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2024 | OriginalPaper | Chapter

Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces

Authors : Zhi-Cheng Deng, Feng Wei

Published in: Advances in Ring Theory and Applications

Publisher: Springer Nature Switzerland

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Abstract

Let X be a Banach space over the field \(\mathbb {F}\) (\(\mathbb {F}\) is either the real field \(\mathbb {R}\) or the complex field \(\mathbb {C}\)). Let B(X) be the set of all bounded linear operators on X and F(X) be the set of all finite rank operators in B(X). A subalgebra \(\mathcal {A}\) of B(X) is called a standard operator algebra if \(\mathcal {A}\) contains F(X). Suppose that \(\delta \) is a map from \(\mathcal {A}\) into B(X). Firstly, we prove that if \(\delta \) is a Lie-type derivation, then \(\delta \) has the standard form. Furthermore, we show that if \(\delta \) is a local Lie-type derivation, then \(\delta \) is a Lie-type derivation. Finally, we prove that if \(\delta \) is a 2-local Lie n-derivation, then \(\delta =d+\tau \), where d is a derivation, and \(\tau \) is homogeneous map from \(\mathcal {A}\) into \(\mathbb {F}I\) such that \(\tau (A+B)=\tau (A)\) for each A, B in \(\mathcal {A}\) where B is a sum of \((n-1)\)-th commutators.

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previous chapter Generalized Skew Derivation of Order 2 in Prime Ring with Multilinear Polynomial
next chapter The Noncommutative Singer-Wermer Conjecture and Generalized Skew Derivations
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Metadata
Title
Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces
Authors
Zhi-Cheng Deng
Feng Wei
Copyright Year
2024
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_13

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