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2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

Part of: Linear spaces and algebras of operators Rings and algebras with additional structure

Published online by Cambridge University Press:  24 July 2020

WENBO HUANG  and
JIANKUI LI
WENBO HUANG
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China e-mail: [email protected]
JIANKUI LI*
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
*
e-mail: [email protected]
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Abstract

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$, then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$-subspace lattice algebras, is a derivation.

Keywords

derivation2-local derivationmodular annihilator algebrasocletrace

MSC classification

Primary: 47L35: Nest algebras, CSL algebras
Secondary: 16W25: Derivations, actions of Lie algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 3 , June 2021 , pp. 321 - 332
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

This paper was partially supported by the National Natural Science Foundation of China (Grant No. 11871021).

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