar, Mohd Tasleem

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

The structure of a prime Banach algebra \(\mathfrak {A}\) containing the identity element e with the help of a nonzero continuous linear generalized derivation \(\mathfrak {D}\) satisfying one of the following conditions, is investigated in this paper : (i) \(x\Big (\mathfrak {D}((ab)^m)\pm a^mb^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (ii) \(x\Big (\mathfrak {D}((ab)^m)\pm b^ma^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (iii) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(a^mb^m)\Big )\in \mathcal {Z}_\mathfrak {A}\); (iv) \(x\Big (\mathfrak {D}((ab)^m)\pm \mathfrak {D}(b^ma^m)\Big )\in \mathcal {Z}_\mathfrak {A}\); (v) \(x\Big (\mathfrak {D}((ab)^m)\pm [a^m, b^m]\Big )\in \mathcal {Z}_\mathfrak {A}\); (vi) \(x\Big (\mathfrak {D}((ab)^m)\pm [b^m, a^m]\Big )\in \mathcal {Z}_\mathfrak {A}\); (vii) \(x\Big (\mathfrak {D}((ab)^m)\pm (ab)^m\Big )\in \mathcal {Z}_\mathfrak {A}\); (viii) \(x\Big (\mathfrak {D}((ab)^m)\Big )\in \mathcal {Z}_\mathfrak {A}\) for all a, b in some suitable non empty subset of \(\mathfrak {A}\), where m is a positive integer such that \(m=m(a, b)>1\). Moreover, we give a characterization of the map \(\mathfrak {D}\). Finally it is shown that our theorems do not hold in case \(\mathfrak {A}\) is not prime.

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previous chapter Some Results on Left-sided Ideals of Semiprime Rings with Symmetric n-Derivations

next chapter Generalized Skew Derivation of Order 2 in Prime Ring with Multilinear Polynomial

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Title: Central Power Values of Generalized Derivation and Structure of Unital Banach Algebra
Authors: Asma Ali
Kapil Kumar
Mohd Tasleem
Publisher: Springer Nature Switzerland
Book: Advances in Ring Theory and Applications
Print ISBN: 978-3-031-50794-6

Electronic ISBN: 978-3-031-50795-3

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50795-3_11

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